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Statistics and ComputingSeries Editors:J. ChambersD. HandW. Härdle

Statistics and ComputingBrusco/Stahl: Branch and Bound Applications in Combinatorial

Data AnalysisChambers: Software for Data Analysis: Programming with RDalgaard: Introductory Statistics with R, 2nd ed.Gentle: Elements of Computational StatisticsGentle: Numerical Linear Algebra for Applications in StatisticsGentle: Random Number Generation and Monte

Carlo Methods, 2nd ed.Härdle/Klinke/Turlach: XploRe: An Interactive Statistical

Computing EnvironmentHörmann/Leydold/Derflinger: Automatic Nonuniform Random

Variate GenerationKrause/Olson: The Basics of S-PLUS, 4th ed.Lange: Numerical Analysis for StatisticiansLemmon/Schafer: Developing Statistical Software in Fortran 95Loader: Local Regression and LikelihoodMarasinghe/Kennedy: SAS for Data Analysis: Intermediate

Statistical MethodsÓ Ruanaidh/Fitzgerald: Numerical Bayesian Methods Applied to

Signal ProcessingPannatier: VARIOWIN: Software for Spatial Data Analysis in 2DPinheiro/Bates: Mixed-Effects Models in S and S-PLUSUnwin/Theus/Hofmann: Graphics of Large Datasets:

Visualizing a MillionVenables/Ripley: Modern Applied Statistics with S, 4th ed.Venables/Ripley: S ProgrammingWilkinson: The Grammar of Graphics, 2nd ed.

Peter Dalgaard

Introductory Statistics with R

Second Edition

123

Peter DalgaardDepartment of BiostatisticsUniversity of [email protected]

ISSN: 1431-8784ISBN: 978-0-387-79053-4 e-ISBN: 978-0-387-79054-1DOI: 10.1007/978-0-387-79054-1

Library of Congress Control Number: 2008932040

c© 2008 Springer Science+Business Media, LLCAll rights reserved. This work may not be translated or copied in whole or in part without the writtenpermission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York,NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Usein connection with any form of information storage and retrieval, electronic adaptation, computersoftware, or by similar or dissimilar methodology now known or hereafter developed is forbidden.The use in this publication of trade names, trademarks, service marks, and similar terms, even if theyare not identified as such, is not to be taken as an expression of opinion as to whether or not they aresubject to proprietary rights.

Printed on acid-free paper

springer.com

To Grete, for putting up with me for so long

Preface

R is a statistical computer program made available through the Internetunder the General Public License (GPL). That is, it is supplied with a li-cense that allows you to use it freely, distribute it, or even sell it, as long asthe receiver has the same rights and the source code is freely available. Itexists for Microsoft Windows XP or later, for a variety of Unix and Linuxplatforms, and for Apple Macintosh OS X.

R provides an environment in which you can perform statistical analysisand produce graphics. It is actually a complete programming language,although that is only marginally described in this book. Here we contentourselves with learning the elementary concepts and seeing a number ofcookbook examples.

R is designed in such a way that it is always possible to do furthercomputations on the results of a statistical procedure. Furthermore, thedesign for graphical presentation of data allows both no-nonsense meth-ods, for example plot(x,y), and the possibility of fine-grained controlof the output’s appearance. The fact that R is based on a formal computerlanguage gives it tremendous flexibility. Other systems present simplerinterfaces in terms of menus and forms, but often the apparent user-friendliness turns into a hindrance in the longer run. Although elementarystatistics is often presented as a collection of fixed procedures, analysisof moderately complex data requires ad hoc statistical model building,which makes the added flexibility of R highly desirable.

viii Preface

R owes its name to typical Internet humour. You may be familiar withthe programming language C (whose name is a story in itself). Inspiredby this, Becker and Chambers chose in the early 1980s to call their newlydeveloped statistical programming language S. This language was furtherdeveloped into the commercial product S-PLUS, which by the end of thedecade was in widespread use among statisticians of all kinds. Ross Ihakaand Robert Gentleman from the University of Auckland, New Zealand,chose to write a reduced version of S for teaching purposes, and what wasmore natural than choosing the immediately preceding letter? Ross’ andRobert’s initials may also have played a role.

In 1995, Martin Maechler persuaded Ross and Robert to release the sourcecode for R under the GPL. This coincided with the upsurge in Open Sourcesoftware spurred by the Linux system. R soon turned out to fill a gap forpeople like me who intended to use Linux for statistical computing buthad no statistical package available at the time. A mailing list was set upfor the communication of bug reports and discussions of the developmentof R.

In August 1997, I was invited to join an extended international core teamwhose members collaborate via the Internet and that has controlled thedevelopment of R since then. The core team was subsequently expandedseveral times and currently includes 19 members. On February 29, 2000,version 1.0.0 was released. As of this writing, the current version is 2.6.2.

This book was originally based upon a set of notes developed for thecourse in Basic Statistics for Health Researchers at the Faculty of HealthSciences of the University of Copenhagen. The course had a primary tar-get of students for the Ph.D. degree in medicine. However, the materialhas been substantially revised, and I hope that it will be useful for a largeraudience, although some biostatistical bias remains, particularly in thechoice of examples.

In later years, the course in Statistical Practice in Epidemiology, which hasbeen held yearly in Tartu, Estonia, has been a major source of inspirationand experience in introducing young statisticians and epidemiologists toR.

This book is not a manual for R. The idea is to introduce a number of basicconcepts and techniques that should allow the reader to get started withpractical statistics.

In terms of the practical methods, the book covers a reasonable curriculumfor first-year students of theoretical statistics as well as for engineeringstudents. These groups will eventually need to go further and studymore complex models as well as general techniques involving actualprogramming in the R language.

Preface ix

For fields where elementary statistics is taught mainly as a tool, the bookgoes somewhat further than what is commonly taught at the under-graduate level. Multiple regression methods or analysis of multifactorialexperiments are rarely taught at that level but may quickly become essen-tial for practical research. I have collected the simpler methods near thebeginning to make the book readable also at the elementary level. How-ever, in order to keep technical material together, Chapters 1 and 2 doinclude material that some readers will want to skip.

The book is thus intended to be useful for several groups, but I will notpretend that it can stand alone for any of them. I have included brieftheoretical sections in connection with the various methods, but morethan as teaching material, these should serve as reminders or perhaps asappetizers for readers who are new to the world of statistics.

Notes on the 2nd edition

The original first chapter was expanded and broken into two chapters,and a chapter on more advanced data handling tasks was inserted afterthe coverage of simpler statistical methods. There are also two new chap-ters on statistical methodology, covering Poisson regression and nonlinearcurve fitting, and a few items have been added to the section on de-scriptive statistics. The original methodological chapters have been quiteminimally revised, mainly to ensure that the text matches the actual out-put of the current version of R. The exercises have been revised, andsolution sketches now appear in Appendix D.

Acknowledgements

Obviously, this book would not have been possible without the efforts ofmy friends and colleagues on the R Core Team, the authors of contributedpackages, and many of the correspondents of the e-mail discussion lists.

I am deeply grateful for the support of my colleagues and co-teachersLene Theil Skovgaard, Bendix Carstensen, Birthe Lykke Thomsen, HelleRootzen, Claus Ekstrøm, Thomas Scheike, and from the Tartu courseKrista Fischer, Esa Läära, Martyn Plummer, Mark Myatt, and MichaelHills, as well as the feedback from several students. In addition, sev-eral people, including Bill Venables, Brian Ripley, and David James, gavevaluable advice on early drafts of the book.

Finally, profound thanks are due to the free software community at large.The R project would not have been possible without their effort. For the

x Preface

typesetting of this book, TEX, LATEX, and the consolidating efforts of theLATEX2e project have been indispensable.

Peter DalgaardCopenhagen

April 2008

Contents

Preface vii

1 Basics 11.1 First steps . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 An overgrown calculator . . . . . . . . . . . . . . 31.1.2 Assignments . . . . . . . . . . . . . . . . . . . . . 31.1.3 Vectorized arithmetic . . . . . . . . . . . . . . . . 41.1.4 Standard procedures . . . . . . . . . . . . . . . . 61.1.5 Graphics . . . . . . . . . . . . . . . . . . . . . . . 7

1.2 R language essentials . . . . . . . . . . . . . . . . . . . . 91.2.1 Expressions and objects . . . . . . . . . . . . . . . 91.2.2 Functions and arguments . . . . . . . . . . . . . 111.2.3 Vectors . . . . . . . . . . . . . . . . . . . . . . . . 121.2.4 Quoting and escape sequences . . . . . . . . . . 131.2.5 Missing values . . . . . . . . . . . . . . . . . . . . 141.2.6 Functions that create vectors . . . . . . . . . . . . 141.2.7 Matrices and arrays . . . . . . . . . . . . . . . . . 161.2.8 Factors . . . . . . . . . . . . . . . . . . . . . . . . 181.2.9 Lists . . . . . . . . . . . . . . . . . . . . . . . . . . 191.2.10 Data frames . . . . . . . . . . . . . . . . . . . . . 201.2.11 Indexing . . . . . . . . . . . . . . . . . . . . . . . 211.2.12 Conditional selection . . . . . . . . . . . . . . . . 221.2.13 Indexing of data frames . . . . . . . . . . . . . . 231.2.14 Grouped data and data frames . . . . . . . . . . 25

xii Contents

1.2.15 Implicit loops . . . . . . . . . . . . . . . . . . . . 261.2.16 Sorting . . . . . . . . . . . . . . . . . . . . . . . . 27

1.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2 The R environment 312.1 Session management . . . . . . . . . . . . . . . . . . . . 31

2.1.1 The workspace . . . . . . . . . . . . . . . . . . . . 312.1.2 Textual output . . . . . . . . . . . . . . . . . . . . 322.1.3 Scripting . . . . . . . . . . . . . . . . . . . . . . . 332.1.4 Getting help . . . . . . . . . . . . . . . . . . . . . 342.1.5 Packages . . . . . . . . . . . . . . . . . . . . . . . 352.1.6 Built-in data . . . . . . . . . . . . . . . . . . . . . 352.1.7 attach and detach . . . . . . . . . . . . . . 362.1.8 subset, transform, and within . . . . . . . . 37

2.2 The graphics subsystem . . . . . . . . . . . . . . . . . . . 392.2.1 Plot layout . . . . . . . . . . . . . . . . . . . . . . 392.2.2 Building a plot from pieces . . . . . . . . . . . . . 402.2.3 Using par . . . . . . . . . . . . . . . . . . . . . . 422.2.4 Combining plots . . . . . . . . . . . . . . . . . . . 42

2.3 R programming . . . . . . . . . . . . . . . . . . . . . . . 442.3.1 Flow control . . . . . . . . . . . . . . . . . . . . . 442.3.2 Classes and generic functions . . . . . . . . . . . 46

2.4 Data entry . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.4.1 Reading from a text file . . . . . . . . . . . . . . . 472.4.2 Further details on read.table . . . . . . . . . . 502.4.3 The data editor . . . . . . . . . . . . . . . . . . . 512.4.4 Interfacing to other programs . . . . . . . . . . . 52

2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3 Probability and distributions 553.1 Random sampling . . . . . . . . . . . . . . . . . . . . . . 553.2 Probability calculations and combinatorics . . . . . . . . 563.3 Discrete distributions . . . . . . . . . . . . . . . . . . . . 573.4 Continuous distributions . . . . . . . . . . . . . . . . . . 583.5 The built-in distributions in R . . . . . . . . . . . . . . . 59

3.5.1 Densities . . . . . . . . . . . . . . . . . . . . . . . 593.5.2 Cumulative distribution functions . . . . . . . . 623.5.3 Quantiles . . . . . . . . . . . . . . . . . . . . . . . 633.5.4 Random numbers . . . . . . . . . . . . . . . . . . 64

3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4 Descriptive statistics and graphics 674.1 Summary statistics for a single group . . . . . . . . . . . 674.2 Graphical display of distributions . . . . . . . . . . . . . 71

4.2.1 Histograms . . . . . . . . . . . . . . . . . . . . . . 71

Contents xiii

4.2.2 Empirical cumulative distribution . . . . . . . . 734.2.3 Q–Q plots . . . . . . . . . . . . . . . . . . . . . . 744.2.4 Boxplots . . . . . . . . . . . . . . . . . . . . . . . 75

4.3 Summary statistics by groups . . . . . . . . . . . . . . . 754.4 Graphics for grouped data . . . . . . . . . . . . . . . . . 79

4.4.1 Histograms . . . . . . . . . . . . . . . . . . . . . . 794.4.2 Parallel boxplots . . . . . . . . . . . . . . . . . . . 804.4.3 Stripcharts . . . . . . . . . . . . . . . . . . . . . . 81

4.5 Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.5.1 Generating tables . . . . . . . . . . . . . . . . . . 834.5.2 Marginal tables and relative frequency . . . . . . 87

4.6 Graphical display of tables . . . . . . . . . . . . . . . . . 894.6.1 Barplots . . . . . . . . . . . . . . . . . . . . . . . . 894.6.2 Dotcharts . . . . . . . . . . . . . . . . . . . . . . . 914.6.3 Piecharts . . . . . . . . . . . . . . . . . . . . . . . 92

4.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5 One- and two-sample tests 955.1 One-sample t test . . . . . . . . . . . . . . . . . . . . . . 955.2 Wilcoxon signed-rank test . . . . . . . . . . . . . . . . . 995.3 Two-sample t test . . . . . . . . . . . . . . . . . . . . . . 1005.4 Comparison of variances . . . . . . . . . . . . . . . . . . 1035.5 Two-sample Wilcoxon test . . . . . . . . . . . . . . . . . 1035.6 The paired t test . . . . . . . . . . . . . . . . . . . . . . . 1045.7 The matched-pairs Wilcoxon test . . . . . . . . . . . . . 1065.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6 Regression and correlation 1096.1 Simple linear regression . . . . . . . . . . . . . . . . . . . 1096.2 Residuals and fitted values . . . . . . . . . . . . . . . . . 1136.3 Prediction and confidence bands . . . . . . . . . . . . . . 1176.4 Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . 120

6.4.1 Pearson correlation . . . . . . . . . . . . . . . . . 1216.4.2 Spearman’s ρ . . . . . . . . . . . . . . . . . . . . . 1236.4.3 Kendall’s τ . . . . . . . . . . . . . . . . . . . . . . 124

6.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

7 Analysis of variance and the Kruskal–Wallis test 1277.1 One-way analysis of variance . . . . . . . . . . . . . . . 127

7.1.1 Pairwise comparisons and multiple testing . . . 1317.1.2 Relaxing the variance assumption . . . . . . . . . 1337.1.3 Graphical presentation . . . . . . . . . . . . . . . 1347.1.4 Bartlett’s test . . . . . . . . . . . . . . . . . . . . . 136

7.2 Kruskal–Wallis test . . . . . . . . . . . . . . . . . . . . . 1367.3 Two-way analysis of variance . . . . . . . . . . . . . . . 137

xiv Contents

7.3.1 Graphics for repeated measurements . . . . . . . 1407.4 The Friedman test . . . . . . . . . . . . . . . . . . . . . . 1417.5 The ANOVA table in regression analysis . . . . . . . . . 1417.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

8 Tabular data 1458.1 Single proportions . . . . . . . . . . . . . . . . . . . . . . 1458.2 Two independent proportions . . . . . . . . . . . . . . . 1478.3 k proportions, test for trend . . . . . . . . . . . . . . . . . 1498.4 r × c tables . . . . . . . . . . . . . . . . . . . . . . . . . . 1518.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

9 Power and the computation of sample size 1559.1 The principles of power calculations . . . . . . . . . . . 155

9.1.1 Power of one-sample and paired t tests . . . . . . 1569.1.2 Power of two-sample t test . . . . . . . . . . . . . 1589.1.3 Approximate methods . . . . . . . . . . . . . . . 1589.1.4 Power of comparisons of proportions . . . . . . . 159

9.2 Two-sample problems . . . . . . . . . . . . . . . . . . . . 1599.3 One-sample problems and paired tests . . . . . . . . . . 1619.4 Comparison of proportions . . . . . . . . . . . . . . . . . 1619.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

10 Advanced data handling 16310.1 Recoding variables . . . . . . . . . . . . . . . . . . . . . . 163

10.1.1 The cut function . . . . . . . . . . . . . . . . . . 16310.1.2 Manipulating factor levels . . . . . . . . . . . . . 16510.1.3 Working with dates . . . . . . . . . . . . . . . . . 16610.1.4 Recoding multiple variables . . . . . . . . . . . . 169

10.2 Conditional calculations . . . . . . . . . . . . . . . . . . 17010.3 Combining and restructuring data frames . . . . . . . . 171

10.3.1 Appending frames . . . . . . . . . . . . . . . . . 17210.3.2 Merging data frames . . . . . . . . . . . . . . . . 17310.3.3 Reshaping data frames . . . . . . . . . . . . . . . 175

10.4 Per-group and per-case procedures . . . . . . . . . . . . 17810.5 Time splitting . . . . . . . . . . . . . . . . . . . . . . . . . 17910.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

11 Multiple regression 18511.1 Plotting multivariate data . . . . . . . . . . . . . . . . . . 18511.2 Model specification and output . . . . . . . . . . . . . . 18711.3 Model search . . . . . . . . . . . . . . . . . . . . . . . . . 19011.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

Contents xv

12 Linear models 19512.1 Polynomial regression . . . . . . . . . . . . . . . . . . . . 19612.2 Regression through the origin . . . . . . . . . . . . . . . 19812.3 Design matrices and dummy variables . . . . . . . . . . 20012.4 Linearity over groups . . . . . . . . . . . . . . . . . . . . 20212.5 Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . 20612.6 Two-way ANOVA with replication . . . . . . . . . . . . 20712.7 Analysis of covariance . . . . . . . . . . . . . . . . . . . 208

12.7.1 Graphical description . . . . . . . . . . . . . . . . 20912.7.2 Comparison of regression lines . . . . . . . . . . 212

12.8 Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . 21812.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

13 Logistic regression 22713.1 Generalized linear models . . . . . . . . . . . . . . . . . 22813.2 Logistic regression on tabular data . . . . . . . . . . . . 229

13.2.1 The analysis of deviance table . . . . . . . . . . . 23413.2.2 Connection to test for trend . . . . . . . . . . . . 235

13.3 Likelihood profiling . . . . . . . . . . . . . . . . . . . . . 23713.4 Presentation as odds-ratio estimates . . . . . . . . . . . . 23913.5 Logistic regression using raw data . . . . . . . . . . . . . 23913.6 Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . 24113.7 Model checking . . . . . . . . . . . . . . . . . . . . . . . 24213.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

14 Survival analysis 24914.1 Essential concepts . . . . . . . . . . . . . . . . . . . . . . 24914.2 Survival objects . . . . . . . . . . . . . . . . . . . . . . . 25014.3 Kaplan–Meier estimates . . . . . . . . . . . . . . . . . . . 25114.4 The log-rank test . . . . . . . . . . . . . . . . . . . . . . . 25414.5 The Cox proportional hazards model . . . . . . . . . . . 25614.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

15 Rates and Poisson regression 25915.1 Basic ideas . . . . . . . . . . . . . . . . . . . . . . . . . . 259

15.1.1 The Poisson distribution . . . . . . . . . . . . . . 26015.1.2 Survival analysis with constant hazard . . . . . . 260

15.2 Fitting Poisson models . . . . . . . . . . . . . . . . . . . 26215.3 Computing rates . . . . . . . . . . . . . . . . . . . . . . . 26615.4 Models with piecewise constant intensities . . . . . . . . 27015.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

16 Nonlinear curve fitting 27516.1 Basic usage . . . . . . . . . . . . . . . . . . . . . . . . . . 27616.2 Finding starting values . . . . . . . . . . . . . . . . . . . 278

xvi Contents

16.3 Self-starting models . . . . . . . . . . . . . . . . . . . . . 28416.4 Profiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28516.5 Finer control of the fitting algorithm . . . . . . . . . . . 28716.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 288

A Obtaining and installing R and the ISwR package 289

B Data sets in the ISwR package 293

C Compendium 325

D Answers to exercises 337

Bibliography 355

Index 357

1Basics

The purpose of this chapter is to get you started using R. It is assumed thatyou have a working installation of the software and of the ISwR packagethat contains the data sets for this book. Instructions for obtaining andinstalling the software are given in Appendix A.

The text that follows describes R version 2.6.2. As of this writing, that isthe latest version of R. As far as possible, I present the issues in a waythat is independent of the operating system in use and assume that thereader has the elementary operational knowledge to select from menus,move windows around, etc. I do, however, make exceptions where I amaware of specific difficulties with a particular platform or specific featuresof it.

1.1 First steps

This section gives an introduction to the R computing environment andwalks you through its most basic features.

Starting R is straightforward, but the method will depend on your com-puting platform. You will be able to launch it from a system menu, bydouble-clicking an icon, or by entering the command “R” at the systemcommand line. This will either produce a console window or cause Rto start up as an interactive program in the current terminal window. In

P. Dalgaard, Introductory Statistics with R,DOI: 10.1007/978-0-387-79054-1_1, © Springer Science+Business Media, LLC 2008

2 1. Basics

Figure 1.1. Screen image of R for Windows.

either case, R works fundamentally by the question-and-answer model:You enter a line with a command and press Enter (←↩). Then the programdoes something, prints the result if relevant, and asks for more input.When R is ready for input, it prints out its prompt, a “>”. It is possi-ble to use R as a text-only application, and also in batch mode, but forthe purposes of this chapter, I assume that you are sitting at a graphicalworkstation.

All the examples in this book should run if you type them in exactly asprinted, provided that you have the ISwR package not only installed butalso loaded into your current search path. This is done by entering

> library(ISwR)

at the command prompt. You do not need to understand what thecommand does at this point. It is explained in Section 2.1.5.

For a first impression of what R can do, try typing the following:

> plot(rnorm(1000))

This command draws 1000 numbers at random from the normal distri-bution (rnorm = random normal) and plots them in a pop-up graphicswindow. The result on a Windows machine can be seen in Figure 1.1.

Of course, you are not expected at this point to guess that you would ob-tain this result in that particular way. The example is chosen because itshows several components of the user interface in action. Before the style

1.1 First steps 3

of commands will fall naturally, it is necessary to introduce some conceptsand conventions through simpler examples.

Under Windows, the graphics window will have taken the keyboard focusat this point. Click on the console to make it accept further commands.

1.1.1 An overgrown calculator

One of the simplest possible tasks in R is to enter an arithmetic expressionand receive a result. (The second line is the answer from the machine.)

> 2 + 2[1] 4

So the machine knows that 2 plus 2 makes 4. Of course, it also knows howto do other standard calculations. For instance, here is how to computee−2 :

> exp(-2)[1] 0.1353353

The [1] in front of the result is part of R’s way of printing numbers andvectors. It is not useful here, but it becomes so when the result is a longervector. The number in brackets is the index of the first number on thatline. Consider the case of generating 15 random numbers from a normaldistribution:

> rnorm(15)[1] -0.18326112 -0.59753287 -0.67017905 0.16075723 1.28199575[6] 0.07976977 0.13683303 0.77155246 0.85986694 -1.01506772[11] -0.49448567 0.52433026 1.07732656 1.09748097 -1.09318582

Here, for example, the [6] indicates that 0.07976977 is the sixth element inthe vector. (For typographical reasons, the examples in this book are madewith a shortened line width. If you try it on your own machine, you willsee the values printed with six numbers per line rather than five. The num-bers themselves will also be different since random number generation isinvolved.)

1.1.2 Assignments

Even on a calculator, you will quickly need some way to store intermedi-ate results, so that you do not have to key them in over and over again.R, like other computer languages, has symbolic variables, that is names that

4 1. Basics

can be used to represent values. To assign the value 2 to the variable x,you can enter

> x <- 2

The two characters <- should be read as a single symbol: an arrow point-ing to the variable to which the value is assigned. This is known as theassignment operator. Spacing around operators is generally disregardedby R, but notice that adding a space in the middle of a <- changes themeaning to “less than” followed by “minus” (conversely, omitting thespace when comparing a variable to a negative number has unexpectedconsequences!).

There is no immediately visible result, but from now on, x has the value 2and can be used in subsequent arithmetic expressions.

> x[1] 2> x + x[1] 4

Names of variables can be chosen quite freely in R. They can be built fromletters, digits, and the period (dot) symbol. There is, however, the limita-tion that the name must not start with a digit or a period followed by adigit. Names that start with a period are special and should be avoided.A typical variable name could be height.1yr, which might be used todescribe the height of a child at the age of 1 year. Names are case-sensitive:WT and wt do not refer to the same variable.

Some names are already used by the system. This can cause some con-fusion if you use them for other purposes. The worst cases are thesingle-letter names c, q, t, C, D, F, I, and T, but there are also diff, df,and pt, for example. Most of these are functions and do not usually causetrouble when used as variable names. However, F and T are the standardabbreviations for FALSE and TRUE and no longer work as such if youredefine them.

1.1.3 Vectorized arithmetic

You cannot do much statistics on single numbers! Rather, you will look atdata from a group of patients, for example. One strength of R is that it canhandle entire data vectors as single objects. A data vector is simply an arrayof numbers, and a vector variable can be constructed like this:

> weight <- c(60, 72, 57, 90, 95, 72)> weight[1] 60 72 57 90 95 72

1.1 First steps 5

The construct c(…) is used to define vectors. The numbers are madeup but might represent the weights (in kg) of a group of normal men.

This is neither the only way to enter data vectors into R nor is it gen-erally the preferred method, but short vectors are used for many otherpurposes, and the c(…) construct is used extensively. In Section 2.4,we discuss alternative techniques for reading data. For now, we stick to asingle method.

You can do calculations with vectors just like ordinary numbers, as longas they are of the same length. Suppose that we also have the heights thatcorrespond to the weights above. The body mass index (BMI) is definedfor each person as the weight in kilograms divided by the square of theheight in meters. This could be calculated as follows:

> height <- c(1.75, 1.80, 1.65, 1.90, 1.74, 1.91)> bmi <- weight/height^2> bmi[1] 19.59184 22.22222 20.93664 24.93075 31.37799 19.73630

Notice that the operation is carried out elementwise (that is, the first valueof bmi is 60/1.752 and so forth) and that the ^ operator is used for raisinga value to a power. (On some keyboards, ^ is a “dead key” and you willhave to press the spacebar afterwards to make it show.)

It is in fact possible to perform arithmetic operations on vectors of differ-ent length. We already used that when we calculated the height^2 partabove since 2 has length 1. In such cases, the shorter vector is recycled.This is mostly used with vectors of length 1 (scalars) but sometimes alsoin other cases where a repeating pattern is desired. A warning is issued ifthe longer vector is not a multiple of the shorter in length.

These conventions for vectorized calculations make it very easy to specifytypical statistical calculations. Consider, for instance, the calculation of themean and standard deviation of the weight variable.

First, calculate the mean, x̄ = ∑ xi /n:

> sum(weight)[1] 446> sum(weight)/length(weight)[1] 74.33333

Then save the mean in a variable xbar and proceed with the calculationof SD =

√(∑(xi − x̄)2)/(n − 1). We do this in steps to see the individual

components. The deviations from the mean are

> xbar <- sum(weight)/length(weight)> weight – xbar

6 1. Basics

[1] -14.333333 -2.333333 -17.333333 15.666667 20.666667[6] -2.333333

Notice how xbar, which has length 1, is recycled and subtracted fromeach element of weight. The squared deviations will be

> (weight – xbar)^2[1] 205.444444 5.444444 300.444444 245.444444 427.111111[6] 5.444444

Since this command is quite similar to the one before it, it is convenientto enter it by editing the previous command. On most systems running R,the previous command can be recalled with the up-arrow key.

The sum of squared deviations is similarly obtained with

> sum((weight – xbar)^2)[1] 1189.333

and all in all the standard deviation becomes

> sqrt(sum((weight – xbar)^2)/(length(weight) – 1))[1] 15.42293

Of course, since R is a statistical program, such calculations are alreadybuilt into the program, and you get the same results just by entering

> mean(weight)[1] 74.33333> sd(weight)[1] 15.42293

1.1.4 Standard procedures

As a slightly more complicated example of what R can do, consider thefollowing: The rule of thumb is that the BMI for a normal-weight indi-vidual should be between 20 and 25, and we want to know if our datadeviate systematically from that. You might use a one-sample t test to as-sess whether the six persons’ BMI can be assumed to have mean 22.5 giventhat they come from a normal distribution. To this end, you can use thefunction t.test. (You might not know the theory of the t test yet. Theexample is included here mainly to give some indication of what “real”statistical output looks like. A thorough description of t.test is given inChapter 5.)

1.1 First steps 7

> t.test(bmi, mu=22.5)One Sample t-test

data: bmit = 0.3449, df = 5, p-value = 0.7442alternative hypothesis: true mean is not equal to 22.595 percent confidence interval:18.41734 27.84791sample estimates:mean of x23.13262

The argument mu=22.5 attaches a value to the formal argument mu,which represents the Greek letter µ conventionally used for the theoret-ical mean. If this is not given, t.test would use the default mu=0, whichis not of interest here.

For a test like this, we get a more extensive printout than in the earlierexamples. The details of the output are explained in Chapter 5, but youmight focus on the p-value which is used for testing the hypothesis thatthe mean is 22.5. The p-value is not small, indicating that it is not at all un-likely to get data like those observed if the mean were in fact 22.5. (Looselyspeaking; actually p is the probability of obtaining a t value bigger than0.3449 or less than −0.3449.) However, you might also look at the 95% con-fidence interval for the true mean. This interval is quite wide, indicatingthat we really have very little information about the true mean.

1.1.5 Graphics

One of the most important aspects of the presentation and analysis of datais the generation of proper graphics. R — like S before it — has a modelfor constructing plots that allows simple production of standard plots aswell as fine control over the graphical components.

If you want to investigate the relation between weight and height, thefirst idea is to plot one versus the other. This is done by

> plot(height,weight)

leading to Figure 1.2.

You will often want to modify the drawing in various ways. To that end,there are a wealth of plotting parameters that you can set. As an example,let us try changing the plotting symbol using the keyword pch (“plottingcharacter”) like this:

> plot(height, weight, pch=2)

8 1. Basics

1.65 1.70 1.75 1.80 1.85 1.90

height

weig

Figure 1.2. A simple x–y plot.

This gives the plot in Figure 1.3, with the points now marked with littletriangles.

The idea behind the BMI calculation is that this value should be inde-pendent of the person’s height, thus giving you a single number as anindication of whether someone is overweight and by how much. Sincea normal BMI should be about 22.5, you would expect that weight ≈22.5 × height2. Accordingly, you can superimpose a curve of expectedweights at BMI 22.5 on the figure:

> hh <- c(1.65, 1.70, 1.75, 1.80, 1.85, 1.90)> lines(hh, 22.5 * hh^2)

yielding Figure 1.4. The function lines will add (x, y) values joined bystraight lines to an existing plot.

The reason for defining a new variable (hh) with heights rather than usingthe original height vector is twofold. First, the relation between heightand weight is a quadratic one and hence nonlinear, although it can be diffi-cult to see on the plot. Since we are approximating a nonlinear curve witha piecewise linear one, it will be better to use points that are spread evenlyalong the x-axis than to rely on the distribution of the original data. Sec-

1.2 R language essentials 9

ond, since the values of height are not sorted, the line segments wouldnot connect neighbouring points but would run back and forth betweendistant points.

1.2 R language essentials

This section outlines the basic aspects of the R language. It is necessaryto do this in a slightly superficial manner, with some of the finer pointsglossed over. The emphasis is on items that are useful to know in interac-tive usage as opposed to actual programming, although a brief section onprogramming is included.

1.2.1 Expressions and objects

The basic interaction mode in R is one of expression evaluation. The userenters an expression; the system evaluates it and prints the result. Someexpressions are evaluated not for their result but for side effects such as

1.65 1.70 1.75 1.80 1.85 1.90

height

weig

Figure 1.3. Plot with pch = 2.

10 1. Basics

1.65 1.70 1.75 1.80 1.85 1.90

height

weig

Figure 1.4. Superimposed reference curve, using lines(…).

putting up a graphics window or writing to a file. All R expressions returna value (possibly NULL), but sometimes it is “invisible” and not printed.

Expressions typically involve variable references, operators such as +, andfunction calls, as well as some other items that have not been introducedyet.

Expressions work on objects. This is an abstract term for anything that canbe assigned to a variable. R contains several different types of objects. Sofar, we have almost exclusively seen numeric vectors, but several othertypes are introduced in this chapter.

Although objects can be discussed abstractly, it would make a rather bor-ing read without some indication of how to generate them and what to dowith them. Conversely, much of the expression syntax makes little sensewithout knowledge of the objects on which it is intended to work. There-fore, the subsequent sections alternate between introducing new objectsand introducing new language elements.

1.2 R language essentials 11

1.2.2 Functions and arguments

At this point, you have obtained an impression of the way R works, andwe have already used some of the special terminology when talking aboutthe plot function, etc. That is exactly the point: Many things in R are doneusing function calls, commands that look like an application of a math-ematical function of one or several variables; for example, log(x) orplot(height, weight).

The format is that a function name is followed by a set of parentheses con-taining one or more arguments. For instance, in plot(height,weight)the function name is plot and the arguments are height and weight.These are the actual arguments, which apply only to the current call. A func-tion also has formal arguments, which get connected to actual arguments inthe call.

When you write plot(height, weight), R assumes that the first argu-ment corresponds to the x-variable and the second one to the y-variable.This is known as positional matching. This becomes unwieldy if a func-tion has a large number of arguments since you have to supply everyone of them and remember their position in the sequence. Fortunately,R has methods to avoid this: Most arguments have sensible defaults andcan be omitted in the standard cases, and there are nonpositional ways ofspecifying them when you need to depart from the default settings.

The plot function is in fact an example of a function that has a largeselection of arguments in order to be able to modify symbols, linewidths, titles, axis type, and so forth. We used the alternative form ofspecifying arguments when setting the plot symbol to triangles withplot(height, weight, pch=2).

The pch=2 form is known as a named actual argument, whose name canbe matched against the formal arguments of the function and therebyallow keyword matching of arguments. The keyword pch was used tosay that the argument is a specification of the plotting character. Thistype of function argument can be specified in arbitrary order. Thus, youcan write plot(y=weight,x=height) and get the same plot as withplot(x=height,y=weight).

The two kinds of argument specification — positional and named — canbe mixed in the same call.

Even if there are no arguments to a function call, you have to write, forexample, ls() for displaying the contents of the workspace. A commonerror is to leave off the parentheses, which instead results in the display ofa piece of R code since ls entered by itself indicates that you want to seethe definition of the function rather than execute it.

12 1. Basics

The formal arguments of a function are part of the function definition. Theset of formal arguments to a function, for instance plot.default (whichis the function that gets called when you pass plot an x argument forwhich no special plot method exists), may be seen with

> args(plot.default)function (x, y = NULL, type = "p", xlim = NULL, ylim = NULL,

log = "", main = NULL, sub = NULL, xlab = NULL, ylab = NULL,ann = par("ann"), axes = TRUE, frame.plot = axes,panel.first = NULL, panel.last = NULL, asp = NA, …)

Notice that most of the arguments have defaults, meaning that if you donot specify (say) the type argument, the function will behave as if youhad passed type="p". The NULL defaults for many of the arguments re-ally serve as indicators that the argument is unspecified, allowing specialbehaviour to be defined inside the function. For instance, if they are notspecified, the xlab and ylab arguments are constructed from the actualarguments passed as x and y. (There are some very fine points associatedwith this, but we do not go further into the topic.)

The triple-dot (…) argument indicates that this function will acceptadditional arguments of unspecified name and number. These are of-ten meant to be passed on to other functions, although some functionstreat it specially. For instance, in data.frame and c, the names of the…-arguments become the names of the elements of the result.

1.2.3 Vectors

We have already seen numeric vectors. There are two further types,character vectors and logical vectors.

A character vector is a vector of text strings, whose elements are specifiedand printed in quotes:

> c("Huey","Dewey","Louie")[1] "Huey" "Dewey" "Louie"

It does not matter whether you use single- or double-quote symbols, aslong as the left quote is the same as the right quote:

> c(’Huey’,’Dewey’,’Louie’)[1] "Huey" "Dewey" "Louie"

However, you should avoid the acute accent key (´), which is present onsome keyboards. Double quotes are used throughout this book to preventmistakes. Logical vectors can take the value TRUE or FALSE (or NA; seebelow). In input, you may use the convenient abbreviations T and F (if you

1.2 R language essentials 13

are careful not to redefine them). Logical vectors are constructed using thec function just like the other vector types:

> c(T,T,F,T)[1] TRUE TRUE FALSE TRUE

Actually, you will not often have to specify logical vectors in the mannerabove. It is much more common to use single logical values to turn anoption on or off in a function call. Vectors of more than one value mostoften result from relational expressions:

> bmi > 25[1] FALSE FALSE FALSE FALSE TRUE FALSE

We return to relational expressions and logical operations in the contextof conditional selection in Section 1.2.12.

1.2.4 Quoting and escape sequences

Quoted character strings require some special considerations: How, forinstance, do you put a quote symbol inside a string? And what about spe-cial characters such as newlines? This is done using escape sequences. Weshall look at those in a moment, but first it will be useful to observe thefollowing.

There is a distinction between a text string and the way it is printed. When,for instance, you give the string "Huey", it is a string of four characters,not six. The quotes are not actually part of the string, they are just thereso that the system can tell the difference between a string and a variablename.

If you print a character vector, it usually comes out with quotes added toeach element. There is a way to avoid this, namely to use the cat function.For instance,

> cat(c("Huey","Dewey","Louie"))Huey Dewey Louie>

This prints the strings without quotes, just separated by a space character.There is no newline following the string, so the prompt (>) for the nextline of input follows directly at the end of the line. (Notice that when thecharacter vector is printed by cat there is no way of telling the differencefrom the single string "Huey Dewey Louie".)

To get the system prompt onto the next line, you must include a newlinecharacter

14 1. Basics

> cat("Huey","Dewey","Louie", "n")Huey Dewey Louie>

Here, n is an example of an escape sequence. It actually represents a sin-gle character, the linefeed (LF), but is represented as two. The backslash() is known as the escape character. In a similar vein, you can insert quotecharacters with ", as in

> cat("What is "R"?n")What is "R"?

There are also ways to insert other control characters and special glyphs,but it would lead us too far astray to discuss it in full detail. One impor-tant thing, though: What about the escape character itself? This, too, mustbe escaped, so to put a backslash in a string, you must double it. Thisis important to know when specifying file paths on Windows, see alsoSection 2.4.1.

1.2.5 Missing values

In practical data analysis, a data point is frequently unavailable (the pa-tient did not show up, an experiment failed, etc.). Statistical softwareneeds ways to deal with this. R allows vectors to contain a special NAvalue. This value is carried through in computations so that operations onNA yield NA as the result. There are some special issues associated with thehandling of missing values; we deal with them as we encounter them (see“missing values” in the index).

1.2.6 Functions that create vectors

Here we introduce three functions, c, seq, and rep, that are used to createvectors in various situations.

The first of these, c, has already been introduced. It is short for “con-catenate”, joining items end to end, which is exactly what the functiondoes:

> c(42,57,12,39,1,3,4)[1] 42 57 12 39 1 3 4

You can also concatenate vectors of more than one element as in

> x <- c(1, 2, 3)> y <- c(10, 20)

1.2 R language essentials 15

> c(x, y, 5)[1] 1 2 3 10 20 5

However, you do not need to use c to create vectors of length 1. Peoplesometimes type, for example, c(1), but it is the same as plain 1.

It is also possible to assign names to the elements. This modifies the waythe vector is printed and is often used for display purposes.

> x <- c(red="Huey", blue="Dewey", green="Louie")> x

red blue green"Huey" "Dewey" "Louie"

(In this case, it does of course make sense to use c even for single-elementvectors.)

The names can be extracted or set using names:

> names(x)[1] "red" "blue" "green"

All elements of a vector have the same type. If you concatenate vectors ofdifferent types, they will be converted to the least “restrictive” type:

> c(FALSE, 3)[1] 0 3> c(pi, "abc")[1] "3.14159265358979" "abc"> c(FALSE, "abc")[1] "FALSE" "abc"

That is, logical values may be converted to 0/1 or "FALSE"/"TRUE" andnumbers converted to their printed representations.

The second function, seq (“sequence”), is used for equidistant series ofnumbers. Writing

> seq(4,9)[1] 4 5 6 7 8 9

yields, as shown, the integers from 4 to 9. If you want a sequence in jumpsof 2, write

> seq(4,10,2)[1] 4 6 8 10

This kind of vector is frequently needed, particularly for graphics. For ex-ample, we previously used c(1.65,1.70,1.75,1.80,1.85,1.90) todefine the x-coordinates for a curve, something that could also have been

16 1. Basics

written seq(1.65,1.90,0.05) (the advantage of using seq might havebeen more obvious if the heights had been in steps of 1 cm rather than5 cm!).

The case with step size equal to 1 can also be written using a specialsyntax:

> 4:9[1] 4 5 6 7 8 9

The above is exactly the same as seq(4,9), only easier to read.

The third function, rep (“replicate”), is used to generate repeated values.It is used in two variants, depending on whether the second argument isa vector or a single number:

> oops <- c(7,9,13)> rep(oops,3)[1] 7 9 13 7 9 13 7 9 13> rep(oops,1:3)[1] 7 9 9 13 13 13

The first of the function calls above repeats the entire vector oops threetimes. The second call has the number 3 replaced by a vector with thethree values (1, 2, 3); these values correspond to the elements of the oopsvector, indicating that 7 should be repeated once, 9 twice, and 13 threetimes. The rep function is often used for things such as group codes: If itis known that the first 10 observations are men and the last 15 are women,you can use

> rep(1:2,c(10,15))[1] 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

to form a vector that for each observation indicates whether it is from aman or a woman.

The special case where there are equally many replications of each valuecan be obtained using the each argument. E.g., rep(1:2,each=10) isthe same as rep(1:2,c(10,10)).

1.2.7 Matrices and arrays

A matrix in mathematics is just a two-dimensional array of numbers. Ma-trices are used for many purposes in theoretical and practical statistics,but it is not assumed that the reader is familiar with matrix algebra,so many special operations on matrices, including matrix multiplication,are skipped. (The document “An Introduction to R”, which comes with

1.2 R language essentials 17

the installation, outlines these items quite well.) However, matrices andalso higher-dimensional arrays do get used for simpler purposes as well,mainly to hold tables, so an elementary description is in order.

In R, the matrix notion is extended to elements of any type, so you couldhave, for instance, a matrix of character strings. Matrices and arrays arerepresented as vectors with dimensions:

> x <- 1:12> dim(x) <- c(3,4)> x

[,1] [,2] [,3] [,4][1,] 1 4 7 10[2,] 2 5 8 11[3,] 3 6 9 12

The dim assignment function sets or changes the dimension attribute of x,causing R to treat the vector of 12 numbers as a 3 × 4 matrix. Notice thatthe storage is column-major; that is, the elements of the first column arefollowed by those of the second, etc.

A convenient way to create matrices is to use the matrix function:

> matrix(1:12,nrow=3,byrow=T)[,1] [,2] [,3] [,4]

[1,] 1 2 3 4[2,] 5 6 7 8[3,] 9 10 11 12

Notice how the byrow=T switch causes the matrix to be filled in a rowwisefashion rather than columnwise.

Useful functions that operate on matrices include rownames, colnames,and the transposition function t (notice the lowercase t as opposed touppercase T for TRUE), which turns rows into columns and vice versa:

> x <- matrix(1:12,nrow=3,byrow=T)> rownames(x) <- LETTERS[1:3]> x[,1] [,2] [,3] [,4]

A 1 2 3 4B 5 6 7 8C 9 10 11 12> t(x)

A B C[1,] 1 5 9[2,] 2 6 10[3,] 3 7 11[4,] 4 8 12

18 1. Basics

The character vector LETTERS is a built-in variable that contains the cap-ital letters A–Z. Similar useful vectors are letters, month.name, andmonth.abb with lowercase letters, month names, and abbreviated monthnames.

You can “glue” vectors together, columnwise or rowwise, using the cbindand rbind functions.

> cbind(A=1:4,B=5:8,C=9:12)A B C

[1,] 1 5 9[2,] 2 6 10[3,] 3 7 11[4,] 4 8 12> rbind(A=1:4,B=5:8,C=9:12)[,1] [,2] [,3] [,4]

A 1 2 3 4B 5 6 7 8C 9 10 11 12

We return to table operations in Section 4.5, which discusses tabulation ofvariables in a data set.

1.2.8 Factors

It is common in statistical data to have categorical variables, indicatingsome subdivision of data, such as social class, primary diagnosis, tu-mor stage, Tanner stage of puberty, etc. Typically, these are input usinga numeric code.

Such variables should be specified as factors in R. This is a data structurethat (among other things) makes it possible to assign meaningful namesto the categories.

There are analyses where it is essential for R to be able to distinguishbetween categorical codes and variables whose values have a directnumerical meaning (see Chapter 7).

The terminology is that a factor has a set of levels — say four levels for con-creteness. Internally, a four-level factor consists of two items: (a) a vector ofintegers between 1 and 4 and (b) a character vector of length 4 containingstrings describing what the four levels are. Let us look at an example:

> pain <- c(0,3,2,2,1)> fpain <- factor(pain,levels=0:3)> levels(fpain) <- c("none","mild","medium","severe")

1.2 R language essentials 19

The first command creates a numeric vector pain, encoding the pain lev-els of five patients. We wish to treat this as a categorical variable, so wecreate a factor fpain from it using the function factor. This is calledwith one argument in addition to pain, namely levels=0:3, which in-dicates that the input coding uses the values 0–3. The latter can in principlebe left out since R by default uses the values in pain, suitably sorted, butit is a good habit to retain it; see below. The effect of the final line is thatthe level names are changed to the four specified character strings.

The result should be apparent from the following:

> fpain[1] none severe medium medium mildLevels: none mild medium severe> as.numeric(fpain)[1] 1 4 3 3 2> levels(fpain)[1] "none" "mild" "medium" "severe"

The function as.numeric extracts the numerical coding as numbers1–4 and levels extracts the names of the levels. Notice that the origi-nal input coding in terms of numbers 0–3 has disappeared; the internalrepresentation of a factor always uses numbers starting at 1.

R also allows you to create a special kind of factor in which the lev-els are ordered. This is done using the ordered function, which workssimilarly to factor. These are potentially useful in that they distinguishnominal and ordinal variables from each other (and arguably text.painabove ought to have been an ordered factor). Unfortunately, R defaultsto treating the levels as if they were equidistant in the modelling code (bygenerating polynomial contrasts), so it may be better to ignore orderedfactors at this stage.

1.2.9 Lists

It is sometimes useful to combine a collection of objects into a largercomposite object. This can be done using lists.

You can construct a list from its components with the function list.

As an example, consider a set of data from Altman (1991, p. 183) concern-ing pre- and postmenstrual energy intake in a group of women. We canplace these data in two vectors as follows:

> intake.pre <- c(5260,5470,5640,6180,6390,+ 6515,6805,7515,7515,8230,8770)> intake.post <- c(3910,4220,3885,5160,5645,+ 4680,5265,5975,6790,6900,7335)

20 1. Basics

Notice how input lines can be broken and continue on the next line. Ifyou press the Enter key while an expression is syntactically incomplete, Rwill assume that the expression continues on the next line and will changeits normal > prompt to the continuation prompt +. This often happens in-advertently due to a forgotten parenthesis or a similar problem; in suchcases, either complete the expression on the next line or press ESC (Win-dows and Macintosh) or Ctrl-C (Unix). The “Stop” button can also be usedunder Windows.

To combine these individual vectors into a list, you can say

> mylist <- list(before=intake.pre,after=intake.post)> mylist$before[1] 5260 5470 5640 6180 6390 6515 6805 7515 7515 8230 8770

$after[1] 3910 4220 3885 5160 5645 4680 5265 5975 6790 6900 7335

The components of the list are named according to the argument namesused in list. Named components may be extracted like this:

> mylist$before[1] 5260 5470 5640 6180 6390 6515 6805 7515 7515 8230 8770

Many of R’s built-in functions compute more than a single vector of valuesand return their results in the form of a list.

1.2.10 Data frames

A data frame corresponds to what other statistical packages call a “datamatrix” or a “data set”. It is a list of vectors and/or factors of the samelength that are related “across” such that data in the same position comefrom the same experimental unit (subject, animal, etc.). In addition, it hasa unique set of row names.

You can create data frames from preexisting variables:

> d <- data.frame(intake.pre,intake.post)> d

intake.pre intake.post1 5260 39102 5470 42203 5640 38854 6180 51605 6390 56456 6515 46807 6805 5265

1.2 R language essentials 21

8 7515 59759 7515 679010 8230 690011 8770 7335

Notice that these data are paired, that is, the same woman has an intakeof 5260 kJ premenstrually and 3910 kJ postmenstrually.

As with lists, components (i.e., individual variables) can be accessed usingthe $ notation:

> d$intake.pre[1] 5260 5470 5640 6180 6390 6515 6805 7515 7515 8230 8770

1.2.11 Indexing

If you need a particular element in a vector, for instance the premenstrualenergy intake for woman no. 5, you can do

> intake.pre[5][1] 6390

The brackets are used for selection of data, also known as indexing or sub-setting. This also works on the left-hand side of an assignment (so that youcan say, for instance, intake.pre[5] <- 6390) if you want to modifyelements of a vector.

If you want a subvector consisting of data for more than one woman, forinstance nos. 3, 5, and 7, you can index with a vector:

> intake.pre[c(3,5,7)][1] 5640 6390 6805

Note that it is necessary to use the c(…)-construction to define the vec-tor consisting of the three numbers 3, 5, and 7. intake.pre[3,5,7]would mean something completely different. It would specify indexinginto a three-dimensional array.

Of course, indexing with a vector also works if the index vector is storedin a variable. This is useful when you need to index several variables inthe same way.

> v <- c(3,5,7)> intake.pre[v][1] 5640 6390 6805

It is also worth noting that to get a sequence of elements, for instance thefirst five, you can use the a:b notation:

22 1. Basics

> intake.pre[1:5][1] 5260 5470 5640 6180 6390

A neat feature of R is the possibility of negative indexing. You can get allobservations except nos. 3, 5, and 7 by writing

> intake.pre[-c(3,5,7)][1] 5260 5470 6180 6515 7515 7515 8230 8770

It is not possible to mix positive and negative indices. That would behighly ambiguous.

1.2.12 Conditional selection

We saw in Section 1.2.11 how to extract data using one or several indices.In practice, you often need to extract data that satisfy certain criteria, suchas data from the males or the prepubertal or those with chronic diseases,etc. This can be done simply by inserting a relational expression insteadof the index,

> intake.post[intake.pre > 7000][1] 5975 6790 6900 7335

yielding the postmenstrual energy intake for the four women who had anenergy intake above 7000 kJ premenstrually.

Of course, this kind of expression makes sense only if the variables that gointo the relational expression have the same length as the variable beingindexed.

The comparison operators available are < (less than), > (greater than), ==(equal to), <= (less than or equal to), >= (greater than or equal to), and !=(not equal to). Notice that a double equal sign is used for testing equality.This is to avoid confusion with the = symbol used to match keywords withfunction arguments. Also, the != operator is new to some; the ! symbolindicates negation. The same operators are used in the C programminglanguage.

To combine several expressions, you can use the logical operators & (log-ical “and”), | (logical “or”), and ! (logical “not”). For instance, we findthe postmenstrual intake for women with a premenstrual intake between7000 and 8000 kJ with

> intake.post[intake.pre > 7000 & intake.pre <= 8000][1] 5975 6790

1.2 R language essentials 23

There are also && and ||, which are used for flow control in Rprogramming. However, their use is beyond what we discuss here.

It may be worth taking a closer look at what actually happens when youuse a logical expression as an index. The result of the logical expression isa logical vector as described in Section 1.2.3:

> intake.pre > 7000 & intake.pre <= 8000[1] FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE FALSE[11] FALSE

Indexing with a logical vector implies that you pick out the values wherethe logical vector is TRUE, so in the preceding example we got the 8th and9th values in intake.post.

If missing values (NA; see Section 1.2.5) appear in an indexing vector, thenR will create the corresponding elements in the result but set the values toNA.

In addition to the relational and logical operators, there are a series offunctions that return a logical value. A particularly important one isis.na(x), which is used to find out which elements of x are recordedas missing (NA).

Notice that there is a real need for is.na because you cannot makecomparisons of the form x==NA. That simply gives NA as the result forany value of x. The result of a comparison with an unknown value isunknown!

1.2.13 Indexing of data frames

We have already seen how it is possible to extract variables from adata frame by typing, for example, d$intake.post. However, it is alsopossible to use a notation that uses the matrix-like structure directly:

> d <- data.frame(intake.pre,intake.post)> d[5,1][1] 6390

gives fifth row, first column (that is, the “pre” measurement for womanno. 5), and

> d[5,]intake.pre intake.post

5 6390 5645

gives all measurements for woman no. 5. Notice that the comma in d[5,]is required; without the comma, for example d[2], you get the data frame

24 1. Basics

consisting of the second column of d (that is, more like d[,2], which is thecolumn itself).

Other indexing techniques also apply. In particular, it can be useful to ex-tract all data for cases that satisfy some criterion, such as women with apremenstrual intake above 7000 kJ:

> d[d$intake.pre>7000,]intake.pre intake.post

8 7515 59759 7515 679010 8230 690011 8770 7335

Here we extracted the rows of the data frame where intake.pre>7000.Notice that the row names are those of the original data frame.

If you want to understand the details of this, it may be a little easier if it isdivided into smaller steps. It could also have been done like this:

> sel <- d$intake.pre>7000> sel[1] FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE TRUE[11] TRUE> d[sel,]

intake.pre intake.post8 7515 59759 7515 679010 8230 690011 8770 7335

What happens is that sel (select) becomes a logical vector with the valueTRUE for to the four women consuming more than 7000 kJ premenstrually.Indexing as d[sel,] yields data from the rows where sel is TRUE andfrom all columns because of the empty field after the comma.

It is often convenient to look at the first few cases in a data set. This can bedone with indexing, like this:

> d[1:2,]intake.pre intake.post

1 5260 39102 5470 4220

This is such a frequent occurrence that a convenience function called headexists. By default, it shows the first six lines.

> head(d)intake.pre intake.post

1 5260 39102 5470 4220

1.2 R language essentials 25

3 5640 38854 6180 51605 6390 56456 6515 4680

Similarly, tail shows the last part.

1.2.14 Grouped data and data frames

The natural way of storing grouped data in a data frame is to have the datathemselves in one vector and parallel to that have a factor telling whichdata are from which group. Consider, for instance, the following data seton energy expenditure for lean and obese women.

> energyexpend stature

1 9.21 obese2 7.53 lean3 7.48 lean4 8.08 lean5 8.09 lean6 10.15 lean7 8.40 lean8 10.88 lean9 6.13 lean10 7.90 lean11 11.51 obese12 12.79 obese13 7.05 lean14 11.85 obese15 9.97 obese16 7.48 lean17 8.79 obese18 9.69 obese19 9.68 obese20 7.58 lean21 9.19 obese22 8.11 lean

This is a convenient format since it generalizes easily to data classifiedby multiple criteria. However, sometimes it is desirable to have data in aseparate vector for each group. Fortunately, it is easy to extract these fromthe data frame:

> exp.lean <- energy$expend[energy$stature=="lean"]> exp.obese <- energy$expend[energy$stature=="obese"]

Alternatively, you can use the split function, which generates a list ofvectors according to a grouping.

26 1. Basics

> l <- split(energy$expend, energy$stature)> l$lean[1] 7.53 7.48 8.08 8.09 10.15 8.40 10.88 6.13 7.90 7.05[11] 7.48 7.58 8.11

$obese[1] 9.21 11.51 12.79 11.85 9.97 8.79 9.69 9.68 9.19

1.2.15 Implicit loops

The looping constructs of R are described in Section 2.3.1. For the purposesof this book, you can largely ignore their existence. However, there is agroup of R functions that it will be useful for you to know about.

A common application of loops is to apply a function to each element ofa set of values or vectors and collect the results in a single structure. InR this is abstracted by the functions lapply and sapply. The formeralways returns a list (hence the ‘l’), whereas the latter tries to simplify(hence the ‘s’) the result to a vector or a matrix if possible. So, to computethe mean of each variable in a data frame of numeric vectors, you can dothe following:

> lapply(thuesen, mean, na.rm=T)$blood.glucose[1] 10.3

$short.velocity[1] 1.325652

> sapply(thuesen, mean, na.rm=T)blood.glucose short.velocity

10.300000 1.325652

Notice how both forms attach meaningful names to the result, whichis another good reason to prefer to use these functions rather than ex-plicit loops. The second argument to lapply/sapply is the function thatshould be applied, here mean. Any further arguments are passed on to thefunction; in this case we pass na.rm=T to request that missing values beremoved (see Section 4.1).

Sometimes you just want to repeat something a number of times but stillcollect the results as a vector. Obviously, this makes sense only when therepeated computations actually give different results, the common casebeing simulation studies. This can be done using sapply, but there is asimplified version called replicate, in which you just have to give acount and the expression to evaluate:

1.2 R language essentials 27

> replicate(10,mean(rexp(20)))[1] 1.0677019 1.2166898 0.8923216 1.1281207 0.9636017 0.8406877[7] 1.3357814 0.8249408 0.9488707 0.5724575

A similar function, apply, allows you to apply a function to the rowsor columns of a matrix (or over indices of a multidimensional array ingeneral) as in

> m <- matrix(rnorm(12),4)> m

[,1] [,2] [,3][1,] -2.5710730 0.2524470 -0.16886795[2,] 0.5509498 1.5430648 0.05359794[3,] 2.4002722 0.1624704 -1.23407417[4,] 1.4791103 0.9484525 -0.84670929> apply(m, 2, min)[1] -2.5710730 0.1624704 -1.2340742

The second argument is the index (or vector of indices) that defines whatthe function is applied to; in this case we get the columnwise minima.

Also, the function tapply allows you to create tables (hence the ‘t’) of thevalue of a function on subgroups defined by its second argument, whichcan be a factor or a list of factors. In the latter case a cross-classified tableis generated. (The grouping can also be defined by ordinary vectors. Theywill be converted to factors internally.)

> tapply(energy$expend, energy$stature, median)lean obese7.90 9.69

1.2.16 Sorting

It is trivial to sort a vector. Just use the sort function. (We use the built-in data set intake here; it contains the same data that were used inSection 1.2.9.)

> intake$post[1] 3910 4220 3885 5160 5645 4680 5265 5975 6790 6900 7335> sort(intake$post)[1] 3885 3910 4220 4680 5160 5265 5645 5975 6790 6900 7335

(intake$pre could not be used for this example since it is sortedalready!)

However, sorting a single vector is not always what is required. Oftenyou need to sort a series of variables according to the values of some othervariables — blood pressures sorted by sex and age, for instance. For this

28 1. Basics

purpose, there is a construction that may look somewhat abstract at firstbut is really very powerful. You first compute an ordering of a variable.

> order(intake$post)[1] 3 1 2 6 4 7 5 8 9 10 11

The result is the numbers 1 to 11 (or whatever the length of the vec-tor is), sorted according to the size of the argument to order (hereintake$post). Interpreting the result of order is a bit tricky — it shouldbe read as follows: You sort intake$post by placing its values in theorder no. 3, no. 1, no. 2, no. 6, etc.

The point is that, by indexing with this vector, other variables can besorted by the same criterion. Note that indexing with a vector containingthe numbers from 1 to the number of elements exactly once correspondsto a reordering of the elements.

> o <- order(intake$post)> intake$post[o][1] 3885 3910 4220 4680 5160 5265 5645 5975 6790 6900 7335> intake$pre[o][1] 5640 5260 5470 6515 6180 6805 6390 7515 7515 8230 8770

What has happened here is that intake$post has been sorted — just asin sort(intake$post) — while intake$pre has been sorted by thesize of the corresponding intake$post.

It is of course also possible to sort the entire data frame intake

> intake.sorted <- intake[o,]

Sorting by several criteria is done simply by having several arguments toorder; for instance, order(sex,age) will give a main division into menand women, and within each sex an ordering by age. The second variableis used when the order cannot be decided from the first variable. Sortingin reverse order can be handled by, for example, changing the sign of thevariable.

1.3 Exercises

1.1 How would you check whether two vectors are the same if theymay contain missing (NA) values? (Use of the identical function isconsidered cheating!)

1.2 If x is a factor with n levels and y is a length n vector, what happensif you compute y[x]?

1.3 Exercises 29

1.3 Write the logical expression to use to extract girls between 7 and 14years of age in the juul data set.

1.4 What happens if you change the levels of a factor (with levels) andgive the same value to two or more levels?

1.5 On p. 27, replicate was used to simulate the distribution of themean of 20 random numbers from the exponential distribution by re-peating the operation 10 times. How would you do the same thing withsapply?

2The R environment

This chapter collects some practical aspects of working with R. It de-scribes issues regarding the structure of the workspace, graphical devicesand their parameters, and elementary programming, and includes a fairlyextensive, although far from complete, discussion of data entry.

2.1 Session management

2.1.1 The workspace

All variables created in R are stored in a common workspace. To see whichvariables are defined in the workspace, you can use the function ls (list).It should look as follows if you have run all the examples in the precedingchapter:

> ls()[1] "bmi" "d" "exp.lean"[4] "exp.obese" "fpain" "height"[7] "hh" "intake.post" "intake.pre"[10] "intake.sorted" "l" "m"[13] "mylist" "o" "oops"[16] "pain" "sel" "v"[19] "weight" "x" "xbar"[22] "y"

P. Dalgaard, Introductory Statistics with R,DOI: 10.1007/978-0-387-79054-1_2, © Springer Science+Business Media, LLC 2008

32 2. The R environment

Remember that you cannot omit the parentheses in ls().

If at some point things begin to look messy, you can delete some of theobjects. This is done using rm (remove), so that

> rm(height, weight)

deletes the variables height and weight.

The entire workspace can be cleared using rm(list=ls()) and also viathe “Remove all objects” or “Clear Workspace” menu entries in the Win-dows and Macintosh GUIs. This does not remove variables whose namebegins with a dot because they are not listed by ls() — you would needls(all=T) for that, but it could be dangerous because such names areused for system purposes.

If you are acquainted with the Unix operating system, for which the S lan-guage, which preceded R, was originally written, then you will know thatthe commands for listing and removing files in Unix are called preciselyls and rm.

It is possible to save the workspace to a file at any time. If you just write

save.image()

then it will be saved to a file called .RData in your working directory.The Windows version also has this on the File menu. When you exit R,you will be asked whether to save the workspace image; if you accept,the same thing will happen. It is also possible to specify an alternativefilename (within quotes). You can also save selected objects with save.The .RData file is loaded by default when R is started in its directory.Other save files can be loaded into your current workspace using load.

2.1.2 Textual output

It is important to note that the workspace consists only of R objects, not ofany of the output that you have generated during a session. If you wantto save your output, use “Save to File” from the File menu in Windows oruse standard cut-and-paste facilities. You can also use ESS (Emacs SpeaksStatistics), which works on all platforms. It is a “mode” for the Emacseditor where you can run your entire session in an Emacs buffer. You canget ESS and installation instructions for it from CRAN (see Appendix A).

An alternative way of diverting output to a file is to use the sink func-tion. This is largely a relic from the days of the 80×25 computer terminal,where cut-and-paste techniques were not available, but it can still be use-

2.1 Session management 33

ful at times. In particular, it can be used in batch processing. The way itworks is as follows:

> sink("myfile")> ls()

No output appears! This is because the output goes into the file myfile inthe current directory. The system will remain in a state where commandsare processed, but the output (apparently) goes into the drain until thenormal state of affairs is reestablished by

> sink()

The current working directory can be obtained by getwd() and changedby setwd(mydir), where mydir is a character string. The initial workingdirectory is system-dependent; for instance, the Windows GUI sets it tothe user’s home directory, and command line versions use the directoryfrom which you start R.

2.1.3 Scripting

Beyond a certain level of complexity, you will not want to work with R ona line-by-line basis. For instance, if you have entered an 8 × 8 matrix overeight lines and realize that you made a mistake, you will find yourselfusing the up-arrow key 64 times to reenter it! In such cases, it is better towork with R scripts, collections of lines of R code stored either in a file orin computer memory somehow.

One option is to use the source function, which is sort of the opposite ofsink. It takes the input (i.e., the commands from a file) and runs them.Notice, though, that the entire file is syntax-checked before anything isexecuted. It is often useful to set echo=T in the call so that commands areprinted along with the output.

Another option is more interactive in nature. You can work with a scripteditor window, which allows you to submit one or more lines of the scriptto a running R, which will then behave as if the same lines had beenentered at the prompt. The Windows and Macintosh versions of R havesimple scripting windows built-in, and a number of text editors also havefeatures for sending commands to R; popular choices on Windows in-clude TINN-R and WinEdt. This is also available as part of ESS (see thepreceding section).

The history of commands entered in a session can be saved and reloadedusing the savehistory and loadhistory commands, which are alsomapped to menu entries in Windows. Saved histories can be useful as a

34 2. The R environment

starting point for writing scripts; notice also that the history() functionwill show the last commands entered at the console (up to a maximum of25 lines by default).

2.1.4 Getting help

R can do a lot more than what a typical beginner can be expected to needor even understand. This book is written so that most of the code you arelikely to need in relation to the statistical procedures is described in thetext, and the compendium in Appendix C is designed to provide a basicoverview. However, it is obviously not possible to cover everything.

R also comes with extensive online help in text form as well as in the formof a series of HTML files that can be read using a Web browser such asNetscape or Internet Explorer. The help pages can be accessed via “help”in the menu bar on Windows and by entering help.start() on anyplatform. You will find that the pages are of a technical nature. Preci-sion and conciseness here take precedence over readability and pedagogy(something one learns to appreciate after exposure to the opposite).

From the command line, you can always enter help(aggregate) to gethelp on the aggregate function or use the prefix form ?aggregate. Ifthe HTML viewer is running, then the help page is shown there. Other-wise it is shown as text either through a pager to the terminal window orin a separate window.

Notice that the HTML version of the help system features a very use-ful “Search Engine and Keywords” and that the apropos functionallows you to get a list of command names that contain a given pat-tern. The function help.search is similar but uses fuzzy matching andsearches deeper into the help pages, so that it will be able to locate,for example, Kendall’s correlation coefficient in cor.test if you usehelp.search("kendal").

Also available with the R distributions is a set of documents in variousformats. Of particular interest is “An Introduction to R”, originally basedon a set of notes for S-PLUS by Bill Venables and David Smith and modi-fied for R by various people. It contains an introduction to the R languageand environment in a rather more language-centric fashion than this book.On the Windows platform, you can choose to install PDF documents aspart of the installation procedure so that — provided the Adobe AcrobatReader program is also installed — it can be accessed via the Help menu.An HTML version (without pictures) can be accessed via the browserinterface on all platforms.

2.1 Session management 35

2.1.5 Packages

An R installation contains one or more libraries of packages. Some of thesepackages are part of the basic installation. Others can be downloaded fromCRAN (see Appendix A), which currently hosts over 1000 packages forvarious purposes. You can even create your own packages.

A library is generally just a folder on your disk. A system library is createdwhen R is installed. In some installations, users may be prohibited frommodifying the system library. It is possible to set up private user libraries;see help(".Library") for details.

A package can contain functions written in the R language, dynamicallyloaded libraries of compiled code (written in C or Fortran mostly), anddata sets. It generally implements functionality that most users will prob-ably not need to have loaded all the time. A package is loaded into R usingthe library command, so to load the survival package you shouldenter

> library(survival)

The loaded packages are not considered part of the user workspace. Ifyou terminate your R session and start a new session with the savedworkspace, then you will have to load the packages again. For the samereason, it is rarely necessary to remove a package that you have loaded,but it can be done if desired with

> detach("package:survival")

(see also Section 2.1.7).

2.1.6 Built-in data

Many packages, both inside and outside the standard R distribution, comewith built-in data sets. Such data sets can be rather large, so it is not agood idea to keep them all in computer memory at all times. A mecha-nism for on-demand loading is required. In many packages, this worksvia a mechanism called lazy loading, which allows the system to “pretend”that the data are in memory, but in fact they are not loaded until they arereferenced for the first time.

With this mechanism, data are “just there”. For example, if you type “thue-sen”, the data frame of that name is displayed. Some packages still requireexplicit calls to the data function. Most often, this loads a data frame withthe name that its argument specifies; data(thuesen) will, for instance,load the thuesen data frame.

36 2. The R environment

What data does is to go through the data directories associated with eachpackage (see Section 2.1.5) and look for files whose basename matches thegiven name. Depending on the file extension, several things can then hap-pen. Files with a .tab extension are read using read.table (Section 2.4),whereas files with a .R extension are executed as source files (and could,in general, do anything!), to give two common examples.

If there is a subdirectory of the current directory called data, then itis searched as well. This can be quite a handy way of organizing yourpersonal projects.

2.1.7 attach and detach

The notation for accessing variables in data frames gets rather heavy ifyou repeatedly have to write longish commands like

plot(thuesen$blood.glucose,thuesen$short.velocity)

Fortunately, you can make R look for objects among the variables in agiven data frame, for example thuesen. You write

> attach(thuesen)

and then thuesen’s data are available without the clumsy $-notation:

> blood.glucose[1] 15.3 10.8 8.1 19.5 7.2 5.3 9.3 11.1 7.5 12.2 6.7 5.2[13] 19.0 15.1 6.7 8.6 4.2 10.3 12.5 16.1 13.3 4.9 8.8 9.5

What happens is that the data frame thuesen is placed in the system’ssearch path. You can view the search path with search:

> search()[1] ".GlobalEnv" "thuesen" "package:ISwR"[4] "package:stats" "package:graphics" "package:grDevices"[7] "package:utils" "package:datasets" "package:methods"[10] "Autoloads" "package:base"

Notice that thuesen is placed as no. 2 in the search path. .GlobalEnvis the workspace and package:base is the system library whereall standard functions are defined. Autoloads is not described here.package:stats and onwards contains the basic statistical routines suchas the Wilcoxon test, and the other packages similarly contain vari-ous functions and data sets. (The package system is modular, and youcan run R with a minimal set of packages for specific uses.) Finally,package:ISwR contains the data sets used for this book.

2.1 Session management 37

There may be several objects of the same name in different parts of thesearch path. In that case, R chooses the first one (that is, it searches first in.GlobalEnv, then in thuesen, and so forth). For this reason, you needto be a little careful with “loose” objects that are defined in the workspaceoutside a data frame since they will be used before any vectors and factorsof the same name in an attached data frame. For the same reason, it is not agood idea to give a data frame the same name as one of the variables insideit. Note also that changing a data frame after attaching it will not affect thevariables available since attach involves a (virtual) copy operation of thedata frame.

It is not possible to attach data frames in front of .GlobalEnv or fol-lowing package:base. However, it is possible to attach more than onedata frame. New data frames are inserted into position 2 by default, andeverything except .GlobalEnv moves one step to the right. It is, how-ever, possible to specify that a data frame should be searched before.GlobalEnv by using constructions of the form

with(thuesen, plot(blood.glucose, short.velocity))

In some contexts, R uses a slightly different method when looking for ob-jects. If looking for a variable of a specific type (usually a function), R willskip those of other types. This is what saves you from the worst conse-quences of accidentally naming a variable (say) c, even though there is asystem function of the same name.

You can remove a data frame from the search path with detach. If noarguments are given, the data frame in position 2 is removed, which isgenerally what is desired. .GlobalEnv and package:base cannot bedetach’ed.

> detach()> search()[1] ".GlobalEnv" "package:ISwR" "package:stats"[4] "package:graphics" "package:grDevices" "package:utils"[7] "package:datasets" "package:methods" "Autoloads"[10] "package:base"

2.1.8 subset, transform, and within

You can attach a data frame to avoid the cumbersome indexing of everyvariable inside of it. However, this is less helpful for selecting subsets ofdata and for creating new data frames with transformed variables. A cou-ple of functions exist to make these operations easier. They are used asfollows:

38 2. The R environment

> thue2 <- subset(thuesen,blood.glucose<7)> thue2

blood.glucose short.velocity6 5.3 1.4911 6.7 1.2512 5.2 1.1915 6.7 1.5217 4.2 1.1222 4.9 1.03> thue3 <- transform(thuesen,log.gluc=log(blood.glucose))> thue3

blood.glucose short.velocity log.gluc1 15.3 1.76 2.7278532 10.8 1.34 2.3795463 8.1 1.27 2.0918644 19.5 1.47 2.9704145 7.2 1.27 1.974081…22 4.9 1.03 1.58923523 8.8 1.12 2.17475224 9.5 1.70 2.251292

Notice that the variables used in the expressions for new variables or forsubsetting are evaluated with variables taken from the data frame.

subset also works on single vectors. This is nearly the same as indexingwith a logical vector (such as short.velocity[blood.glucose<7]),except that observations with missing values in the selection criterion areexcluded.

subset also has a select argument which can be used to extractvariables from the data frame. We shall return to this in Section 10.3.1.

The transform function has a couple of drawbacks, the most serious ofwhich is probably that it does not allow chained calculations where someof the new variables depend on the others. The = signs in the syntax arenot assignments, but indicate names, which are assigned to the computedvectors in the last step.

An alternative to transform is the within function, which can be usedlike this:

> thue4 <- within(thuesen,{+ log.gluc <- log(blood.glucose)+ m <- mean(log.gluc)+ centered.log.gluc <- log.gluc – m+ rm(m)+ })> thue4

blood.glucose short.velocity centered.log.gluc log.gluc1 15.3 1.76 0.481879807 2.7278532 10.8 1.34 0.133573113 2.379546

2.2 The graphics subsystem 39

3 8.1 1.27 -0.154108960 2.0918644 19.5 1.47 0.724441444 2.9704145 7.2 1.27 -0.271891996 1.974081…22 4.9 1.03 -0.656737817 1.58923523 8.8 1.12 -0.071221300 2.17475224 9.5 1.70 0.005318777 2.251292

Notice that the second argument is an arbitrary expression (here a com-pound expression, see p. 45). The function is similar to with, but insteadof just returning the computed value, it collects all new and modifiedvariables into a modified data frame, which is then returned. As shown,variables containing intermediate results can be discarded with rm. (It isparticularly important to do this if the contents are incompatible with thedata frame.)

2.2 The graphics subsystem

In Section 1.1.5, we saw how to generate a simple plot and superimposea curve on it. It is quite common in statistical graphics for you to want tocreate a plot that is slightly different from the default: Sometimes you willwant to add annotation, sometimes you want the axes to be different —labels instead of numbers, irregular placement of tick marks, etc. All thesethings can be obtained in R. The methods for doing them may feel slightlyunusual at first, but offers a very flexible and powerful approach.

In this section, we look deeper into the structure of a typical plot and givesome indication of how you can work with plots to achieve your desiredresults. Beware, though, that this is a large and complex area and it is notwithin the scope of this book to cover it completely. In fact, we completelyignore important newer tools in the grid and lattice packages.

2.2.1 Plot layout

In the graphics model that R uses, there is (for a single plot) a figure regioncontaining a central plotting region surrounded by margins. Coordinatesinside the plotting region are specified in data units (the kind generallyused to label the axes). Coordinates in the margins are specified in linesof text as you move in a direction perpendicular to a side of the plottingregion but in data units as you move along the side. This is useful sinceyou generally want to put text in the margins of a plot.

A standard x–y plot has an x and a y title label generated from the ex-pressions being plotted. You may, however, override these labels and also

40 2. The R environment

add two further titles, a main title above the plot and a subtitle at the verybottom, in the plot call.

> x <- runif(50,0,2)> y <- runif(50,0,2)> plot(x, y, main="Main title", sub="subtitle",+ xlab="x-label", ylab="y-label")

Inside the plotting region, you can place points and lines that are eitherspecified in the plot call or added later with points and lines. Youcan also place a text with

> text(0.6,0.6,"text at (0.6,0.6)")> abline(h=.6,v=.6)

Here, the abline call is just to show how the text is centered on the point(0.6, 0.6). (Normally, abline plots the line y = a + bx when given a and bas arguments, but it can also be used to draw horizontal and vertical linesas shown.)

The margin coordinates are used by the mtext function. They can bedemonstrated as follows:

> for (side in 1:4) mtext(-1:4,side=side,at=.7,line=-1:4)> mtext(paste("side",1:4), side=1:4, line=-1,font=2)

The for loop (see Section 2.3.1) places the numbers −1 to 4 on corre-sponding lines in each of the four margins at an off-center position of 0.7measured in user coordinates. The subsequent call places a label on eachside, giving the side number. The argument font=2 means that a boldfacefont is used. Notice in Figure 2.1 that not all the margins are wide enoughto hold all the numbers and that it is possible to use negative line numbersto place text within the plotting region.

2.2.2 Building a plot from pieces

High-level plots are composed of elements, each of which can also bedrawn separately. The separate drawing commands often allow finer con-trol of the element, so a standard strategy to achieve a given effect is firstto draw the plot without that element and add the element subsequently.As an extreme case, the following command will plot absolutely nothing:

> plot(x, y, type="n", xlab="", ylab="", axes=F)

Here type="n" causes the points not to be drawn. axes=F suppressesthe axes and the box around the plot, and the x and y title labels are set toempty strings.

2.2 The graphics subsystem 41

0.0 0.5 1.0 1.5 2.0

0.0

0.5

1.0

1.5

2.0

Main title

subtitlex−label

y−la

bel

text at (0.6,0.6)

−101234

−10123

−1 0 1

side 1

sid

e 2

side 3

sid

e 4

Figure 2.1. The layout of a standard plot.

However, the fact that nothing is plotted does not mean that nothing hap-pened. The command sets up the plotting region and coordinate systemsjust as if it had actually plotted the data. To add the plot elements, evaluatethe following:

> points(x,y)> axis(1)> axis(2,at=seq(0.2,1.8,0.2))> box()> title(main="Main title", sub="subtitle",+ xlab="x-label", ylab="y-label")

Notice how the second axis call specifies an alternative set of tick marks(and labels). This is a common technique used to create special axes on aplot and might also be used to create nonequidistant axes as well as axeswith nonnumeric labelling.

Plotting with type="n" is sometimes a useful technique because it hasthe side effect of dimensioning the plot area. For instance, to create a plotwith different colours for different groups, you could first plot all datawith type="n", ensuring that the plot region is large enough, and then

42 2. The R environment

add the points for each group using points. (Passing a vector argumentfor col is more expedient in this particular case.)

2.2.3 Using par

The par function allows incredibly fine control over the details of a plot,although it can be quite confusing to the beginner (and even to experi-enced users at times). The best strategy for learning it may well be simplyto try and pick up a few useful tricks at a time and once in a while try tosolve a particular problem by poring over the help page.

Some of the parameters, but not all, can also be set via arguments to plot-ting functions, which also have some arguments that cannot be set by par.When a parameter can be set by both methods, the difference is generallythat if something is set via par, then it stays set subsequently.

The par settings allow you to control line width and type, character sizeand font, colour, style of axis calculation, size of the plot and figure re-gions, clipping, etc. It is possible to divide a figure into several subfiguresby using the mfrow and mfcol parameters.

For instance, the default margin sizes are just over 5, 4, 4, and 2 lines.You might set par(mar=c(4,4,2,2)+0.1) before plotting. This shavesone line off the bottom margin and two lines off the top margin of theplot, which will reduce the amount of unused whitespace when there isno main title or subtitle. If you look carefully, you will in fact notice thatFigure 2.1 has a somewhat smaller plotting region than the other plots inthis book. This is because the other plots have been made with reducedmargins for typesetting reasons.

However, it is quite pointless to describe the graphics parameters com-pletely at this point. Instead, we return to them as they are used for specificplots.

2.2.4 Combining plots

Some special considerations arise when you wish to put several elementstogether in the same plot. Consider overlaying a histogram with a normaldensity (see Sections 4.2 and 4.4.1 for information on histograms and Sec-tion 3.5.1 for density). The following is close, but only nearly good enough(figure not shown).

> x <- rnorm(100)> hist(x,freq=F)> curve(dnorm(x),add=T)

2.2 The graphics subsystem 43

The freq=F argument to hist ensures that the histogram is in terms ofdensities rather than absolute counts. The curve function graphs an ex-pression (in terms of x) and its add=T allows it to overplot an existingplot. So things are generally set up correctly, but sometimes the top of thedensity function gets chopped off. The reason is of course that the heightof the normal density played no role in the setting of the y-axis for the his-togram. It will not help to reverse the order and draw the curve first andadd the histogram because then the highest bars might get clipped.

The solution is first to get hold of the magnitude of the y values for bothplot elements and make the plot big enough to hold both (Figure 2.2):

> h <- hist(x, plot=F)> ylim <- range(0, h$density, dnorm(0))> hist(x, freq=F, ylim=ylim)> curve(dnorm(x), add=T)

Histogram of x

Densi

−2 −1 0 1 2 3

0.0

0.1

0.2

0.3

0.4

Figure 2.2. Histogram with normal density overlaid.

When called with plot=F, hist will not plot anything, but it will re-turn a structure containing the bar heights on the density scale. This andthe fact that the maximum of dnorm(x) is dnorm(0) allows us to cal-culate a range covering both the bars and the normal density. The zero in

44 2. The R environment

the range call ensures that the bottom of the bars will be in range, too.The range of y values is then passed to the hist function via the ylimargument.

2.3 R programming

It is possible to write your own R functions. In fact, this is a major as-pect and attraction of working with the system in the long run. This booklargely avoids the issue in favour of covering a larger set of basic statisticalprocedures that can be executed from the command line. However, to giveyou a feel for what can be done, consider the following function, whichwraps the code from the example of Section 2.2.4 so that you can justsay hist.with.normal(rnorm(200)). It has been slightly extendedso that it now uses the empirical mean and standard deviation of the datainstead of just 0 and 1.

> hist.with.normal <- function(x, xlab=deparse(substitute(x)),…)+ {+ h <- hist(x, plot=F, …)+ s <- sd(x)+ m <- mean(x)+ ylim <- range(0,h$density,dnorm(0,sd=s))+ hist(x, freq=F, ylim=ylim, xlab=xlab, …)+ curve(dnorm(x,m,s), add=T)+ }

Notice the use of a default argument for xlab. If xlab is not specified,then it is obtained from this expression, which evaluates to a characterform of the expression given for x; that is, if you pass rnorm(100) forx, then the x label becomes “rnorm(100)”. Notice also the use of a …argument, which collects any additional arguments and passes them onto hist in the two calls.

You can learn more about programming in R by studying the built-infunctions, starting with simple ones like log10 or weighted.mean.

2.3.1 Flow control

Until now, we have seen components of the R language that cause evalua-tion of single expressions. However, R is a true programming languagethat allows conditional execution and looping constructs as well. Con-sider, for instance, the following code. (The code implements a versionof Newton’s method for calculating the square root of y.)

2.3 R programming 45

> y <- 12345> x <- y/2> while (abs(x*x-y) > 1e-10) x <- (x + y/x)/2> x[1] 111.1081> x^2[1] 12345

Notice the while(condition) expression construction, which saysthat the expression should be evaluated as long as the condition is TRUE.The test occurs at the top of the loop, so the expression might never beevaluated.

A variation of the same algorithm with the test at the bottom of the loopcan be written with a repeat construction:

> x <- y/2> repeat{+ x <- (x + y/x)/2+ if (abs(x*x-y) < 1e-10) break+ }> x[1] 111.1081

This also illustrates three other flow control structures: (a) a compound ex-pression, several expressions held together between curly braces; (b) an ifconstruction for conditional execution; and (c) a break expression, whichcauses the enclosing loop to exit.

Incidentally, the loop could allow for y being a vector simply by changingthe termination condition to

if (all(abs(x*x – y) < 1e-10)) break

This would iterate excessively for some elements, but the vectorizedarithmetic would likely more than make up for that.

However, the most frequently used looping construct is for, which loopsover a fixed set of values as in the following example, which plots a set ofpower curves on the unit interval.

> x <- seq(0, 1,.05)> plot(x, x, ylab="y", type="l")> for ( j in 2:8 ) lines(x, x^j)

Notice the loop variable j, which in turn takes the values of the givensequence when used in the lines call.

46 2. The R environment

2.3.2 Classes and generic functions

Object-oriented programming is about creating coherent systems of dataand methods that work upon them. One purpose is to simplify programsby accommodating the fact that you will have conceptually similar meth-ods for different types of data, even though the implementations will haveto be different. A prototype example is the print method: It makes senseto print many kinds of data objects, but the print layout will depend onwhat the data object is. You will generally have a class of data objects anda print method for that class. There are several object-oriented languagesimplementing these ideas in different ways.

Most of the basic parts of R use the same object system as S version 3. Analternative object system similar to that of S version 4 has been developedin recent years. The new system has several advantages over the old one,but we shall restrict attention to the latter. The S3 object system is a sim-ple system in which an object has a class attribute, which is simply acharacter vector. One example of this is that all the return values of theclassical tests such as t.test have class "htest", indicating that theyare the result of a hypothesis test. When these objects are printed, it isdone by print.htest, which creates the nice layout (see Chapter 5 forexamples). However, from a programmatic viewpoint, these objects arejust lists, and you can, for instance, extract the p-value by writing

> t.test(bmi, mu=22.5)$p.value[1] 0.7442183

The function print is a generic function, one that acts differently depend-ing on its argument. These generally look like this:

> printfunction (x, …)UseMethod("print")<environment: namespace:base>

What UseMethod("print") means is that R should pass control to afunction named according to the object class (print.htest for objects ofclass "htest", etc.) or, if this is not found, to print.default. To see allthe methods available for print, type methods(print) (there are 138of them in R 2.6.2, so the output is not shown here).

2.4 Data entry

Data sets do not have to be very large before it becomes impractical to typethem in with c(…). Most of the examples in this book use data sets in-

2.4 Data entry 47

cluded in the ISwR package, made available to you by library(ISwR).However, as soon as you wish to apply the methods to your own data,you will have to deal with data file formats and the specification thereof.

In this section we discuss how to read data files and how to use the dataeditor module in R. The text has some bias toward Windows systems,mainly because of some special issues that need to be mentioned for thatplatform.

2.4.1 Reading from a text file

The most convenient way of reading data into R is via the function calledread.table. It requires that data be in “ASCII format”; that is, a “flatfile” as created with Windows’ NotePad or any plain-text editor. The resultof read.table is a data frame, and it expects to find data in a corre-sponding layout where each line in the file contains all data from onesubject (or rat or . . . ) in a specific order, separated by blanks or, option-ally, some other separator. The first line of the file can contain a headergiving the names of the variables, a practice that is highly recommended.

Table 11.6 in Altman (1991) contains an example on ventricular circum-ferential shortening velocity versus fasting blood glucose by Thuesen etal. We used those data to illustrate subsetting and use them again in thechapter on correlation and regression. They are among the built-in datasets in the ISwR package and available as the data frame thuesen, butthe point here is to show how to read them from a plain-text file.

Assume that the data are contained in the file thuesen.txt, which looksas follows:

blood.glucose short.velocity15.3 1.7610.8 1.348.1 1.2719.5 1.477.2 1.275.3 1.499.3 1.3111.1 1.097.5 1.1812.2 1.226.7 1.255.2 1.1919.0 1.9515.1 1.286.7 1.528.6 NA4.2 1.12

48 2. The R environment

10.3 1.3712.5 1.1916.1 1.0513.3 1.324.9 1.038.8 1.129.5 1.70

To enter the data into the file, you could start up Windows’ NotePador any other plain-text editor, such as those discussed in Section 2.1.3.Unix/Linux users should just use a standard editor, such as emacs orvi. If you must, you can even use a word processing program with a littlecare.

You should simply type in the data as shown. Notice that the columnsare separated by an arbitrary number of blanks and that NA represents amissing value.

At the end, you should save the data to a text file. Notice that word pro-cessors require special actions in order to save as text. Their normal saveformat is difficult to read from other programs.

Assuming further that the file is in the ISwR folder on the N: drive, thedata can be read using

> thuesen2 <- read.table("N:/ISwR/thuesen.txt",header=T)

Notice header=T specifying that the first line is a header containingthe names of variables contained in the file. Also note that you use for-ward slashes (/), not backslashes (), in the filename, even on a Windowssystem.

The reason for avoiding backslashes in Windows filenames is that thesymbol is used as an escape character (see Section 1.2.4) and thereforeneeds to be doubled. You could have used N:\ISwR\thuesen.txt.

The result is a data frame, which is assigned to the variable thuesen2and looks as follows:

> thuesen2blood.glucose short.velocity

1 15.3 1.762 10.8 1.343 8.1 1.274 19.5 1.475 7.2 1.276 5.3 1.497 9.3 1.318 11.1 1.099 7.5 1.1810 12.2 1.22

2.4 Data entry 49

11 6.7 1.2512 5.2 1.1913 19.0 1.9514 15.1 1.2815 6.7 1.5216 8.6 NA17 4.2 1.1218 10.3 1.3719 12.5 1.1920 16.1 1.0521 13.3 1.3222 4.9 1.0323 8.8 1.1224 9.5 1.70

To read in factor variables (see Section 1.2.8), the easiest way may be toencode them using a textual representation. The read.table functionautodetects whether a vector is text or numeric and converts it to a factorin the former case (but makes no attempt to recognize numerically codedfactors). For instance, the secretin built-in data set is read from a filethat begins like this:

gluc person time repl time20plus time.comb1 92 A pre a pre pre2 93 A pre b pre pre3 84 A 20 a 20+ 204 88 A 20 b 20+ 205 88 A 30 a 20+ 30+6 90 A 30 b 20+ 30+7 86 A 60 a 20+ 30+8 89 A 60 b 20+ 30+9 87 A 90 a 20+ 30+10 90 A 90 b 20+ 30+11 85 B pre a pre pre12 85 B pre b pre pre13 74 B 20 a 20+ 20….

This file can be read directly by read.table with no arguments otherthan the filename. It will recognize the case where the first line is one itemshorter than the rest and will interpret that layout to imply that the firstline contains a header and the first value on all subsequent lines is a rowlabel — that is, exactly the layout generated when printing a data frame.

Reading factors like this may be convenient, but there is a drawback: Thelevel order is alphabetic, so for instance

> levels(secretin$time)[1] "20" "30" "60" "90" "pre"

50 2. The R environment

If this is not what you want, then you may have to manipulate the factorlevels; see Section 10.1.2.

A technical note: The files referenced above are contained in the ISwRpackage in the subdirectory (folder) rawdata. Exactly where the file islocated on your system will depend on where the ISwR package wasinstalled. You can find this out as follows:

> system.file("rawdata", "thuesen.txt", package="ISwR")[1] "/home/pd/Rlibrary/ISwR/rawdata/thuesen.txt"

2.4.2 Further details on read.table

The read.table function is a very flexible tool that is controlled by manyoptions. We shall not attempt a full description here but just give someindication of what it can do.

File format details

We have already seen the use of header=T. A couple of other optionscontrol the detailed format of the input file:

Field separator. This can be specified using sep. Notice that when this isused, as opposed to the default use of whitespace, there must be ex-actly one separator between data fields. Two consecutive separatorswill imply that there is a missing value in between. Conversely, itis necessary to use specific codes to represent missing values in thedefault format and also to use some form of quoting for strings thatcontain embedded spaces.

NA strings. You can specify which strings represent missing values viana.strings. There can be several different strings, although notdifferent strings for different columns. For print files from the SASprogram, you would use na.strings=".".

Quotes and comments. By default, R-style quotes can be used to delimitcharacter strings, and parts of files following the comment character# are ignored. These features can be modified or removed via thequote and comment.char arguments.

Unequal field count. It is normally considered an error if not all lines con-tain the same number of values (the first line can be one item short,as described above for the secretin data). The fill and flusharguments can be used in case lines vary in length.

2.4 Data entry 51

Delimited file types

Applications such as spreadsheets and databases produce text files in for-mats that require multiple options to be adjusted. For such purposes, thereexist “precooked” variants of read.table. Two of these are intendedto handle CSV files and are called read.csv and read.csv2. The for-mer assumes that fields are separated by a comma, and the latter assumesthat they are separated by semicolons but use a comma as the decimalpoint (this format is often generated in European locales). Both formatshave header=T as the default. Further variants are read.delim andread.delim2 for reading delimited files (by default, Tab-delimited files).

Conversion of input

It can be desirable to override the default conversion mechanisms inread.table. By default, nonnumeric input is converted to factors, butit does not always make sense. For instance, names and addresses typi-cally should not be converted. This can be modified either for all columnsusing stringsAsFactors or on a per-item basis using as.is.

Automatic conversion is often convenient, but it is inefficient in termsof computer time and storage; in order to read a numeric column,read.table first reads it as character data, checks whether all elementscan be converted to numeric, and only then performs the conversion. ThecolClasses argument allows you to bypass the mechanism by explic-itly specifying which columns are of which class (the standard classes"character", "numeric", etc., get special treatment). You can also skipunwanted columns by specifying "NULL" as the class.

2.4.3 The data editor

R lets you edit data frames using a spreadsheet-like interface. Theinterface is a bit rough but quite useful for small data sets.

To edit a data frame, you can use the edit function:

> aq <- edit(airquality)

This brings up a spreadsheet-like editor with a column for each vari-able in the data frame. The airquality data set is built into R; seehelp(airquality) for its contents. Inside the editor, you can movearound with the mouse or the cursor keys and edit the current cell by typ-ing in data. The type of variable can be switched between real (numeric)and character (factor) by clicking on the column header, and the name of

52 2. The R environment

the variable can be changed similarly. Note that there is (as of R 2.6.2) noway to delete rows and columns and that new data can be entered only atthe end.

When you close the data editor, the edited data frame is assigned to aq.The original airquality is left intact. Alternatively, if you do not mindoverwriting the original data frame, you can use

> fix(aq)

This is equivalent to aq <- edit(aq).

To enter data into a blank data frame, use

> dd <- data.frame()> fix(dd)

An alternative would be dd <- edit(data.frame()), which worksfine except that beginners tend to reexecute the command when they needto edit dd, which of course destroys all data. It is necessary in either caseto start with an empty data frame since by default edit expects you towant to edit a user-defined function and would bring up a text editor ifyou started it as edit().

2.4.4 Interfacing to other programs

Sometimes you will want to move data between R and other statisticalpackages or spreadsheets. A simple fallback approach is to request thatthe package in question export data as a text file of some sort and useread.table, read.csv, read.csv2, read.delim, or read.delim2,as previously described.

The foreign package is one of the packages labelled “recommended”and should therefore be available with binary distributions of R. Itcontains routines to read files in several formats, including those fromSPSS (.sav format), SAS (export libraries), Epi-Info (.rec), Stata, Systat,Minitab, and some S-PLUS version 3 dump files.

Unix/Linux users sometimes find themselves with data sets written onWindows machines. The foreign package will work there as well forthose formats that it supports. Notice that ordinary SAS data sets arenot among the supported formats. These have to be converted to ex-port libraries on the originating system. Data that have been enteredinto Microsoft Excel spreadsheets are most conveniently extracted using acompatible application such as OOo (OpenOffice.org).

2.5 Exercises 53

An expedient technique is to read from the system clipboard. Say, high-light a rectangular region in a spreadsheet, press Ctrl-C (if on Windows),and inside R use

read.table("clipboard", header=T)

This does require a little caution, though. It may result in loss of accu-racy since you only transfer the data as they appear on the screen. This ismostly a concern if you have data to many significant digits.

For data stored in databases, there exist a number of interface packages onCRAN. Of particular interest on Windows and with some Unix databasesis the RODBC package because you can set up ODBC (“Open DatabaseConnectivity”) connections to data stored by common applications, in-cluding Excel and Access. Some Unix databases (e.g., PostgreSQL) alsoallow ODBC connections.

For up-to-date information on these matters, consult the “R Data Im-port/Export” manual that comes with the system.

2.5 Exercises

2.1 Describe how to insert a value between two elements of a vector at agiven position by using the append function (use the help system to findout). Without append, how would you do it?

2.2 Write the built-in data set thuesen to a Tab-separated text file withwrite.table. View it with a text editor (depending on your system).Change the NA value to . (period), and read the changed file back into Rwith a suitable command. Also try importing the data into other applica-tions of your choice and exporting them to a new file after editing. Youmay have to remove row names to make this work.

3Probability and distributions

The concepts of randomness and probability are central to statistics. Itis an empirical fact that most experiments and investigations are notperfectly reproducible. The degree of irreproducibility may vary: Someexperiments in physics may yield data that are accurate to many decimalplaces, whereas data on biological systems are typically much less reli-able. However, the view of data as something coming from a statisticaldistribution is vital to understanding statistical methods. In this section,we outline the basic ideas of probability and the functions that R has forrandom sampling and handling of theoretical distributions.

3.1 Random sampling

Much of the earliest work in probability theory was about games and gam-bling issues, based on symmetry considerations. The basic notion then isthat of a random sample: dealing from a well-shuffled pack of cards orpicking numbered balls from a well-stirred urn.

In R, you can simulate these situations with the sample function. If youwant to pick five numbers at random from the set 1:40, then you canwrite

> sample(1:40,5)[1] 4 30 28 40 13

P. Dalgaard, Introductory Statistics with R,DOI: 10.1007/978-0-387-79054-1_3, © Springer Science+Business Media, LLC 2008

56 3. Probability and distributions

The first argument (x) is a vector of values to be sampled and the second(size) is the sample size. Actually, sample(40,5) would suffice sincea single number is interpreted to represent the length of a sequence ofintegers.

Notice that the default behaviour of sample is sampling without replace-ment. That is, the samples will not contain the same number twice, andsize obviously cannot be bigger than the length of the vector to be sam-pled. If you want sampling with replacement, then you need to add theargument replace=TRUE.

Sampling with replacement is suitable for modelling coin tosses or throwsof a die. So, for instance, to simulate 10 coin tosses we could write

> sample(c("H","T"), 10, replace=T)[1] "T" "T" "T" "T" "T" "H" "H" "T" "H" "T"

In fair coin-tossing, the probability of heads should equal the probabilityof tails, but the idea of a random event is not restricted to symmetric cases.It could be equally well applied to other cases, such as the successful out-come of a surgical procedure. Hopefully, there would be a better than 50%chance of this. You can simulate data with nonequal probabilities for theoutcomes (say, a 90% chance of success) by using the prob argument tosample, as in

> sample(c("succ", "fail"), 10, replace=T, prob=c(0.9, 0.1))[1] "succ" "succ" "succ" "succ" "succ" "succ" "succ" "succ"[9] "succ" "succ"

This may not be the best way to generate such a sample, though. See thelater discussion of the binomial distribution.

3.2 Probability calculations and combinatorics

Let us return to the case of sampling without replacement, specificallysample(1:40, 5). The probability of obtaining a given number as thefirst one of the sample should be 1/40, the next one 1/39, and so forth. Theprobability of a given sample should then be 1/(40 × 39 × 38 × 37 × 36).In R, use the prod function, which calculates the product of a vector ofnumbers

> 1/prod(40:36)[1] 1.266449e-08

3.3 Discrete distributions 57

However, notice that this is the probability of getting given numbers in agiven order. If this were a Lotto-like game, then you would rather be inter-ested in the probability of guessing a given set of five numbers correctly.Thus you need also to include the cases that give the same numbers ina different order. Since obviously the probability of each such case is go-ing to be the same, all we need to do is to figure out how many such casesthere are and multiply by that. There are five possibilities for the first num-ber, and for each of these there are four possibilities for the second, and soforth; that is, the number is 5 × 4 × 3 × 2 × 1. This number is also writtenas 5! (5 factorial). So the probability of a “winning Lotto coupon” would be

> prod(5:1)/prod(40:36)[1] 1.519738e-06

There is another way of arriving at the same result. Notice that since theactual set of numbers is immaterial, all sets of five numbers must have thesame probability. So all we need to do is to calculate the number of waysto choose 5 numbers out of 40. This is denoted(

405

40!5!35!

= 658008

In R, the choose function can be used to calculate this number, and theprobability is thus

> 1/choose(40,5)[1] 1.519738e-06

3.3 Discrete distributions

When looking at independent replications of a binary experiment, youwould not usually be interested in whether each case is a success or a fail-ure but rather in the total number of successes (or failures). Obviously,this number is random since it depends on the individual random out-comes, and it is consequently called a random variable. In this case it is adiscrete-valued random variable that can take values 0, 1, . . . , n, where n isthe number of replications. Continuous random variables are encounteredlater.

A random variable X has a probability distribution that can be describedusing point probabilities f (x) = P(X = x) or the cumulative distributionfunction F(x) = P(X ≤ x). In the case at hand, the distribution can beworked out as having the point probabilities

f (x) =(

)px(1 − p)n−x

58 3. Probability and distributions

This is known as the binomial distribution, and the (nx) are known as bino-mial coefficients. The parameter p is the probability of a successful outcomein an individual trial. A graph of the point probabilities of the binomialdistribution appears in Figure 3.2 ahead.

We delay describing the R functions related to the binomial distributionuntil we have discussed continuous distributions so that we can presentthe conventions in a unified manner.

Many other distributions can be derived from simple probability models.For instance, the geometric distribution is similar to the binomial distri-bution but records the number of failures that occur before the firstsuccess.

3.4 Continuous distributions

Some data arise from measurements on an essentially continuous scale,for instance temperature, concentrations, etc. In practice, they will berecorded to a finite precision, but it is useful to disregard this in themodelling. Such measurements will usually have a component of randomvariation, which makes them less than perfectly reproducible. However,these random fluctuations will tend to follow patterns; typically they willcluster around a central value, with large deviations being more rare thansmaller ones.

In order to model continuous data, we need to define random variablesthat can obtain the value of any real number. Because there are infinitelymany numbers infinitely close, the probability of any particular value willbe zero, so there is no such thing as a point probability as for discrete-valued random variables. Instead we have the concept of a density. This isthe infinitesimal probability of hitting a small region around x divided bythe size of the region. The cumulative distribution function can be definedas before, and we have the relation

F(x) =∫ x−∞

f (x) dx

There are a number of standard distributions that come up in statisticaltheory and are available in R. It makes little sense to describe them indetail here except for a couple of examples.

The uniform distribution has a constant density over a specified interval (bydefault [0, 1]).

3.5 The built-in distributions in R 59

The normal distribution (also known as the Gaussian distribution) hasdensity

f (x) =1

√2πσ

exp(−

(x − µ)2

2σ2

)depending on its mean µ and standard deviation σ. The normal distribu-tion has a characteristic bell shape (Figure 3.1), and modifying µ and σsimply translates and widens the distribution. It is a standard buildingblock in statistical models, where it is commonly used to describe errorvariation. It also comes up as an approximating distribution in severalcontexts; for instance, the binomial distribution for large sample sizes canbe well approximated by a suitably scaled normal distribution.

3.5 The built-in distributions in R

The standard distributions that turn up in connection with model buildingand statistical tests have been built into R, and it can therefore completelyreplace traditional statistical tables. Here we look only at the normal dis-tribution and the binomial distribution, but other distributions followexactly the same pattern.

Four fundamental items can be calculated for a statistical distribution:

• Density or point probability

• Cumulated probability, distribution function

• Quantiles

• Pseudo-random numbers

For all distributions implemented in R, there is a function for each of thefour items listed above. For example, for the normal distribution, these arenamed dnorm, pnorm, qnorm, and rnorm (density, probability, quantile,and random, respectively).

3.5.1 Densities

The density for a continuous distribution is a measure of the relative prob-ability of “getting a value close to x”. The probability of getting a value ina particular interval is the area under the corresponding part of the curve.

60 3. Probability and distributions

For discrete distributions, the term “density” is used for the point proba-bility — the probability of getting exactly the value x. Technically, this iscorrect: It is a density with respect to counting measure.

−4 −2 0 2 4

0.0

0.1

0.2

0.3

0.4

dnorm

(x)

Figure 3.1. Density of normal distribution.

The density function is likely the one of the four function types that is leastused in practice, but if for instance it is desired to draw the well-knownbell curve of the normal distribution, then it can be done like this:

> x <- seq(-4,4,0.1)> plot(x,dnorm(x),type="l")

(Notice that this is the letter ‘l’, not the digit ‘1’).

The function seq (see p. 15) is used to generate equidistant values, herefrom −4 to 4 in steps of 0.1; that is, (−4.0,−3.9,−3.8, . . . , 3.9, 4.0). The useof type="l" as an argument to plot causes the function to draw linesbetween the points rather than plotting the points themselves.

An alternative way of creating the plot is to use curve as follows:

> curve(dnorm(x), from=-4, to=4)

3.5 The built-in distributions in R 61

This is often a more convenient way of making graphs, but it does requirethat the y-values can be expressed as a simple functional expression in x.

For discrete distributions, where variables can take on only distinct values,it is preferable to draw a pin diagram, here for the binomial distributionwith n = 50 and p = 0.33 (Figure 3.2):

> x <- 0:50> plot(x,dbinom(x,size=50,prob=.33),type="h")

0 10 20 30 40 50

0.0

00.0

20.0

40.0

60.0

80.1

00.1

dbin

(x, si

ze =

50, pro

b =

0.3

Figure 3.2. Point probabilities in binom(50, 0.33).

Notice that there are three arguments to the “d-function” this time. Inaddition to x, you have to specify the number of trials n and the proba-bility parameter p. The distribution drawn corresponds to, for example,the number of 5s or 6s in 50 throws of a symmetrical die. Actually, dnormalso takes more than one argument, namely the mean and standard devia-tion, but they have default values of 0 and 1, respectively, since most oftenit is the standard normal distribution that is requested.

The form 0:50 is a short version of seq(0,50,1): the whole numbersfrom 0 to 50 (see p. 15). It is type="h" (as in histogram-like) that causesthe pins to be drawn.

62 3. Probability and distributions

3.5.2 Cumulative distribution functions

The cumulative distribution function describes the probability of “hitting”x or less in a given distribution. The corresponding R functions begin witha ‘p’ (for probability) by convention.

Just as you can plot densities, you can of course also plot cumulative dis-tribution functions, but that is usually not very informative. More often,actual numbers are desired. Say that it is known that some biochemicalmeasure in healthy individuals is well described by a normal distributionwith a mean of 132 and a standard deviation of 13. Then, if a patient has avalue of 160, there is

> 1-pnorm(160,mean=132,sd=13)[1] 0.01562612

or only about 1.5% of the general population, that has that value or higher.The function pnorm returns the probability of getting a value smallerthan its first argument in a normal distribution with the given mean andstandard deviation.

Another typical application occurs in connection with statistical tests.Consider a simple sign test: Twenty patients are given two treatments each(blindly and in randomized order) and then asked whether treatment A orB worked better. It turned out that 16 patients liked A better. The questionis then whether this can be taken as sufficient evidence that A actually isthe better treatment or whether the outcome might as well have happenedby chance even if the treatments were equally good. If there was no dif-ference between the two treatments, then we would expect the number ofpeople favouring treatment A to be binomially distributed with p = 0.5and n = 20. How (im)probable would it then be to obtain what we haveobserved? As in the normal distribution, we need a tail probability, andthe immediate guess might be to look at

> pbinom(16,size=20,prob=.5)[1] 0.9987116

and subtract it from 1 to get the upper tail — but this would be an error!What we need is the probability of the observed or more extreme, and pbinomis giving the probability of 16 or less. We need to use “15 or less” instead.

> 1-pbinom(15,size=20,prob=.5)[1] 0.005908966

If you want a two-tailed test because you have no prior idea about whichtreatment is better, then you will have to add the probability of obtainingequally extreme results in the opposite direction. In the present case, that

3.5 The built-in distributions in R 63

means the probability that four or fewer people prefer A, giving a totalprobability of

> 1-pbinom(15,20,.5)+pbinom(4,20,.5)[1] 0.01181793

(which is obviously exactly twice the one-tailed probability).

As can be seen from the last command, it is not strictly necessary to usethe size and prob keywords as long as the arguments are given in theright order (positional matching; see Section 1.2.2).

It is quite confusing to keep track of whether or not the observation itselfneeds to be counted. Fortunately, the function binom.test keeps trackof such formalities and performs the correct binomial test. This is furtherdiscussed in Chapter 8.

3.5.3 Quantiles

The quantile function is the inverse of the cumulative distribution func-tion. The p-quantile is the value with the property that there is probabilityp of getting a value less than or equal to it. The median is by definition the50% quantile.

Some details concerning the definition in the case of discontinuous distri-butions are glossed over here. You can fairly easily deduce the behaviourby experimenting with the R functions.

Tables of statistical distributions are almost always given in terms of quan-tiles. For a fixed set of probabilities, the table shows the boundary that atest statistic must cross in order to be considered significant at that level.This is purely for operational reasons; it is almost superfluous when youhave the option of computing p exactly.

Theoretical quantiles are commonly used for the calculation of confi-dence intervals and for power calculations in connection with designingand dimensioning experiments (see Chapter 9). A simple example of aconfidence interval can be given here (see also Chapter 5).

If we have n normally distributed observations with the same mean µand standard deviation σ, then it is known that the average x̄ is normallydistributed around µ with standard deviation σ/

√n. A 95% confidence

interval for µ can be obtained as

x̄ + σ/√

n × N0.025 ≤ µ ≤ x̄ + σ/√

n × N0.975where N0.025 is the 2.5% quantile in the normal distribution. If σ = 12 andwe have measured n = 5 persons and found an average of x̄ = 83, then

64 3. Probability and distributions

we can compute the relevant quantities as (“sem” means standard error ofthe mean)

> xbar <- 83> sigma <- 12> n <- 5> sem <- sigma/sqrt(n)> sem[1] 5.366563> xbar + sem * qnorm(0.025)[1] 72.48173> xbar + sem * qnorm(0.975)[1] 93.51827

and thus find a 95% confidence interval for µ going from 72.48 to 93.52.(Notice that this is based on the assumption that σ is known. This is some-times reasonable in process control applications. The more common caseof estimating σ from the data leads to confidence intervals based on the tdistribution and is discussed in Chapter 5.)

Since it is known that the normal distribution is symmetric, so thatN0.025 = −N0.975, it is common to write the formula for the confidence in-terval as x̄±σ/

√n× N0.975. The quantile itself is often written Φ−1(0.975),

where Φ is standard notation for the cumulative distribution function ofthe normal distribution (pnorm).

Another application of quantiles is in connection with Q–Q plots (seeSection 4.2.3), which can be used to assess whether a set of data canreasonably be assumed to come from a given distribution.

3.5.4 Random numbers

To many people, it sounds like a contradiction in terms to generaterandom numbers on a computer since its results are supposed to be pre-dictable and reproducible. What is in fact possible is to generate sequencesof “pseudo-random” numbers, which for practical purposes behave as ifthey were drawn randomly.

Here random numbers are used to give the reader a feeling for the way inwhich randomness affects the quantities that can be calculated from a setof data. In professional statistics, they are used to create simulated datasets in order to study the accuracy of mathematical approximations andthe effect of assumptions being violated.

The use of the functions that generate random numbers is straightfor-ward. The first argument specifies the number of random numbers tocompute, and the subsequent arguments are similar to those for otherfunctions related to the same distributions. For instance,

3.6 Exercises 65

> rnorm(10)[1] -0.2996466 -0.1718510 -0.1955634 1.2280843 -2.6074190[6] -0.2999453 -0.4655102 -1.5680666 1.2545876 -1.8028839> rnorm(10)[1] 1.7082495 0.1432875 -1.0271750 -0.9246647 0.6402383[6] 0.7201677 -0.3071239 1.2090712 0.8699669 0.5882753> rnorm(10,mean=7,sd=5)[1] 8.934983 8.611855 4.675578 3.670129 4.223117 5.484290[7] 12.141946 8.057541 -2.893164 13.590586> rbinom(10,size=20,prob=.5)[1] 12 11 10 8 11 8 11 8 8 13

3.6 Exercises

3.1 Calculate the probability for each of the following events: (a) Astandard normally distributed variable is larger than 3. (b) A normallydistributed variable with mean 35 and standard deviation 6 is larger than42. (c) Getting 10 out of 10 successes in a binomial distribution with prob-ability 0.8. (d) X < 0.9 when X has the standard uniform distribution. (e)X > 6.5 in a χ2 distribution with 2 degrees of freedom.

3.2 A rule of thumb is that 5% of the normal distribution lies outside aninterval approximately ±2s about the mean. To what extent is this true?Where are the limits corresponding to 1%, 0.5%, and 0.1%? What is theposition of the quartiles measured in standard deviation units?

3.3 For a disease known to have a postoperative complication frequencyof 20%, a surgeon suggests a new procedure. He tests it on 10 patientsand there are no complications. What is the probability of operating on 10patients successfully with the traditional method?

3.4 Simulated coin-tossing can be done using rbinom instead of sample.How exactly would you do that?

4Descriptive statistics and graphics

Before going into the actual statistical modelling and analysis of a dataset, it is often useful to make some simple characterizations of the data interms of summary statistics and graphics.

4.1 Summary statistics for a single group

It is easy to calculate simple summary statistics with R. Here is how tocalculate the mean, standard deviation, variance, and median.

> x <- rnorm(50)> mean(x)[1] 0.03301363> sd(x)[1] 1.069454> var(x)[1] 1.143731> median(x)[1] -0.08682795

Notice that the example starts with the generation of an artificial datavector x of 50 normally distributed observations. It is used in examplesthroughout this section. When reproducing the examples, you will not getexactly the same results since your random numbers will differ.

P. Dalgaard, Introductory Statistics with R,DOI: 10.1007/978-0-387-79054-1_4, © Springer Science+Business Media, LLC 2008

68 4. Descriptive statistics and graphics

Empirical quantiles may be obtained with the function quantile likethis:

> quantile(x)0% 25% 50% 75% 100%

-2.60741896 -0.54495849 -0.08682795 0.70018536 2.98872414

As you see, by default you get the minimum, the maximum, and thethree quartiles — the 0.25, 0.50, and 0.75 quantiles — so named becausethey correspond to a division into four parts. Similarly, we have deciles for0.1, 0.2, . . . , 0.9, and centiles or percentiles. The difference between the firstand third quartiles is called the interquartile range (IQR) and is sometimesused as a robust alternative to the standard deviation.

It is also possible to obtain other quantiles; this is done by adding an argu-ment containing the desired percentage points. This, for example, is howto get the deciles:

> pvec <- seq(0,1,0.1)> pvec[1] 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0> quantile(x,pvec)

0% 10% 20% 30% 40%-2.60741896 -1.07746896 -0.70409272 -0.46507213 -0.29976610

50% 60% 70% 80% 90%-0.08682795 0.19436950 0.49060129 0.90165137 1.31873981

100%2.98872414

Be aware that there are several possible definitions of empirical quantiles.The one R uses by default is based on a sum polygon where the ith rankingobservation is the (i − 1)/(n − 1) quantile and intermediate quantiles areobtained by linear interpolation. It sometimes confuses students that ina sample of 10 there will be 3 observations below the first quartile withthis definition. Other definitions are available via the type argument toquantile.

If there are missing values in data, things become a bit more complicated.For illustration, we use the following example.

The data set juul contains variables from an investigation performed byAnders Juul (Rigshospitalet, Department for Growth and Reproduction)concerning serum IGF-I (insulin-like growth factor) in a group of healthyhumans, primarily schoolchildren. The data set is contained in the ISwRpackage and contains a number of variables, of which we only use igf1(serum IGF-I) for now, but later in the chapter we also use tanner (Tan-ner stage of puberty, a classification into five groups based on appearance

4.1 Summary statistics for a single group 69

of primary and secondary sexual characteristics), sex, and menarche(indicating whether or not a girl has had her first period).

Attempting to calculate the mean of igf1 reveals a problem.

> attach(juul)> mean(igf1)[1] NA

R will not skip missing values unless explicitly requested to do so. Themean of a vector with an unknown value is unknown. However, you cangive the na.rm argument (not available, remove) to request that missingvalues be removed:

> mean(igf1,na.rm=T)[1] 340.168

There is one slightly annoying exception: The length function will notunderstand na.rm, so we cannot use it to count the number of nonmissingmeasurements of igf1. However, you can use

> sum(!is.na(igf1))[1] 1018

The construction above uses the fact that if logical values are used inarithmetic, then TRUE is converted to 1 and FALSE to 0.

A nice summary display of a numeric variable is obtained from thesummary function:

> summary(igf1)Min. 1st Qu. Median Mean 3rd Qu. Max. NA’s25.0 202.2 313.5 340.2 462.8 915.0 321.0

The 1st Qu. and 3rd Qu. refer to the empirical quartiles (0.25 and 0.75quantiles).

In fact, it is possible to summarize an entire data frame with

> summary(juul)age menarche sex

Min. : 0.170 Min. : 1.000 Min. :1.0001st Qu.: 9.053 1st Qu.: 1.000 1st Qu.:1.000Median :12.560 Median : 1.000 Median :2.000Mean :15.095 Mean : 1.476 Mean :1.5343rd Qu.:16.855 3rd Qu.: 2.000 3rd Qu.:2.000Max. :83.000 Max. : 2.000 Max. :2.000NA’s : 5.000 NA’s :635.000 NA’s :5.000

igf1 tanner testvolMin. : 25.0 Min. : 1.000 Min. : 1.0001st Qu.:202.2 1st Qu.: 1.000 1st Qu.: 1.000

70 4. Descriptive statistics and graphics

Median :313.5 Median : 2.000 Median : 3.000Mean :340.2 Mean : 2.640 Mean : 7.8963rd Qu.:462.8 3rd Qu.: 5.000 3rd Qu.: 15.000Max. :915.0 Max. : 5.000 Max. : 30.000NA’s :321.0 NA’s :240.000 NA’s :859.000

The data set has menarche, sex, and tanner coded as numeric variableseven though they are clearly categorical. This can be mended as follows:

> detach(juul)> juul$sex <- factor(juul$sex,labels=c("M","F"))> juul$menarche <- factor(juul$menarche,labels=c("No","Yes"))> juul$tanner <- factor(juul$tanner,+ labels=c("I","II","III","IV","V"))> attach(juul)> summary(juul)

age menarche sex igf1Min. : 0.170 No :369 M :621 Min. : 25.01st Qu.: 9.053 Yes :335 F :713 1st Qu.:202.2Median :12.560 NA’s:635 NA’s: 5 Median :313.5Mean :15.095 Mean :340.23rd Qu.:16.855 3rd Qu.:462.8Max. :83.000 Max. :915.0NA’s : 5.000 NA’s :321.0tanner testvolI :515 Min. : 1.000II :103 1st Qu.: 1.000III : 72 Median : 3.000IV : 81 Mean : 7.896V :328 3rd Qu.: 15.000NA’s:240 Max. : 30.000

NA’s :859.000

Notice how the display changes for the factor variables. Note also thatjuul was detached and reattached after the modification. This is becausemodifying a data frame does not affect any attached version. It was notstrictly necessary to do it here because summary works directly on thedata frame whether attached or not.

In the above, the variables sex, menarche, and tanner were convertedto factors with suitable level names (in the raw data these are representedusing numeric codes). The converted variables were put back into thedata frame juul, replacing the original sex, tanner, and menarchevariables. We might also have used the transform function (or within):

> juul <- transform(juul,+ sex=factor(sex,labels=c("M","F")),+ menarche=factor(menarche,labels=c("No","Yes")),+ tanner=factor(tanner,labels=c("I","II","III","IV","V")))

4.2 Graphical display of distributions 71

Histogram of x

Fre

quency

−3 −2 −1 0 1 2 3

Figure 4.1. Histogram.

4.2 Graphical display of distributions

4.2.1 Histograms

You can get a reasonable impression of the shape of a distribution bydrawing a histogram; that is, a count of how many observations fall withinspecified divisions (“bins”) of the x-axis (Figure 4.1).

> hist(x)

By specifying breaks=n in the hist call, you get approximately n bars inthe histogram since the algorithm tries to create “pretty” cutpoints. Youcan have full control over the interval divisions by specifying breaks asa vector rather than as a number. Altman (1991, pp. 25–26) contains anexample of accident rates by age group. These are given as a count in agegroups 0–4, 5–9, 10–15, 16, 17, 18–19, 20–24, 25–59, and 60–79 years of age.The data can be entered as follows:

> mid.age <- c(2.5,7.5,13,16.5,17.5,19,22.5,44.5,70.5)> acc.count <- c(28,46,58,20,31,64,149,316,103)> age.acc <- rep(mid.age,acc.count)

72 4. Descriptive statistics and graphics

Histogram of age.acc

age.acc

Densi

0 20 40 60 80

0.0

00.0

10.0

20.0

30.0

Figure 4.2. Histogram with unequal divisions.

> brk <- c(0,5,10,16,17,18,20,25,60,80)> hist(age.acc,breaks=brk)

Here the first three lines generate pseudo-data from the table in the book.For each interval, the relevant number of “observations” is generated withan age set to the midpoint of the interval; that is, 28 2.5-year-olds, 46 7.5-year-olds, etc. Then a vector brk of cutpoints is defined (note that theextremes need to be included) and used as the breaks argument to hist,yielding Figure 4.2.

Notice that you automatically got the “correct” histogram where the areaof a column is proportional to the number. The y-axis is in density units(that is, proportion of data per x unit), so that the total area of the his-togram will be 1. If, for some reason, you want the (misleading) histogramwhere the column height is the raw number in each interval, then it canbe specified using freq=T. For equidistant breakpoints, that is the default(because then you can see how many observations have gone into eachcolumn), but you can set freq=F to get densities displayed. This is reallyjust a change of scale on the y-axis, but it has the advantage that it be-comes possible to overlay the histogram with a corresponding theoreticaldensity function.

4.2 Graphical display of distributions 73

−2 −1 0 1 2 3

0.0

0.2

0.4

0.6

0.8

1.0

sort(x)

(1:n

)/n

Figure 4.3. Empirical cumulative distribution function.

4.2.2 Empirical cumulative distribution

The empirical cumulative distribution function is defined as the fraction ofdata smaller than or equal to x. That is, if x is the kth smallest observation,then the proportion k/n of the data is smaller than or equal to x (7/10if x is no. 7 of 10). The empirical cumulative distribution function can beplotted as follows (see Figure 4.3) where x is the simulated data vectorfrom Section 4.1:

> n <- length(x)> plot(sort(x),(1:n)/n,type="s",ylim=c(0,1))

The plotting parameter type="s" gives a step function where (x, y) is theleft end of the steps and ylim is a vector of two elements specifying theextremes of the y-coordinates on the plot. Recall that c(…) is used tocreate vectors.

Some more elaborate displays of empirical cumulative distribution func-tions are available via the ecdf function. This is also more preciseregarding the mathematical definition of the step function.

74 4. Descriptive statistics and graphics

−2 −1 0 1 2

−2

−1

Normal Q−Q Plot

Theoretical Quantiles

Sam

ple

Quantil

Figure 4.4. Q–Q plot using qqnorm(x).

4.2.3 Q–Q plots

One purpose of calculating the empirical cumulative distribution function(c.d.f.) is to see whether data can be assumed normally distributed. Fora better assessment, you might plot the kth smallest observation againstthe expected value of the kth smallest observation out of n in a standardnormal distribution. The point is that in this way you would expect toobtain a straight line if data come from a normal distribution with anymean and standard deviation.

Creating such a plot is slightly complicated. Fortunately, there is a built-in function for doing it, qqnorm. The result of using it can be seen inFigure 4.4. You only have to write

> qqnorm(x)

As the title of the plot indicates, plots of this kind are also called “Q–Qplots” (quantile versus quantile). Notice that x and y are interchanged rel-ative to the empirical c.d.f. — the observed values are now drawn alongthe y-axis. You should notice that with this convention the distribution hasheavy tails if the outer parts of the curve are steeper than the middle part.

4.3 Summary statistics by groups 75

Some readers will have been taught “probability plots”, which are similarbut have the axes interchanged. It can be argued that the way R draws theplot is the better one since the theoretical quantiles are known in advance,while the empirical quantiles depend on data. You would normally chooseto draw fixed values horizontally and variable values vertically.

4.2.4 Boxplots

A “boxplot”, or more descriptively a “box-and-whiskers plot”, is a graph-ical summary of a distribution. Figure 4.5 shows boxplots for IgM and itslogarithm; see the example on page 23 in Altman (1991).

Here is how a boxplot is drawn in R. The box in the middle indicates“hinges” (nearly quartiles; see the help page for boxplot.stats) andmedian. The lines (“whiskers”) show the largest or smallest observationthat falls within a distance of 1.5 times the box size from the nearest hinge.If any observations fall farther away, the additional points are considered“extreme” values and are shown separately.

The practicalities are these:

> par(mfrow=c(1,2))> boxplot(IgM)> boxplot(log(IgM))> par(mfrow=c(1,1))

A layout with two plots side by side is specified using the mfrow graph-ical parameter. It should be read as “multif rame, rowwise, 1 × 2 layout”.Individual plots are organized in one row and two columns. As you mightguess, there is also an mfcol parameter to plot columnwise. In a 2×2 lay-out, the difference is whether plot no. 2 is drawn in the top right or bottomleft corner.

Notice that it is necessary to reset the layout parameter to c(1,1) at theend unless you also want two plots side by side subsequently.

4.3 Summary statistics by groups

When dealing with grouped data, you will often want to have vari-ous summary statistics computed within groups; for example, a table ofmeans and standard deviations. To this end, you can use tapply (see Sec-tion 1.2.15). Here is an example concerning the folate concentration in redblood cells according to three types of ventilation during anesthesia (Alt-

76 4. Descriptive statistics and graphics

−2

−1

Figure 4.5. Boxplots for IgM and log IgM.

man, 1991, p. 208). We return to this example in Section 7.1, which alsocontains the explanation of the category names.

> attach(red.cell.folate)> tapply(folate,ventilation,mean)N2O+O2,24h N2O+O2,op O2,24h316.6250 256.4444 278.0000

The tapply call takes the folate variable, splits it according toventilation, and computes the mean for each group. In the same way,standard deviations and the number of observations in the groups can becomputed.

> tapply(folate,ventilation,sd)N2O+O2,24h N2O+O2,op O2,24h58.71709 37.12180 33.75648

> tapply(folate,ventilation,length)N2O+O2,24h N2O+O2,op O2,24h

8 9 5

Try something like this for a nicer display:

4.3 Summary statistics by groups 77

> xbar <- tapply(folate, ventilation, mean)> s <- tapply(folate, ventilation, sd)> n <- tapply(folate, ventilation, length)> cbind(mean=xbar, std.dev=s, n=n)

mean std.dev nN2O+O2,24h 316.6250 58.71709 8N2O+O2,op 256.4444 37.12180 9O2,24h 278.0000 33.75648 5

For the juul data, we might want the mean igf1 by tanner group, butof course we run into the problem of missing values again:

> tapply(igf1, tanner, mean)I II III IV VNA NA NA NA NA

We need to get tapply to pass na.rm=T as a parameter to mean to makeit exclude the missing values. This is achieved simply by passing it as anadditional argument to tapply.

> tapply(igf1, tanner, mean, na.rm=T)I II III IV V

207.4727 352.6714 483.2222 513.0172 465.3344

The functions aggregate and by are variations on the same topic. Theformer is very much like tapply, except that it works on an entire dataframe and presents its results as a data frame. This is useful for presentingmany variables at once; e.g.,

> aggregate(juul[c("age","igf1")],+ list(sex=juul$sex), mean, na.rm=T)sex age igf1

1 M 15.38436 310.88662 F 14.84363 368.1006

Notice that the grouping argument in this case must be a list, even whenit is one-dimensional, and that the names of the list elements get usedas column names in the output. Notice also that since the function is ap-plied to all columns of the data frame, you may have to choose a subset ofcolumns, in this case the numeric variables.

The indexing variable is not necessarily part of the data frame that is beingaggregated, and there is no attempt at “smart evaluation” as there is insubset, so you have to spell out juul$sex. You can also use the factthat data frames are list-like and say

> aggregate(juul[c("age","igf1")], juul["sex"], mean, na.rm=T)sex age igf1

1 M 15.38436 310.88662 F 14.84363 368.1006

78 4. Descriptive statistics and graphics

(the “trick” being that indexing a data frame with single brackets yields adata frame as the result).

The by function is again similar, but different. The difference is that thefunction now takes an entire (sub-) data frame as its argument, so thatyou can for instance summarize the Juul data by sex as follows:

> by(juul, juul["sex"], summary)sex: M

age menarche sex igf1 tannerMin. : 0.17 No : 0 M:621 Min. : 29.0 I :2911st Qu.: 8.85 Yes : 0 F: 0 1st Qu.:176.0 II : 55Median :12.38 NA’s:621 Median :280.0 III : 34Mean :15.38 Mean :310.9 IV : 413rd Qu.:16.77 3rd Qu.:430.2 V :124Max. :83.00 Max. :915.0 NA’s: 76

NA’s :145.0testvol

Min. : 1.0001st Qu.: 1.000Median : 3.000Mean : 7.8963rd Qu.: 15.000Max. : 30.000NA’s :141.000————————————————-sex: F

age menarche sex igf1 tannerMin. : 0.25 No :369 M: 0 Min. : 25.0 I :2241st Qu.: 9.30 Yes :335 F:713 1st Qu.:233.0 II : 48Median :12.80 NA’s: 9 Median :352.0 III : 38Mean :14.84 Mean :368.1 IV : 403rd Qu.:16.93 3rd Qu.:483.0 V :204Max. :75.12 Max. :914.0 NA’s:159

NA’s :176.0testvol

Min. : NA1st Qu.: NAMedian : NAMean :NaN3rd Qu.: NAMax. : NANA’s :713

The result of the call to by is actually a list of objects that has has beenwrapped as an object of class "by" and printed using a print method forthat class. You can assign the result to a variable and access the result foreach subgroup using standard list indexing.

The same technique can also be used to generate more elaborate statisticalanalyses for each group.

4.4 Graphics for grouped data 79

4.4 Graphics for grouped data

In dealing with grouped data, it is important to be able not only to createplots for each group but also to compare the plots between groups. Inthis section we review some general graphical techniques that allow us todisplay similar plots for several groups on the same page. Some functionshave specific features for displaying data from more than one group.

4.4.1 Histograms

We have already seen in Section 4.2.1 how to obtain a histogram simply bytyping hist(x), where x is the variable containing the data. R will thenchoose a number of groups so that a reasonable number of data points fallin each bin while at the same time ensuring that the cutpoints are “pretty”numbers on the x-axis.

It is also mentioned there that an alternative number of intervals can beset via the argument breaks, although you do not always get exactlythe number you asked for since R reserves the right to choose “pretty”column boundaries. For instance, multiples of 0.5 MJ are chosen in thefollowing example using the energy data introduced in Section 1.2.14 onthe 24-hour energy expenditure for two groups of women.

In this example, some further techniques of general use are illustrated.The end result is seen in Figure 4.6, but first we must fetch the data:

> attach(energy)> expend.lean <- expend[stature=="lean"]> expend.obese <- expend[stature=="obese"]

Notice how we separate the expend vector in the energy data frame intotwo vectors according to the value of the factor stature.

Now we do the actual plotting:

> par(mfrow=c(2,1))> hist(expend.lean,breaks=10,xlim=c(5,13),ylim=c(0,4),col="white")> hist(expend.obese,breaks=10,xlim=c(5,13),ylim=c(0,4),col="grey")> par(mfrow=c(1,1))

We set par(mfrow=c(2,1)) to get the two histograms in the same plot.In the hist commands themselves, we used the breaks argument asalready mentioned and col, whose effect should be rather obvious. Wealso used xlim and ylim to get the same x and y axes in the two plots.However, it is a coincidence that the columns have the same width.

80 4. Descriptive statistics and graphics

Histogram of expend.lean

expend.lean

Fre

quency

6 8 10 12

Histogram of expend.obese

expend.obese

Fre

quency

6 8 10 12

Figure 4.6. Histograms with refinements.

As a practical remark, when working with plots like the above, wheremore than a single line of code is required, it gets cumbersome to use com-mand recall in the R console window every time something needs to bechanged. A better idea may be to start up a script window or a plain-texteditor and cut and paste entire blocks of code from there (see Section 2.1.3).You might also take it as an incentive to start writing simple functions.

4.4.2 Parallel boxplots

You might want a set of boxplots from several groups in the same frame.boxplot can handle this both when data are given in the form of sepa-rate vectors from each group and when data are in one long vector anda parallel vector or factor defines the grouping. To illustrate the latter, weuse the energy data introduced in Section 1.2.14.

Figure 4.7 is created as follows:

> boxplot(expend ~ stature)

4.4 Graphics for grouped data 81

lean obese

Figure 4.7. Parallel boxplot.

We could also have based the plot on the separate vectors expend.leanand expend.obese. In that case, a syntax is used that specifies thevectors as two separate arguments:

> boxplot(expend.lean,expend.obese)

The plot is not shown here, but the only difference lies in the labellingof the x-axis. There is also a third form, where data are given as a singleargument that is a list of vectors.

The bottom plot has been made using the complete expend vector andthe grouping variable fstature.

Notation of the type y ~ x should be read “y described using x”. This isthe first example we see of a model formula. We see many more examplesof model formulas later on.

4.4.3 Stripcharts

The boxplots made in the preceding section show a “Laurel & Hardy”effect that is not really well founded in the data. The cause is that the in-

82 4. Descriptive statistics and graphics

6 7 8 9 10 11 12 13

lean

ese

6 7 8 9 10 11 12 13

lean

ese

6 7 8 9 10 11 12 13

lea

6 7 8 9 10 11 12 13

lean

obe

Figure 4.8. Stripcharts in four variations.

terquartile range is quite a bit larger in one group than in the other, makingthe boxplot appear “fatter”. With groups as small as these, the quartileswill be quite inaccurately determined, and it may therefore be more desir-able to plot the raw data. If you were to do this by hand, you might draw adot diagram where every number is marked with a dot on a number line.R’s automated variant of this is the function stripchart. Four variantsof stripcharts are shown in Figure 4.8.

The four plots were created as follows:

> opar <- par(mfrow=c(2,2), mex=0.8, mar=c(3,3,2,1)+.1)> stripchart(expend ~ stature)> stripchart(expend ~ stature, method="stack")> stripchart(expend ~ stature, method="jitter")> stripchart(expend ~ stature, method="jitter", jitter=.03)> par(opar)

Notice that a little par magic was used to reduce the spacing between thefour plots. The mex setting reduces the interline distance, and mar reducesthe number of lines that surround the plot region. This can be done forthese plots since they have neither main title, subtitle, nor x and y labels.

4.5 Tables 83

All the original values of the changed settings can be stored in a variable(here opar) and reestablished with par(opar).

The first plot is a standard stripchart, where the points are simply plottedon a line. The problem with this is that some points can become invisiblebecause they are overplotted. This is why there is a method argument,which can be set to either "stack" or "jitter".

The former method stacks points with identical values, but it only doesso if data are completely identical, so in the upper right plot, it is only thetwo replicates of 7.48 that get stacked, whereas 8.08, 8.09, and 8.11 are stillplotted in almost the same spot.

The “jitter” method offsets all points a random amount vertically. Thestandard jittering on plot no. 3 (bottom left) is a bit large; it may be prefer-able to make it clearer that data are placed along a horizontal line. For thatpurpose, you can set jitter lower than the default of 0.1, which is donein the fourth plot.

In this example we have not bothered to specify data in several forms aswe did for boxplot but used expend~stature throughout. We couldalso have written

stripchart(list(lean=expend.lean, obese=expend.obese))

but stripchart(expend.lean, expend.obese) cannot be used.

4.5 Tables

Categorical data are usually described in the form of tables. This sectionoutlines how you can create tables from your data and calculate relativefrequencies.

4.5.1 Generating tables

We deal mainly with two-way tables. In the first example, we enter a tabledirectly, as is required for tables taken from a book or a journal article.

A two-way table can be entered as a matrix object (Section 1.2.7). Altman(1991, p. 242) contains an example on caffeine consumption by maritalstatus among women giving birth. That table may be input as follows:

84 4. Descriptive statistics and graphics

> caff.marital <- matrix(c(652,1537,598,242,36,46,38,21,218+ ,327,106,67),+ nrow=3,byrow=T)> caff.marital

[,1] [,2] [,3] [,4][1,] 652 1537 598 242[2,] 36 46 38 21[3,] 218 327 106 67

The matrix function needs an argument containing the table values asa single vector and also the number of rows in the argument nrow. Bydefault, the values are entered columnwise; if rowwise entry is desired,then you need to specify byrow=T.

You might also give the number of columns instead of rows using ncol.If exactly one of ncol and nrow is given, R will compute the other one sothat it fits the number of values. If both ncol and nrow are given and itdoes not fit the number of values, the values will be “recycled”, which insome (other!) circumstances can be useful.

To get readable printouts, you can add row and column names to thematrices.

> colnames(caff.marital) <- c("0","1-150","151-300",">300")> rownames(caff.marital) <- c("Married","Prev.married","Single")> caff.marital

0 1-150 151-300 >300Married 652 1537 598 242Prev.married 36 46 38 21Single 218 327 106 67

Furthermore, you can name the row and column names as follows. Thisis particularly useful if you are generating many tables with similarclassification criteria.

> names(dimnames(caff.marital)) <- c("marital","consumption")> caff.marital

consumptionmarital 0 1-150 151-300 >300Married 652 1537 598 242Prev.married 36 46 38 21Single 218 327 106 67

Actually, I glossed over something. Tables are not completely equivalentto matrices. There is a "table" class for which special methods exist,and you can convert to that class using as.table(caff.marital). Thetable function below returns an object of class "table".

4.5 Tables 85

For most elementary purposes, you can use matrices where two-dimensio-nal tables are expected. One important case where you do need as.tableis when converting a table to a data frame of counts:

> as.data.frame(as.table(caff.marital))marital consumption Freq

1 Married 0 6522 Prev.married 0 363 Single 0 2184 Married 1-150 15375 Prev.married 1-150 466 Single 1-150 3277 Married 151-300 5988 Prev.married 151-300 389 Single 151-300 10610 Married >300 24211 Prev.married >300 2112 Single >300 67

In practice, the more frequent case is that you have a data frame withvariables for each person in a data set. In that case, you should do thetabulation with table, xtabs, or ftable. These functions will gener-ally work for tabulating numeric vectors as well as factor variables, butthe latter will have their levels used for row and column names automati-cally. Hence, it is recommended to convert numerically coded categoricaldata into factors. The table function is the oldest and most basic of thethree. The two others offer formula-based interfaces and better printing ofmultiway tables.

The data set juul was introduced on p. 68. Here we look at some othervariables in that data set, namely sex and menarche; the latter indicateswhether or not a girl has had her first period. We can generate some simpletables as follows:

> table(sex)sexM F

621 713> table(sex,menarche)

menarchesex No YesM 0 0F 369 335

> table(menarche,tanner)tanner

menarche I II III IV VNo 221 43 32 14 2Yes 1 1 5 26 202

86 4. Descriptive statistics and graphics

Of course, the table of menarche versus sex is just a check on internal con-sistency of the data. The table of menarche versus Tanner stage of pubertyis more interesting.

There are also tables with more than two sides, but not many simple sta-tistical functions use them. Briefly, to tabulate such data, just write, forexample, table(factor1,factor2,factor3). To input a table of cellcounts, use the array function (an analogue of matrix).

The xtabs function is quite similar to table except that it uses a modelformula interface. This most often uses a one-sided formula where youjust list the classification variables separated by +.

> xtabs(~ tanner + sex, data=juul)sex

tanner M FI 291 224II 55 48III 34 38IV 41 40V 124 204

Notice how the interface allows you to refer to variables in a data framewithout attaching it. The empty left-hand side can be replaced by a vectorof counts in order to handle pretabulated data.

The formatting of multiway tables from table or xtabs is not really nice;e.g.,

> xtabs(~ dgn + diab + coma, data=stroke), , coma = No

diabdgn No YesICH 53 6ID 143 21INF 411 64SAH 38 0

, , coma = Yes

diabdgn No YesICH 19 1ID 23 3INF 23 2SAH 9 0

As you add dimensions, you get more of these two-sided subtables andit becomes rather easy to lose track. This is where ftable comes in. Thisfunction creates “flat” tables; e.g., like this:

4.5 Tables 87

> ftable(coma + diab ~ dgn, data=stroke)coma No Yesdiab No Yes No Yes

dgnICH 53 6 19 1ID 143 21 23 3INF 411 64 23 2SAH 38 0 9 0

That is, variables on the left-hand side tabulate across the page and thoseon the right tabulate downwards. ftable works on raw data as shown,but its data argument can also be a table as generated by one of the otherfunctions.

Like any matrix, a table can be transposed with the t function:

> t(caff.marital)marital

consumption Married Prev.married Single0 652 36 2181-150 1537 46 327151-300 598 38 106>300 242 21 67

For multiway tables, exchanging indices (generalized transposition) isdone by aperm.

4.5.2 Marginal tables and relative frequency

It is often desired to compute marginal tables; that is, the sums of thecounts along one or the other dimension of a table. Due to missing val-ues, this might not coincide with just tabulating a single factor. This isdone fairly easily using the apply function (Section 1.2.15), but there isalso a simplified version called margin.table, described below.

First, we need to generate the table itself:

> tanner.sex <- table(tanner,sex)

(tanner.sex is an arbitrarily chosen name for the crosstable.)

> tanner.sexsex

tanner M FI 291 224II 55 48III 34 38IV 41 40V 124 204

88 4. Descriptive statistics and graphics

Then we compute the marginal tables:

> margin.table(tanner.sex,1)tannerI II III IV V

515 103 72 81 328> margin.table(tanner.sex,2)sexM F

545 554

The second argument to margin.table is the number of the marginalindex: 1 and 2 give row and column totals, respectively.

Relative frequencies in a table are generally expressed as proportions ofthe row or column totals. Tables of relative frequencies can be constructedusing prop.table as follows:

> prop.table(tanner.sex,1)sex

tanner M FI 0.5650485 0.4349515II 0.5339806 0.4660194III 0.4722222 0.5277778IV 0.5061728 0.4938272V 0.3780488 0.6219512

Note that the rows (1st index) sum to 1. If a table of percentages is desired,just multiply the entire table by 100.

prop.table cannot be used to express the numbers relative to the grandtotal of the table, but you can of course always write

> tanner.sex/sum(tanner.sex)sex

tanner M FI 0.26478617 0.20382166II 0.05004550 0.04367607III 0.03093722 0.03457689IV 0.03730664 0.03639672V 0.11282985 0.18562329

The functions margin.table and prop.table also work on multiwaytables — the margin argument can be a vector if the relevant margin hastwo or more dimensions.

4.6 Graphical display of tables 89

0 1−150 151−300 >300

0500

1000

1500

Figure 4.9. Simple barplot of total caffeine consumption.

4.6 Graphical display of tables

For presentation purposes, it may be desirable to display a graph ratherthan a table of counts or percentages. In this section, the main methods fordoing this are described.

4.6.1 Barplots

Barplots are made using barplot. This function takes an argument,which can be a vector or a matrix. The simplest variant goes as follows(Figure 4.9):

> total.caff <- margin.table(caff.marital,2)> total.caffconsumption

0 1-150 151-300 >300906 1910 742 330

> barplot(total.caff, col="white")

90 4. Descriptive statistics and graphics

0 1−150 >300

0500

Married Single

000

200

0300

Married Single

0400

01400

Married Single

0.0

0.2

0.4

Figure 4.10. Four variants of barplot on a two-way table.

Without the col="white" argument, the plot comes out in colour, butthis is not suitable for a black and white book illustration.

If the argument is a matrix, then barplot creates by default a “stackedbarplot”, where the columns are partitioned according to the contri-butions from different rows of the table. If you want to place therow contributions beside each other instead, you can use the argu-ment beside=T. A series of variants is found in Figure 4.10, which isconstructed as follows:

> par(mfrow=c(2,2))> barplot(caff.marital, col="white")> barplot(t(caff.marital), col="white")> barplot(t(caff.marital), col="white", beside=T)> barplot(prop.table(t(caff.marital),2), col="white", beside=T)> par(mfrow=c(1,1))

In the last three plots, we switched rows and columns with the trans-position function t. In the very last one, the columns are expressed asproportions of the total number in the group. Thus, information is lost onthe relative sizes of the marital status groups, but the group of previouslymarried women (recall that the data set deals with women giving birth)

4.6 Graphical display of tables 91

Married Prev.married Single

01−150151−300>300

0.0

0.1

0.2

0.3

0.4

0.5

Figure 4.11. Bar plot with specified colours and legend.

is so small that it otherwise becomes almost impossible to compare theircaffeine consumption profile with those of the other groups.

As usual, there are a multitude of ways to “prettify” the plots. Here is onepossibility (Figure 4.11):

> barplot(prop.table(t(caff.marital),2),beside=T,+ legend.text=colnames(caff.marital),+ col=c("white","grey80","grey50","black"))

Notice that the legend overlaps the top of one of the columns. R is notdesigned to be able to find a clear area in which to place the legend.However, you can get full control of the legend’s position if you insertit explicitly with the legend function. For that purpose, it will be help-ful to use locator(), which allows you to click a mouse button over theplot and have the coordinates returned. See p. 209 for more about this.

4.6.2 Dotcharts

The Cleveland dotcharts, named after William S. Cleveland (1994), can beemployed to study a table from both sides at the same time. They contain

92 4. Descriptive statistics and graphics

01−150151−300>300

Married

Prev.married

Single

0 500 1000 1500

Figure 4.12. Dotchart of caffeine consumption.

the same information as barplots with beside=T but give quite a differ-ent visual impression. We content ourselves with a single example here(Figure 4.12):

> dotchart(t(caff.marital), lcolor="black")

(The line colour was changed from the default "gray" because it tends tobe hard to see in print.)

4.6.3 Piecharts

Piecharts are traditionally frowned upon by statisticians because they areso often used to make trivial data look impressive and are difficult todecode for the human mind. They very rarely contain information thatwould not have been at least as effectively conveyed in a barplot. Oncein a while they are useful, though, and it is no problem to get R to drawthem. Here is a way to represent the table of caffeine consumption versusmarital status (Figure 4.13; see Section 4.4.3 for an explanation of the “parmagic” used to reduce the space between the subplots):

4.7 Exercises 93

1−150

151−300

>300

Married

01−150

151−300

>300

Previously married

1−150

151−300

>300

Single

Figure 4.13. Pie charts of caffeine consumption according to marital status.

> opar <- par(mfrow=c(2,2),mex=0.8, mar=c(1,1,2,1))> slices <- c("white","grey80","grey50","black")> pie(caff.marital["Married",], main="Married", col=slices)> pie(caff.marital["Prev.married",],+ main="Previously married", col=slices)> pie(caff.marital["Single",], main="Single", col=slices)> par(opar)

The col argument sets the colour of the pie slices.

There are more possibilities with piechart. The help page for pie con-tains an illustrative example concerning the distribution of pie sales (!) bypie type.

4.7 Exercises

4.1 Explore the possibilities for different kinds of line and point plots.Vary the plot symbol, line type, line width, and colour.

94 4. Descriptive statistics and graphics

4.2 If you make a plot like plot(rnorm(10),type="o") with over-plotted lines and points, the lines will be visible inside the plottingsymbols. How can this be avoided?

4.3 How can you overlay two qqnorm plots in the same plotting area?What goes wrong if you try to generate the plot using type="l", andhow do you avoid that?

4.4 Plot a histogram for the react data set. Since these data are highlydiscretized, the histogram will be biased. Why? You may want to trytruehist from the MASS package as a replacement.

4.5 Generate a sample vector z of five random numbers from the uni-form distribution, and plot quantile(z,x) as a function of x (usecurve, for instance).

5One- and two-sample tests

Most of the rest of this book describes applications of R for actual sta-tistical analysis. The focus to some extent shifts from explanation of thesyntax to description of the output and specific arguments to the relevantfunctions.

Some of the most basic statistical tests deal with comparing continuousdata either between two groups or against an a priori stipulated value.This is the topic for this chapter.

Two functions are introduced here, namely t.test and wilcox.testfor t tests and Wilcoxon tests, respectively. Both can be applied to one-and two-sample problems as well as paired data. Notice that the “two-sample Wilcoxon test” is the same as the one called the “Mann–Whitneytest” in many textbooks.

5.1 One-sample t test

The t tests are based on an assumption that data come from the nor-mal distribution. In the one-sample case we thus have data x1, . . . , xnassumed to be independent realizations of random variables with distri-bution N(µ, σ2), which denotes the normal distribution with mean µ andvariance σ2, and we wish to test the null hypothesis that µ = µ0. We canestimate the parameters µ and σ by the empirical mean x̄ and standard

P. Dalgaard, Introductory Statistics with R,DOI: 10.1007/978-0-387-79054-1_5, © Springer Science+Business Media, LLC 2008

96 5. One- and two-sample tests

deviation s, although we must realize that we could never pinpoint theirvalues exactly.

The key concept is that of the standard error of the mean, or SEM. This de-scribes the variation of the average of n random values with mean µ andvariance σ2. This value is

SEM = σ/√

and means that if you were to repeat the entire experiment several timesand calculate an average for each experiment, then these averages wouldfollow a distribution that is narrower than that of the original distribu-tion. The crucial point is that even based on a single sample, it is possibleto calculate an empirical SEM as s/

√n using the empirical standard de-

viation of the sample. This value will tell us how far the observed meanmay reasonably have strayed from its true value. For normally distributeddata, the rule of thumb is that there is a 95% probability of staying withinµ ± 2σ, so we would expect that if µ0 were the true mean, then x̄ shouldbe within 2 SEMs of it. Formally, you calculate

t =x̄ − µ0SEM

and see whether this falls within an acceptance region outside which tshould fall with probability equal to a specified significance level. This isoften chosen as 5%, in which case the acceptance region is almost, but notexactly, the interval from −2 to 2.

In small samples, it is necessary to correct for the fact that an empiricalSEM is used and that the distribution of t therefore has somewhat “heaviertails” than the N(0, 1): Large deviations happen more frequently than inthe normal distribution since they can result from normalizing with anSEM that is too small. The correct values for the acceptance region canbe looked up as quantiles in the t distribution with f = n − 1 degrees offreedom.

If t falls outside the acceptance region, then we reject the null hypothesisat the chosen significance level. Alternatively (and equivalently), you cancalculate the p-value, which is the probability of obtaining a value as nu-merically large as or larger than the observed t and reject the hypothesisif the p-value is less than the significance level.

Sometimes you have prior information on the direction of an effect; forinstance, that all plausible mechanisms that would cause µ not to equal µ0would tend to make it bigger. In those cases, you may choose to reject thehypothesis only if t falls in the upper tail of the distribution. This is knownas testing against a one-sided alternative. Since removing the lower tail fromthe rejection region effectively halves the significance level, a one-sidedtest at a given level will have a smaller cutoff point. Similarly, p-values

5.1 One-sample t test 97

are calculated as the probability of a larger value than observed ratherthan a numerically larger one, effectively halving the p-value as long asthe observed effect is in the stipulated direction. One-sided tests shouldbe used with some care, preferably only when there is a clear statementof the intent to use them in the study protocol. Switching to a one-sidedtest to make an otherwise nonsignificant result significant could easily beregarded as dishonest.

Here is an example concerning daily energy intake in kJ for 11 women(Altman, 1991, p. 183). First, the values are placed in a data vector:

> daily.intake <- c(5260,5470,5640,6180,6390,6515,+ 6805,7515,7515,8230,8770)

Let us first look at some simple summary statistics, even though these arehardly necessary for such a small data set:

> mean(daily.intake)[1] 6753.636> sd(daily.intake)[1] 1142.123> quantile(daily.intake)0% 25% 50% 75% 100%

5260 5910 6515 7515 8770

You might wish to investigate whether the women’s energy intake devi-ates systematically from a recommended value of 7725 kJ. Assuming thatdata come from a normal distribution, the object is to test whether thisdistribution might have mean µ = 7725. This is done with t.test asfollows:

> t.test(daily.intake,mu=7725)

One Sample t-test

data: daily.intaket = -2.8208, df = 10, p-value = 0.01814alternative hypothesis: true mean is not equal to 772595 percent confidence interval:5986.348 7520.925sample estimates:mean of x6753.636

This is an example of the exact same type as used in the introductory Sec-tion 1.1.4. The description of the output is quite superficial there. Here itis explained more thoroughly.

The layout is common to many of the standard statistical tests, and a“dissection” is given in the following:

98 5. One- and two-sample tests

One Sample t-test

This should be self-explanatory. It is simply a description of the test thatwe have asked for. Notice that, by looking at the format of the functioncall, t.test has automatically found out that a one-sample test is desired.

data: daily.intake

This tells us which data are being tested. Of course, this will be obviousunless output has been separated from the command that generated it.This can happen, for example, when using the source function to readcommands from an external file.

t = -2.8208, df = 10, p-value = 0.01814

This is where it begins to get interesting. We get the t statistic, the asso-ciated degrees of freedom, and the exact p-value. We do not need to usea table of the t distribution to look up which quantiles the t-value can befound between. You can immediately see that p < 0.05 and thus that (us-ing the customary 5% level of significance) data deviate significantly fromthe hypothesis that the mean is 7725.

alternative hypothesis: true mean is not equal to 7725

This contains two important pieces of information: (a) the value wewanted to test whether the mean could be equal to (7725 kJ) and (b) thatthe test is two-sided (“not equal to”).

95 percent confidence interval:5986.348 7520.925

This is a 95% confidence interval for the true mean; that is, the set of (hy-pothetical) mean values from which the data do not deviate significantly.It is based on inverting the t test by solving for the values of µ0 that causet to lie within its acceptance region. For a 95% confidence interval, thesolution is

x̄ − t0.975( f ) × SEM < µ < x̄ + t0.975( f ) × SEM

sample estimates:mean of x6753.636

This final item is the observed mean; that is, the (point) estimate of thetrue mean.

The function t.test has a number of optional arguments, three of whichare relevant in one-sample problems. We have already seen the use of mu

5.2 Wilcoxon signed-rank test 99

to specify the mean value µ under the null hypothesis (default is mu=0). Inaddition, you can specify that a one-sided test is desired against alterna-tives greater than µ by using alternative="greater" or alternativesless than µ using alternative="less". The third item that can be spec-ified is the confidence level used for the confidence intervals; you wouldwrite conf.level=0.99 to get a 99% interval.

Actually, it is often allowable to abbreviate a longish argument specifica-tion; for instance, it is sufficient to write alt="g" to get the test againstgreater alternatives.

5.2 Wilcoxon signed-rank test

The t tests are fairly robust against departures from the normal distribu-tion especially in larger samples, but sometimes you wish to avoid makingthat assumption. To this end, the distribution-free methods are convenient.These are generally obtained by replacing data with corresponding orderstatistics.

For the one-sample Wilcoxon test, the procedure is to subtract the the-oretical µ0 and rank the differences according to their numerical value,ignoring the sign, and then calculate the sum of the positive or negativeranks. The point is that, assuming only that the distribution is symmetricaround µ0, the test statistic corresponds to selecting each number from 1to n with probability 1/2 and calculating the sum. The distribution of thetest statistic can be calculated exactly, at least in principle. It becomes com-putationally excessive in large samples, but the distribution is then verywell approximated by a normal distribution.

Practical application of the Wilcoxon signed-rank test is done almostexactly like the t test:

> wilcox.test(daily.intake, mu=7725)Wilcoxon signed rank test with continuity correction

data: daily.intakeV = 8, p-value = 0.0293alternative hypothesis: true location is not equal to 7725

Warning message:In wilcox.test.default(daily.intake, mu = 7725) :cannot compute exact p-value with ties

There is not quite as much output as from t.test due to the fact thatthere is no such thing as a parameter estimate in a nonparametric test andtherefore no confidence limits, etc., either. It is, however, possible under

100 5. One- and two-sample tests

some assumptions to define a location measure and calculate confidenceintervals for it. See the help files for wilcox.test for details.

The relative merits of distribution-free (or nonparametric) versus para-metric methods such as the t test are a contentious issue. If the modelassumptions of the parametric test are fulfilled, then it will be somewhatmore efficient, on the order of 5% in large samples, although the differencecan be larger in small samples. Notice, for instance, that unless the samplesize is 6 or above, the signed-rank test simply cannot become significant atthe 5% level. This is probably not too important, though; what is more im-portant is that the apparent lack of assumptions for these tests sometimesmisleads people into using them for data where the observations are notindependent or where a comparison is biased by an important covariate.

The Wilcoxon tests are susceptible to the problem of ties, where several ob-servations share the same value. In such cases, you simply use the averageof the tied ranks; for example, if there are four identical values corre-sponding to places 6 to 9, they will all be assigned the value 7.5. This isnot a problem for the large-sample normal approximations, but the exactsmall-sample distributions become much more difficult to calculate andwilcox.test cannot do so.

The test statistic V is the sum of the positive ranks. In the example, thep-value is computed from the normal approximation because of the tie at7515.

The function wilcox.test takes arguments mu and alternative,just like t.test. In addition, it has correct, which turns a continu-ity correction on or off (the default is “on”, as seen from the outputtitle; correct=F turns it off), and exact, which specifies whether exacttests should be calculated. Recall that “on/off” options such as these arespecified using logical values that can be either TRUE or FALSE.

5.3 Two-sample t test

The two-sample t test is used to test the hypothesis that two samples maybe assumed to come from distributions with the same mean.

The theory for the two-sample t test is not very different in principle fromthat of the one-sample test. Data are now from two groups, x11, . . . , x1n1and x21, . . . , x2n2 , which we assume are sampled from the normal distribu-tions N(µ1, σ21 ) and N(µ2, σ

22 ), and it is desired to test the null hypothesis

µ1 = µ2. You then calculate

t =x̄2 − x̄1SEDM

5.3 Two-sample t test 101

where the standard error of difference of means is

SEDM =√

SEM21 + SEM22

There are two ways of calculating the SEDM depending on whether or notyou assume that the two groups have the same variance. The “classical”approach is to assume that the variances are identical. With this approach,you first calculate a pooled s based on the standard deviations from thetwo groups and plug that value into the SEM. Under the null hypothesis,the t value will follow a t distribution with n1 + n2 −2 degrees of freedom.

An alternative procedure due to Welch is to calculate the SEMs from theseparate group standard deviations s1 and s2. With this procedure, t isactually not t-distributed, but its distribution may be approximated by at distribution with a number of degrees of freedom that can be calculatedfrom s1, s2, and the group sizes. This is generally not an integer.

The Welch procedure is generally considered the safer one. Usually, thetwo procedures give very similar results unless both the group sizes andthe standard deviations are very different.

We return to the daily energy expenditure data (see Section 1.2.14) andconsider the problem of comparing energy expenditures between lean andobese women.

> attach(energy)> energy

expend stature1 9.21 obese2 7.53 lean3 7.48 lean…20 7.58 lean21 9.19 obese22 8.11 lean

Notice that the necessary information is contained in two parallel columnsof a data frame. The factor stature contains the group and the numericvariable expend the energy expenditure in mega-Joules. R allows data inthis format to be analyzed by t.test and wilcox.test using a modelformula specification. An older format (still available) requires you tospecify data from each group in a separate variable, but the newer for-mat is much more convenient for data that are kept in data frames andis also more flexible if you later want to group the same response dataaccording to other criteria.

The object is to see whether there is a shift in level between the two groups,so we apply a t test as follows:

102 5. One- and two-sample tests

> t.test(expend~stature)

Welch Two Sample t-test

data: expend by staturet = -3.8555, df = 15.919, p-value = 0.001411alternative hypothesis: true difference in means is not equal to 095 percent confidence interval:-3.459167 -1.004081sample estimates:mean in group lean mean in group obese

8.066154 10.297778

Notice the use of the tilde (~) operator to specify that expend is describedby stature.

The output is not much different from that of the one-sample test. Theconfidence interval is for the difference in means and does not contain 0,which is in accordance with the p-value indicating a significant differenceat the 5% level.

It is Welch’s variant of the t test that is calculated by default. This is the testwhere you do not assume that the variance is the same in the two groups,which (among other things) results in the fractional degrees of freedom.

To get the usual (textbook) t test, you must specify that you are willingto assume that the variances are the same. This is done via the optionalargument var.equal=T; that is:

> t.test(expend~stature, var.equal=T)

Two Sample t-test

data: expend by staturet = -3.9456, df = 20, p-value = 0.000799alternative hypothesis: true difference in means is not equal to 095 percent confidence interval:-3.411451 -1.051796sample estimates:mean in group lean mean in group obese

8.066154 10.297778

Notice that the degrees of freedom now has become a whole number,namely 13 + 9 − 2 = 20. The p-value has dropped slightly (from 0.14%to 0.08%) and the confidence interval is a little narrower, but overall thechanges are slight.

5.4 Comparison of variances 103

5.4 Comparison of variances

Even though it is possible in R to perform the two-sample t test withoutthe assumption that the variances are the same, you may still be interestedin testing that assumption, and R provides the var.test function for thatpurpose, implementing an F test on the ratio of the group variances. It iscalled the same way as t.test:

> var.test(expend~stature)

F test to compare two variances

data: expend by statureF = 0.7844, num df = 12, denom df = 8, p-value = 0.6797alternative hypothesis: true ratio of variances is not equal to 195 percent confidence interval:0.1867876 2.7547991sample estimates:ratio of variances

0.784446

The test is not significant, so there is no evidence against the assumptionthat the variances are identical. However, the confidence interval is verywide. For small data sets such as this one, the assumption of constant vari-ance is largely a matter of belief. It may also be noted that this test is notrobust against departures from a normal distribution. The stats packagecontains several alternative tests for variance homogeneity, each with itsown assumptions, benefits, and drawbacks, but we shall not discuss themat length.

Notice that the test is based on the assumption that the groups areindependent. You should not apply this test to paired data.

5.5 Two-sample Wilcoxon test

You might prefer a nonparametric test if you doubt the normal distribu-tion assumptions of the t test. The two-sample Wilcoxon test is based onreplacing the data by their rank (without regard to grouping) and calcu-lating the sum of the ranks in one group, thus reducing the problem to oneof sampling n1 values without replacement from the numbers 1 to n1 + n2.

This is done using wilcox.test, which behaves similarly to t.test:

104 5. One- and two-sample tests

> wilcox.test(expend~stature)

Wilcoxon rank sum test with continuity correction

data: expend by statureW = 12, p-value = 0.002122alternative hypothesis: true location shift is not equal to 0

Warning message:In wilcox.test.default(x = c(7.53, 7.48, 8.08, 8.09, 10.15, 8.4, :cannot compute exact p-value with ties

The test statistic W is the sum of ranks in the first group minus its theoreti-cal minimum (i.e., it is zero if all the smallest values fall in the first group).Some textbooks use a statistic that is the sum of ranks in the smallest groupwith no minimum correction, which is of course equivalent. Notice that,as in the one-sample example, we are having problems with ties and relyon the approximate normal distribution of W.

5.6 The paired t test

Paired tests are used when there are two measurements on the same ex-perimental unit. The theory is essentially based on taking differences andthus reducing the problem to that of a one-sample test. Notice, though,that it is implicitly assumed that such differences have a distribution thatis independent of the level. A useful graphical check is to make a scatter-plot of the pairs with the line of identity added or to plot the differenceagainst the average of the pair (sometimes called a Bland–Altman plot).If there seems to be a tendency for the dispersion to change with thelevel, then it may be useful to transform the data; frequently the stan-dard deviation is proportional to the level, in which case a logarithmictransformation is useful.

The data on pre- and postmenstrual energy intake in a group of womenare considered several times in Chapter 1 (and you may notice that the firstcolumn is identical to daily.intake, which was used in Section 5.1).There data are entered from the command line, but they are also availableas a data set in the ISwR package:

> attach(intake)> intake

pre post1 5260 39102 5470 42203 5640 38854 6180 5160

5.6 The paired t test 105

5 6390 56456 6515 46807 6805 52658 7515 59759 7515 679010 8230 690011 8770 7335

The point is that the same 11 women are measured twice, so it makes senseto look at individual differences:

> post – pre[1] -1350 -1250 -1755 -1020 -745 -1835 -1540 -1540 -725 -1330[11] -1435

It is immediately seen that they are all negative. All the women have alower energy intake postmenstrually than premenstrually. The paired ttest is obtained as follows:

> t.test(pre, post, paired=T)

Paired t-test

data: pre and postt = 11.9414, df = 10, p-value = 3.059e-07alternative hypothesis: true difference in means is not equal to 095 percent confidence interval:1074.072 1566.838sample estimates:mean of the differences

1320.455

There is not much new to say about the output; it is virtually identical tothat of a one-sample t test on the elementwise differences.

Notice that you have to specify paired=T explicitly in the call, indicat-ing that you want a paired test. In the old-style interface for the unpairedt test, the two groups are specified as separate vectors and you wouldrequest that analysis by omitting paired=T. If data are actually paired,then it would be seriously inappropriate to analyze them without takingthe pairing into account.

Even though it might be considered pedagogically dubious to show whatyou should not do, the following shows the results of an unpaired t teston the same data for comparison:

106 5. One- and two-sample tests

> t.test(pre, post) #WRONG!

Welch Two Sample t-test

data: pre and postt = 2.6242, df = 19.92, p-value = 0.01629alternative hypothesis: true difference in means is not equal to 095 percent confidence interval:270.5633 2370.3458

sample estimates:mean of x mean of y6753.636 5433.182

The number symbol (or “hash”) # introduces a comment in R. The rest ofthe line is skipped.

It is seen that t has become considerably smaller, although still significantat the 5% level. The confidence interval has become almost four timeswider than in the correct paired analysis. Both illustrate the loss of ef-ficiency caused by not using the information that the “pre” and “post”measurements are from the same person. Alternatively, you could say thatit demonstrates the gain in efficiency obtained by planning the experi-ment with two measurements on the same person, rather than having twoindependent groups of pre- and postmenstrual women.

5.7 The matched-pairs Wilcoxon test

The paired Wilcoxon test is the same as a one-sample Wilcoxon signed-rank test on the differences. The call is completely analogous to t.test:

> wilcox.test(pre, post, paired=T)Wilcoxon signed rank test with continuity correction

data: pre and postV = 66, p-value = 0.00384alternative hypothesis: true location shift is not equal to 0

Warning message:In wilcox.test.default(pre, post, paired = T) :cannot compute exact p-value with ties

The result does not show any material difference from that of the t test.The p-value is not quite so extreme, which is not too surprising since theWilcoxon rank sum cannot get any larger than it does when all differenceshave the same sign, whereas the t statistic can become arbitrarily extreme.

5.8 Exercises 107

Again, we have trouble with tied data invalidating the exact p calcula-tions. This time it is the two identical differences of −1540.

In the present case it is actually very easy to calculate the exact p-valuefor the Wilcoxon test. It is the probability of 11 positive differences + theprobability of 11 negative ones, 2 × (1/2)11 = 1/1024 = 0.00098, so theapproximate p-value is almost four times too large.

5.8 Exercises

5.1 Do the values of the react data set (notice that this is a single vector,not a data frame) look reasonably normally distributed? Does the meandiffer significantly from zero according to a t test?

5.2 In the data set vitcap, use a t test to compare the vital capacity forthe two groups. Calculate a 99% confidence interval for the difference. Theresult of this comparison may be misleading. Why?

5.3 Perform the analyses of the react and vitcap data using nonpara-metric techniques.

5.4 Perform graphical checks of the assumptions for a paired t test in theintake data set.

5.5 The function shapiro.test computes a test of normality based onthe degree of linearity of the Q–Q plot. Apply it to the react data. Doesit help to remove the outliers?

5.6 The crossover trial in ashina can be analyzed for a drug effect ina simple way (how?) if you ignore a potential period effect. However,you can do better. Hint: Consider the intra-individual differences; if therewere only a period effect present, how should the differences behave in thetwo groups? Compare the results of the simple method and the improvedmethod.

5.7 Perform 10 one-sample t tests on simulated normally distributeddata sets of 25 observations each. Repeat the experiment, but instead sim-ulate samples from a different distribution; try the t distribution with 2degrees of freedom and the exponential distribution (in the latter case,test for the mean being equal to 1). Can you find a way to automate thisso that you can have a larger number of replications?

6Regression and correlation

The main object of this chapter is to show how to perform basic regressionanalyses, including plots for model checking and display of confidenceand prediction intervals. Furthermore, we describe the related topic ofcorrelation in both its parametric and nonparametric variants.

6.1 Simple linear regression

We consider situations where you want to describe the relation be-tween two variables using linear regression analysis. You may, forinstance, be interested in describing short.velocity as a function ofblood.glucose. This section deals only with the very basics, whereasseveral more complicated issues are postponed until Chapter 12.

The linear regression model is given by

yi = α + βxi + ei

in which the ei are assumed independent and N(0, σ2). The nonrandompart of the equation describes the yi as lying on a straight line. The slopeof the line (the regression coefficient) is β, the increase per unit change in x.The line intersects the y-axis at the intercept α.

The parameters α, β, and σ2 can be estimated using the method of leastsquares. Find the values of α and β that minimize the sum of squared

P. Dalgaard, Introductory Statistics with R,DOI: 10.1007/978-0-387-79054-1_6, © Springer Science+Business Media, LLC 2008

110 6. Regression and correlation

residuals

SSres = ∑i(yi − (α + βxi))2

This is not actually done by trial and error. One can find closed-formexpressions for the choice of parameters that gives the smallest value ofSSres:

β̂ =∑(xi − x̄)(yi − ȳ)

∑(xi − x̄)2

α̂ = ȳ − β̂x̄

The residual variance is estimated as SSres/(n − 2), and the residualstandard deviation is of course the square root of that.

The empirical slope and intercept will deviate somewhat from the truevalues due to sampling variation. If you were to generate several sets of yiat the same set of xi, you would observe a distribution of empirical slopesand intercepts. Just as you could calculate the SEM to describe the vari-ability of the empirical mean, it is also possible from a single sample of(xi, yi) to calculate the standard error of the computed estimates, s.e.(α̂)and s.e.(β̂). These standard errors can be used to compute confidence in-tervals for the parameters and tests for whether a parameter has a specificvalue.

It is usually of prime interest to test the null hypothesis that β = 0 sincethat would imply that the line was horizontal and thus that the ys have adistribution that is the same, whatever the value of x. You can compute at test for that hypothesis simply by dividing the estimate by its standarderror

t =β̂

s.e.(β̂)

which follows a t distribution on n − 2 degrees of freedom if the true β iszero. A similar test can be calculated for whether the intercept is zero, butyou should be aware that it is often a meaningless hypothesis either be-cause there is no natural reason to believe that the line should go throughthe origin or because it would involve an extrapolation far outside therange of data.

For the example in this section, we need the data frame thuesen, whichwe attach with

> attach(thuesen)

For linear regression analysis, the function lm (linear model) is used:

6.1 Simple linear regression 111

> lm(short.velocity~blood.glucose)

Call:lm(formula = short.velocity ~ blood.glucose)

Coefficients:(Intercept) blood.glucose

1.09781 0.02196

The argument to lm is a model formula in which the tilde symbol (~) shouldbe read as “described by”. This was seen several times earlier, both inconnection with boxplots and stripcharts and with the t and Wilcoxontests.

The lm function handles much more complicated models than simple lin-ear regression. There can be many other things besides a dependent and adescriptive variable in a model formula. A multiple linear regression anal-ysis (which we discuss in Chapter 11) of, for example, y on x1, x2, and x3is specified as y ~ x1 + x2 + x3.

In its raw form, the output of lm is very brief. All you see is the estimatedintercept (α) and the estimated slope (β). The best-fitting straight line isseen to be short.velocity = 1.098 + 0.0220 ×blood.glucose, butfor instance no tests of significance are given.

The result of lm is a model object. This is a distinctive concept of the S lan-guage (of which R is a dialect). Whereas other statistical systems focus ongenerating printed output that can be controlled by setting options, youget instead the result of a model fit encapsulated in an object from whichthe desired quantities can be obtained using extractor functions. An lm ob-ject does in fact contain much more information than you see when it isprinted.

A basic extractor function is summary:

> summary(lm(short.velocity~blood.glucose))

Call:lm(formula = short.velocity ~ blood.glucose)

Residuals:Min 1Q Median 3Q Max

-0.40141 -0.14760 -0.02202 0.03001 0.43490

Coefficients:Estimate Std. Error t value Pr(>|t|)

(Intercept) 1.09781 0.11748 9.345 6.26e-09 ***blood.glucose 0.02196 0.01045 2.101 0.0479 *—Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

112 6. Regression and correlation

Residual standard error: 0.2167 on 21 degrees of freedom(1 observation deleted due to missingness)

Multiple R-squared: 0.1737, Adjusted R-squared: 0.1343F-statistic: 4.414 on 1 and 21 DF, p-value: 0.0479

The format above looks more like what other statistical packages wouldoutput. The following is a “dissection” of the output:

Call:lm(formula = short.velocity ~ blood.glucose)

As in t.test, etc., the output starts with something that is essentiallya repeat of the function call. This is not very interesting when one hasjust given it as a command to R, but it is useful if the result is saved in avariable that is printed later.

Residuals:Min 1Q Median 3Q Max

-0.40141 -0.14760 -0.02202 0.03001 0.43490

This gives a superficial view of the distribution of the residuals that maybe used as a quick check of the distributional assumptions. The averageof the residuals is zero by definition, so the median should not be far fromzero, and the minimum and maximum should be roughly equal in ab-solute value. In the example, it can be noticed that the third quartile isremarkably close to zero, but in view of the small number of observations,this is not really something to worry about.

Coefficients:Estimate Std. Error t value Pr(>|t|)

(Intercept) 1.09781 0.11748 9.345 6.26e-09 ***blood.glucose 0.02196 0.01045 2.101 0.0479 *—Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Here we see the regression coefficient and the intercept again, but this timewith accompanying standard errors, t tests, and p-values. The symbolsto the right are graphical indicators of the level of significance. The linebelow the table shows the definition of these indicators; one star means0.01 < p < 0.05.

The graphical indicators have been the target of some controversy. Somepeople like to have the possibility of seeing at a glance whether there is“anything interesting” in an analysis, whereas others feel that the indica-tors too often correspond to meaningless tests. For instance, the interceptin the analysis above is hardly a meaningful quantity at all, and the three-star significance of it is certainly irrelevant. If you are bothered by thestars, turn them off with options(show.signif.stars=FALSE).

6.2 Residuals and fitted values 113

Residual standard error: 0.2167 on 21 degrees of freedom(1 observation deleted due to missingness)

This is the residual variation, an expression of the variation of the ob-servations around the regression line, estimating the model parameterσ. The model is not fitted to the entire data set because one value ofshort.velocity is missing.

Multiple R-squared: 0.1737, Adjusted R-squared: 0.1343

The first item above is R2, which in a simple linear regression may be rec-ognized as the squared Pearson correlation coefficient (see Section 6.4.1);that is, R2 = r2. The other one is the adjusted R2; if you multiply it by100%, it can be interpreted as “% variance reduction” (this can, in fact,become negative).

F-statistic: 4.414 on 1 and 21 DF, p-value: 0.0479

This is an F test for the hypothesis that the regression coefficient is zero.This test is not really interesting in a simple linear regression analysis sinceit just duplicates information already given — it becomes more interestingwhen there is more than one explanatory variable. Notice that it gives theexact same result as the t test for a zero slope. In fact, the F test is identicalto the square of the t test: 4.414 = (2.101)2. This is true in any model with1 degree of freedom.

We will see later how to draw residual plots and plots of data with confi-dence and prediction limits. First, we draw just the points and the fittedline. Figure 6.1 has been constructed as follows:

> plot(blood.glucose,short.velocity)> abline(lm(short.velocity~blood.glucose))

abline, meaning (a, b)-line, draws lines based on the intercept andslope, a and b, respectively. It can be used with scalar values as inabline(1.1,0.022), but conveniently it can also extract the informa-tion from a linear model fitted to data with lm.

6.2 Residuals and fitted values

We have seen how summary can be used to extract information aboutthe results of a regression analysis. Two further extraction functions arefitted and resid. They are used as follows. For convenience, wefirst store the value returned by lm under the name lm.velo (short for“velocity”, but you could of course use any other name).

114 6. Regression and correlation

5 10 15 20

1.0

1.2

1.4

1.6

1.8

blood.glucose

short

.velo

city

Figure 6.1. Scatterplot with regression line.

> lm.velo <- lm(short.velocity~blood.glucose)> fitted(lm.velo)

1 2 3 4 5 6 71.433841 1.335010 1.275711 1.526084 1.255945 1.214216 1.302066

8 9 10 11 12 13 141.341599 1.262534 1.365758 1.244964 1.212020 1.515103 1.429449

15 17 18 19 20 21 221.244964 1.190057 1.324029 1.372346 1.451411 1.389916 1.205431

23 241.291085 1.306459> resid(lm.velo)

1 2 3 4 50.326158532 0.004989882 -0.005711308 -0.056084062 0.014054962

6 7 8 9 100.275783754 0.007933665 -0.251598875 -0.082533795 -0.145757649

11 12 13 14 150.005036223 -0.022019994 0.434897199 -0.149448964 0.275036223

17 18 19 20 21-0.070057471 0.045971143 -0.182346406 -0.401411486 -0.069916424

22 23 24-0.175431237 -0.171085074 0.393541161

The function fitted returns fitted values — the y-values that youwould expect for the given x-values according to the best-fitting straight

6.2 Residuals and fitted values 115

line; in the present case, 1.098+0.0220*blood.glucose. The resid-uals shown by resid is the difference between this and the observedshort.velocity.

Note that the fitted values and residuals are labelled with the row namesof the thuesen data frame. Notice in particular that they do not containobservation no. 16, which had a missing value in the response variable.

It is necessary to discuss some awkward aspects that arise when there aremissing values in data.

To put the fitted line on the plot, you might, although it is easier to useabline(lm.velo), get the idea of doing it with lines, but

> plot(blood.glucose,short.velocity)> lines(blood.glucose,fitted(lm.velo))Error in xy.coords(x, y) : ’x’ and ’y’ lengths differCalls: lines -> lines.default -> plot.xy -> xy.coords

which is true. There are 24 observations but only 23 fitted values becauseone of the short.velocity values is NA. Notice, incidentally, that theerror occurs within a series of nested function calls, which are being listedalong with the error message to reduce confusion.

What we needed was blood.glucose, but only for those patients whoseshort.velocity has been recorded.

> lines(blood.glucose[!is.na(short.velocity)],fitted(lm.velo))

Recall that the is.na function yields a vector that is TRUE wherever theargument is NA (missing). One advantage to this method is that the fittedline does not extend beyond the range of data. The technique works butbecomes clumsy if there are missing values in several variables:

…blood.glucose[!is.na(short.velocity) & !is.na(blood.glucose)]…

It becomes easier with the function complete.cases, which can findobservations that are nonmissing on several variables or across an entiredata frame.

> cc <- complete.cases(thuesen)

We could then attach thuesen[cc,] and work on from there. How-ever, there is a better alternative available: You can use the na.excludemethod for NA handling. This can be set either as an argument to lm or asan option; that is,

> options(na.action=na.exclude)> lm.velo <- lm(short.velocity~blood.glucose)

116 6. Regression and correlation

5 10 15 20

1.0

1.2

1.4

1.6

1.8

blood.glucose

short

.velo

city

Figure 6.2. Scatterplot of short.velocity versus blood.glucose with fittedline and residual line segments.

> fitted(lm.velo)1 2 3 4 5 6 7

1.433841 1.335010 1.275711 1.526084 1.255945 1.214216 1.3020668 9 10 11 12 13 14

1.341599 1.262534 1.365758 1.244964 1.212020 1.515103 1.42944915 16 17 18 19 20 21

1.244964 NA 1.190057 1.324029 1.372346 1.451411 1.38991622 23 24

1.205431 1.291085 1.306459

Notice how the missing observation, no. 16, now appears in the fitted val-ues with a missing fitted value. It is necessary to recalculate the lm.veloobject after changing the option.

To create a plot where residuals are displayed by connecting observationsto corresponding points on the fitted line, you can do the following. Thefinal result will look like Figure 6.2. segments draws line segments; itsarguments are the endpoint coordinates in the order (x1, y1, x2, y2).

> segments(blood.glucose,fitted(lm.velo),+ blood.glucose,short.velocity)

6.3 Prediction and confidence bands 117

1.20 1.25 1.30 1.35 1.40 1.45 1.50

−0.4

−0.2

0.0

0.2

0.4

fitted(lm.velo)

resi

d(lm

.velo

)

Figure 6.3. short.velocity and blood.glucose: residuals versus fitted value.

A simple plot of residuals versus fitted values is obtained as (Figure 6.3)

> plot(fitted(lm.velo),resid(lm.velo))

and we can get an indication of whether residuals might have come froma normal distribution by checking for a straight line on a Q–Q plot (seeSection 4.2.3) as follows (Figure 6.4):

> qqnorm(resid(lm.velo))

6.3 Prediction and confidence bands

Fitted lines are often presented with uncertainty bands around them.There are two kinds of bands, often referred to as the “narrow” and“wide” limits.

The narrow bands, confidence bands, reflect the uncertainty about the lineitself, like the SEM expresses the precision with which a mean is known.If there are many observations, the bands will be quite narrow, reflecting

118 6. Regression and correlation

−2 −1 0 1 2

−0.4

−0.2

0.0

0.2

0.4

Normal Q−Q Plot

Theoretical Quantiles

Sam

ple

Quantil

Figure 6.4. short.velocity and blood.glucose: Q–Q plot of residuals.

a well-determined line. These bands often show a marked curvature sincethe line is better determined near the center of the point cloud. This is afact that can be shown mathematically, but you may also understand itintuitively as follows: The predicted value at x̄ will be ȳ, whatever theslope is, and hence the standard error of the fitted value at that point isthe SEM of the ys. At other values of x, there will also be a contributionfrom the variability of the estimated slope, having increasing influence asyou move away from x̄. Technically, you also need to establish that ȳ andβ̂ are uncorrelated.

The wide bands, prediction bands, include the uncertainty about futureobservations. These bands should capture the majority of the observedpoints and will not collapse to a line as the number of observations in-creases. Rather, the limits approach the true line ±2 standard deviations(for 95% limits). In smaller samples, the bands do curve since they includeuncertainty about the line itself, but not as markedly as the confidencebands. Obviously, these limits rely strongly on the assumption of nor-mally distributed errors with a constant variance, so you should notuse such limits unless you believe that the assumption is a reasonableapproximation for the data at hand.

6.3 Prediction and confidence bands 119

Predicted values, with or without prediction and confidence bands, maybe extracted with the function predict. With no arguments, it just givesthe fitted values:

> predict(lm.velo)1 2 3 4 5 6 7

1.433841 1.335010 1.275711 1.526084 1.255945 1.214216 1.3020668 9 10 11 12 13 14

1.341599 1.262534 1.365758 1.244964 1.212020 1.515103 1.42944915 16 17 18 19 20 21

1.244964 NA 1.190057 1.324029 1.372346 1.451411 1.38991622 23 24

1.205431 1.291085 1.306459

If you add interval="confidence" or interval="prediction",then you get the vector of predicted values augmented with limits. Thearguments can be abbreviated:

> predict(lm.velo,int="c")fit lwr upr

1 1.433841 1.291371 1.5763122 1.335010 1.240589 1.429431…23 1.291085 1.191084 1.39108624 1.306459 1.210592 1.402326> predict(lm.velo,int="p")

fit lwr upr1 1.433841 0.9612137 1.9064692 1.335010 0.8745815 1.795439…23 1.291085 0.8294798 1.75269024 1.306459 0.8457315 1.767186Warning message:In predict.lm(lm.velo, int = "p") :Predictions on current data refer to _future_ responses

fit denotes the expected values, here identical to the fitted values(they need not be; read on). lwr and upr are the lower and upperconfidence limits for the expected values, respectively, the predictionlimits for short.velocity for new persons with these values ofblood.glucose. The warning in this case does not really mean that any-thing is wrong, but there is a pitfall: The limits should not be used forevaluating the observed data to which the line has been fitted. These willtend to lie closer to the line for the extreme x values because those datapoints are the more influential; that is, the prediction bands curve thewrong way.

The best way to add prediction and confidence intervals to a scatterplot isto use the matlines function, which plots the columns of a matrix againsta vector.

120 6. Regression and correlation

There are a few snags to this, however: (a) The blood.glucose valuesare in random order; we do not want line segments connecting pointshaphazardly along the confidence curves; (b) the prediction limits, partic-ularly the lower one, extend outside the plot region; and (c) the matlinescommand needs to be prevented from cycling through line styles andcolours. Notice that the na.exclude setting (p. 115) prevents us fromalso having an observation omitted from the predicted values.

The solution is to predict in a new data frame containing suitable x values(here blood.glucose) at which to predict. It is done as follows:

> pred.frame <- data.frame(blood.glucose=4:20)> pp <- predict(lm.velo, int="p", newdata=pred.frame)> pc <- predict(lm.velo, int="c", newdata=pred.frame)> plot(blood.glucose,short.velocity,+ ylim=range(short.velocity, pp, na.rm=T))> pred.gluc <- pred.frame$blood.glucose> matlines(pred.gluc, pc, lty=c(1,2,2), col="black")> matlines(pred.gluc, pp, lty=c(1,3,3), col="black")

What happens is that we create a new data frame in which the variableblood.glucose contains the values at which we want predictions to bemade. pp and pc are then made to contain the result of predict for thenew data in pred.frame with prediction limits and confidence limits,respectively.

For the plotting, we first create a standard scatterplot, except that we en-sure that it has enough room for the prediction limits. This is obtained bysetting ylim=range(short.velocity, pp, na.rm=T). The func-tion range returns a vector of length 2 containing the minimum andmaximum values of its arguments. We need the na.rm=T argument tocause missing values to be skipped for the range computation; notice thatshort.velocity is included to ensure that points outside the predic-tion limits are not missed (although in this case there are none). Finally,the curves are added, using as x-values the blood.glucose used for theprediction and setting the line types and colours to more sensible values.The final result is seen in Figure 6.5.

6.4 Correlation

A correlation coefficient is a symmetric, scale-invariant measure of associ-ation between two random variables. It ranges from −1 to +1, where theextremes indicate perfect correlation and 0 means no correlation. The signis negative when large values of one variable are associated with smallvalues of the other and positive if both variables tend to be large or small

6.4 Correlation 121

5 10 15 20

0.8

1.0

1.2

1.4

1.6

1.8

2.0

blood.glucose

short

.velo

city

Figure 6.5. Plot with confidence and prediction bands.

simultaneously. The reader should be warned that there are many incor-rect uses of correlation coefficients, particularly when they are used inregression-type settings.

This section describes the computation of parametric and nonparametriccorrelation measures in R.

6.4.1 Pearson correlation

The Pearson correlation is rooted in the two-dimensional normal distri-bution where the theoretical correlation describes the contour ellipses forthe density. If both variables are scaled to have a variance of 1, then acorrelation of zero corresponds to circular contours, whereas the ellipsesbecome narrower and finally collapse into a line segment as the correlationapproaches ±1.

The empirical correlation coefficient is

r =∑(xi − x̄)(yi − ȳ)√

∑(xi − x̄)2 ∑(yi − ȳ)2

122 6. Regression and correlation

It can be shown that |r| will be less than 1 unless there is a perfect linearrelation between xi and yi, and for that reason the Pearson correlation issometimes called the “linear correlation”.

It is possible to test the significance of the correlation by transforming it toa t-distributed variable (the formula is not particularly elucidating so weskip it here), which will be identical with the test obtained from testing thesignificance of the slope of either the regression of y on x or vice versa.

The function cor can be used to compute the correlation between twoor more vectors. However, if it is naively applied to the two vectors inthuesen, the following happens:

> cor(blood.glucose,short.velocity)Error in cor(blood.glucose, short.velocity) :

missing observations in cov/cor

All the elementary statistical functions in R require either that all valuesbe nonmissing or that you explicitly state what should be done with thecases with missing values. For mean, var, sd, and similar one-vector func-tions, you can give the argument na.rm=T to indicate that missing valuesshould be removed before the computation. For cor, you can write

> cor(blood.glucose,short.velocity,use="complete.obs")[1] 0.4167546

The reason that cor does not use na.rm=T like the other functions isthat there are more possibilities than just removing incomplete cases orfailing. If more than two variables are in play, it is also possible to use in-formation from all nonmissing pairs of measurements (this might result ina correlation matrix that is not positive definite, though).

You can obtain the entire matrix of correlations between all variables in adata frame by saying, for instance,

> cor(thuesen,use="complete.obs")blood.glucose short.velocity

blood.glucose 1.0000000 0.4167546short.velocity 0.4167546 1.0000000

Of course, this is more interesting when the data frame contains more thantwo vectors!

However, the calculations above give no indication of whether the correla-tion is significantly different from zero. To that end, you need cor.test.It works simply by specifying the two variables:

6.4 Correlation 123

> cor.test(blood.glucose,short.velocity)

Pearson’s product-moment correlation

data: blood.glucose and short.velocityt = 2.101, df = 21, p-value = 0.0479alternative hypothesis: true correlation is not equal to 095 percent confidence interval:0.005496682 0.707429479sample estimates:

cor0.4167546

We also get a confidence interval for the true correlation. Notice that it isexactly the same p-value as in the regression analysis in Section 6.1 andalso that based on the ANOVA table for the regression model, which isdescribed in Section 7.5.

6.4.2 Spearman’s ρ

As with the one- and two-sample problems, you may be interested innonparametric variants. These have the advantage of not dependingon the normal distribution and, indeed, being invariant to monotonetransformations of the coordinates. The main disadvantage is that its in-terpretation is not quite clear. A popular and simple choice is Spearman’srank correlation coefficient ρ. This is obtained quite simply by replac-ing the observations by their rank and computing the correlation. Underthe null hypothesis of independence between the two variables, the exactdistribution of ρ can be calculated.

Unlike group comparisons where there is essentially one function pernamed test, correlation tests are all grouped into cor.test. There is nospecial spearman.test function. Instead, the test is considered one ofseveral possibilities for testing correlations and is therefore specified viaan option to cor.test:

> cor.test(blood.glucose,short.velocity,method="spearman")

Spearman’s rank correlation rho

data: blood.glucose and short.velocityS = 1380.364, p-value = 0.1392alternative hypothesis: true rho is not equal to 0sample estimates:

rho0.318002

Warning message:

124 6. Regression and correlation

In cor.test.default(blood.glucose, short.velocity, method="spearman"):Cannot compute exact p-values with ties

6.4.3 Kendall’s τ

The third correlation method that you can choose is Kendall’s τ, which isbased on counting the number of concordant and discordant pairs. A pairof points is concordant if the difference in the x-coordinate is of the samesign as the difference in the y-coordinate. For a perfect monotone rela-tion, either all pairs will be concordant or all pairs will be discordant.Under independence, there should be as many concordant pairs as thereare discordant ones.

Since there are many pairs of points to check, this is quite a computation-ally intensive procedure compared with the two others. In small data setssuch as the present one, it does not matter at all, though, and the procedureis generally usable up to at least 5000 observations.

The τ coefficient has the advantage of a more direct interpretation overSpearman’s ρ, but apart from that there is little reason to prefer one overthe other.

> cor.test(blood.glucose,short.velocity,method="kendall")

Kendall’s rank correlation tau

data: blood.glucose and short.velocityz = 1.5604, p-value = 0.1187alternative hypothesis: true tau is not equal to 0sample estimates:

tau0.2350616

Warning message:In cor.test.default(blood.glucose, short.velocity, method="kendall"):Cannot compute exact p-value with ties

Notice that neither of the two nonparametric correlations is significantat the 5% level, which the Pearson correlation is, albeit only borderlinesignificant.

6.5 Exercises

6.1 With the rmr data set, plot metabolic rate versus body weight. Fita linear regression model to the relation. According to the fitted model,

6.5 Exercises 125

what is the predicted metabolic rate for a body weight of 70 kg? Give a95% confidence interval for the slope of the line.

6.2 In the juul data set, fit a linear regression model for the square rootof the IGF-I concentration versus age to the group of subjects over 25 yearsold.

6.3 In the malaria data set, analyze the log-transformed antibody levelversus age. Make a plot of the relation. Do you notice anything peculiar?

6.4 One can generate simulated data from the two-dimensional normaldistribution with a correlation of ρ by the following technique: (a) Gen-erate X as a normal variate with mean 0 and standard deviation 1; (b)generate Y with mean ρX and standard deviation

√1 − ρ2. Use this to

create scatterplots of simulated data with a given correlation. Computethe Spearman and Kendall statistics for some of these data sets.

7Analysis of variance and theKruskal–Wallis test

In this section, we consider comparisons among more than two groupsparametrically, using analysis of variance, as well as nonparametrically,using the Kruskal–Wallis test. Furthermore, we look at two-way analysisof variance in the case of one observation per cell.

7.1 One-way analysis of variance

We start this section with a brief sketch of the theory underlying the one-way analysis of variance. A little bit of notation is necessary. Let xij denoteobservation no. j in group i, so that x35 is the fifth observation in group3; x̄i is the mean for group i, and x̄. is the grand mean (average of allobservations).

We can decompose the observations as

xij = x̄. + (x̄i − x̄.)︸︷︷︸deviation ofgroup mean fromgrand mean

+ (xij − x̄i)︸︷︷︸deviation ofobservation fromgroup mean

informally corresponding to the model

Xij = µ + αi + eij, eij ∼ N(0, σ2)

P. Dalgaard, Introductory Statistics with R,DOI: 10.1007/978-0-387-79054-1_7, © Springer Science+Business Media, LLC 2008

128 7. Analysis of variance and the Kruskal–Wallis test

in which the hypothesis that all the groups are the same implies that allαi are zero. Notice that the error terms eij are assumed to be independentand have the same variance.

Now consider the sums of squares of the underbraced terms, known asvariation within groups

SSDW = ∑i

∑j(xij − x̄i)2

and variation between groups

SSDB = ∑i

∑j(x̄i − x̄.)2 = ∑

ini(x̄i − x̄.)2

It is possible to prove that

SSDB + SSDW = SSDtotal = ∑i

∑j(xij − x̄.)2

That is, the total variation is split into a term describing differences be-tween group means and a term describing differences between individualmeasurements within the groups. One says that the grouping explainspart of the total variation, and obviously an informative grouping willexplain a large part of the variation.

However, the sums of squares can only be positive, so even a completelyirrelevant grouping will always “explain” some part of the variation. Thequestion is how small an amount of explained variation can be before itmight as well be due to chance. It turns out that in the absence of anysystematic differences between the groups, you should expect the sum ofsquares to be partitioned according to the degrees of freedom for eachterm, k −1 for SSDB and N −k for SSDW , where k is the number of groupsand N is the total number of observations.

Accordingly, you can normalize the sums of squares by calculating meansquares:

MSW = SSDW /(N − k)MSB = SSDB /(k − 1)

MSW is the pooled variance obtained by combining the individual groupvariances and thus an estimate of σ2. In the absence of a true group effect,MSB will also be an estimate of σ2, but if there is a group effect, then thedifferences between group means and hence MSB will tend to be larger.Thus, a test for significant differences between the group means can beperformed by comparing two variance estimates. This is why the proce-dure is called analysis of variance even though the objective is to comparethe group means.

7.1 One-way analysis of variance 129

A formal test needs to account for the fact that random variation will causesome difference in the mean squares. You calculate

F = MSB /MSW

so that F is ideally 1, but some variation around that value is expected.The distribution of F under the null hypothesis is an F distribution withk − 1 and N − k degrees of freedom. You reject the hypothesis of identicalmeans if F is larger than the 95% quantile in that F distribution (if thesignificance level is 5%). Notice that this test is one-sided; a very small Fwould occur if the group means were very similar, and that will of coursenot signify a difference between the groups.

Simple analyses of variance can be performed in R using the func-tion lm, which is also used for regression analysis. For more elaborateanalyses, there are also the functions aov and lme (linear mixed ef-fects models, from the nlme package). An implementation of Welch’sprocedure, relaxing the assumption of equal variances and generaliz-ing the unequal-variance t test, is implemented in oneway.test (seeSection 7.1.2).

The main example in this section is the “red cell folate” data from Alt-man (1991, p. 208). To use lm, it is necessary to have the data values inone vector and a factor variable (see Section 1.2.8) describing the divisioninto groups. The red.cell.folate data set contains a data frame in theproper format.

> attach(red.cell.folate)> summary(red.cell.folate)

folate ventilationMin. :206.0 N2O+O2,24h:81st Qu.:249.5 N2O+O2,op :9Median :274.0 O2,24h :5Mean :283.23rd Qu.:305.5Max. :392.0

Recall that summary applied to a data frame gives a short summary ofthe distribution of each of the variables contained in it. The format of thesummary is different for numeric vectors and factors, so that provides acheck that the variables are defined correctly.

The category names for ventilation mean “N2O and O2 for 24 hours”,“N2O and O2 during operation”, and “only O2 for 24 hours”.

In the following, the analysis of variance is demonstrated first and then acouple of useful techniques for the presentation of grouped data as tablesand graphs are shown.

130 7. Analysis of variance and the Kruskal–Wallis test

The specification of a one-way analysis of variance is analogous to a re-gression analysis. The only difference is that the descriptive variable needsto be a factor and not a numeric variable. We calculate a model object usinglm and extract the analysis of variance table with anova.

> anova(lm(folate~ventilation))Analysis of Variance Table

Response: folateDf Sum Sq Mean Sq F value Pr(>F)

ventilation 2 15516 7758 3.7113 0.04359 *Residuals 19 39716 2090—Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Here we have SSDB and MSB in the top line and SSDW and MSW in thesecond line.

In statistics textbooks, the sums of squares are most often labelled “be-tween groups” and “within groups”. Like most other statistical software,R uses slightly different labelling. Variation between groups is labelledby the name of the grouping factor (ventilation), and variation withingroups is labelled Residual. ANOVA tables can be used for a wide rangeof statistical models, and it is convenient to use a format that is less linkedto the particular problem of comparing groups.

For a further example, consider the data set juul, introduced in Sec-tion 4.1. Notice that the tanner variable in this data set is a numericvector and not a factor. For purposes of tabulation, this makes little dif-ference, but it would be a serious error to use it in this form in an analysisof variance:

> attach(juul)> anova(lm(igf1~tanner)) ## WRONG!Analysis of Variance Table

Response: igf1Df Sum Sq Mean Sq F value Pr(>F)

tanner 1 10985605 10985605 686.07 < 2.2e-16 ***Residuals 790 12649728 16012—Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

This does not describe a grouping of data but a linear regression on thegroup number! Notice the telltale 1 DF for the effect of tanner.

Things can be fixed as follows:

> juul$tanner <- factor(juul$tanner,+ labels=c("I","II","III","IV","V"))

7.1 One-way analysis of variance 131

> detach(juul)> attach(juul)> summary(tanner)

I II III IV V NA’s515 103 72 81 328 240> anova(lm(igf1~tanner))Analysis of Variance Table

Response: igf1Df Sum Sq Mean Sq F value Pr(>F)

tanner 4 12696217 3174054 228.35 < 2.2e-16 ***Residuals 787 10939116 13900—Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

We needed to reattach the juul data frame in order to use the changeddefinition. An attached data frame is effectively a separate copy of it(although it does not take up extra space as long as the original isunchanged). The Df column now has an entry of 4 for tanner, as itshould.

7.1.1 Pairwise comparisons and multiple testing

If the F test shows that there is a difference between groups, the ques-tion quickly arises of where the difference lies. It becomes necessary tocompare the individual groups.

Part of this information can be found in the regression coefficients. You canuse summary to extract regression coefficients with standard errors and ttests. These coefficients do not have their usual meaning as the slope of aregression line but have a special interpretation, which is described below.

> summary(lm(folate~ventilation))Call:lm(formula = folate ~ ventilation)Residuals:

Min 1Q Median 3Q Max-73.625 -35.361 -4.444 35.625 75.375Coefficients:

Estimate Std. Error t value Pr(>|t|)(Intercept) 316.62 16.16 19.588 4.65e-14 ***ventilationN2O+O2,op -60.18 22.22 -2.709 0.0139 *ventilationO2,24h -38.62 26.06 -1.482 0.1548—Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 45.72 on 19 degrees of freedomMultiple R-squared: 0.2809, Adjusted R-squared: 0.2052F-statistic: 3.711 on 2 and 19 DF, p-value: 0.04359

132 7. Analysis of variance and the Kruskal–Wallis test

The interpretation of the estimates is that the intercept is the mean in thefirst group (N2O+O2,24h), whereas the two others describe the differencebetween the relevant group and the first one.

There are multiple ways of representing the effect of a factor variable inlinear models (and one-way analysis of variance is the simplest example ofa linear model with a factor variable). The representations are in terms ofcontrasts, the choice of which can be controlled either by global options oras part of the model formula. We do not go deeply into this but just men-tion that the contrasts used by default are the so-called treatment contrasts,in which the first group is treated as a baseline and the other groups aregiven relative to that. Concretely, the analysis is performed as a multipleregression analysis (see Chapter 11) by introducing two dummy variables,which are 1 for observations in the relevant group and 0 elsewhere.

Among the t tests in the table, you can immediately find a test for the hy-pothesis that the first two groups have the same true mean (p = 0.0139)and also whether the first and the third might be identical (p = 0.1548).However, a comparison of the last two groups cannot be found. This canbe overcome by modifying the factor definition (see the help page forrelevel), but that gets tedious when there are more than a few groups.

If we want to compare all groups, we ought to correct for multiple testing.Performing many tests will increase the probability of finding one of themto be significant; that is, the p-values tend to be exaggerated. A commonadjustment method is the Bonferroni correction, which is based on the factthat the probability of observing at least one of n events is less than thesum of the probabilities for each event. Thus, by dividing the significancelevel by the number of tests or, equivalently, multiplying the p-values, weobtain a conservative test where the probability of a significant result is lessthan or equal to the formal significance level.

A function called pairwise.t.test computes all possible two-groupcomparisons. It is also capable of making adjustments for multiplecomparisons and works like this:

> pairwise.t.test(folate, ventilation, p.adj="bonferroni")

Pairwise comparisons using t tests with pooled SD

data: folate and ventilation

N2O+O2,24h N2O+O2,opN2O+O2,op 0.042 -O2,24h 0.464 1.000

P value adjustment method: bonferroni

7.1 One-way analysis of variance 133

The output is a table of p-values for the pairwise comparisons. Here, thep-values have been adjusted by the Bonferroni method, where the unad-justed values have been multiplied by the number of comparisons, namely3. If that results in a value bigger than 1, then the adjustment proceduresets the adjusted p-value to 1.

The default method for pairwise.t.test is actually not the Bonferronicorrection but a variant due to Holm. In this method, only the smallest pneeds to be corrected by the full number of tests, the second smallest iscorrected by n − 1, etc., unless that would make it smaller than the pre-vious one, since the order of the p-values should be unaffected by theadjustment.

> pairwise.t.test(folate,ventilation)

Pairwise comparisons using t tests with pooled SD

data: folate and ventilation

N2O+O2,24h N2O+O2,opN2O+O2,op 0.042 -O2,24h 0.310 0.408

P value adjustment method: holm

7.1.2 Relaxing the variance assumption

The traditional one-way ANOVA requires an assumption of equal vari-ances for all groups. There is, however, an alternative procedure thatdoes not require that assumption. It is due to Welch and similar to theunequal-variances t test. This has been implemented in the oneway.testfunction:

> oneway.test(folate~ventilation)

One-way analysis of means (not assuming equal variances)

data: folate and ventilationF = 2.9704, num df = 2.000, denom df = 11.065, p-value = 0.09277

In this case, the p-value increased to a nonsignificant value, presumablyrelated to the fact that the group that seems to differ from the two othersalso has the largest variance.

It is also possible to perform the pairwise t tests so that they do not usea common pooled standard deviation. This is controlled by the argumentpool.sd.

134 7. Analysis of variance and the Kruskal–Wallis test

> pairwise.t.test(folate,ventilation,pool.sd=F)

Pairwise comparisons using t tests with non-pooled SD

data: folate and ventilation

N2O+O2,24h N2O+O2,opN2O+O2,op 0.087 -O2,24h 0.321 0.321

P value adjustment method: holm

Again, it is seen that the significance disappears as we remove theconstraint on the variances.

7.1.3 Graphical presentation

Of course, there are many ways to present grouped data. Here we createa somewhat elaborate plot where the raw data are plotted as a stripchartand overlaid with an indication of means and SEMs (Figure 7.1):

> xbar <- tapply(folate, ventilation, mean)> s <- tapply(folate, ventilation, sd)> n <- tapply(folate, ventilation, length)> sem <- s/sqrt(n)> stripchart(folate~ventilation, method="jitter",+ jitter=0.05, pch=16, vert=T)> arrows(1:3,xbar+sem,1:3,xbar-sem,angle=90,code=3,length=.1)> lines(1:3,xbar,pch=4,type="b",cex=2)

Here we used pch=16 (small plotting dots) in stripchart and putvertical=T to make the “strips” vertical.

The error bars have been made with arrows, which adds arrows to aplot. We slightly abuse the fact that the angle of the arrowhead is ad-justable to create the little crossbars at either end. The first four argumentsspecify the endpoints, (x1, y1, x2, y2); the angle argument gives the an-gle between the lines of the arrowhead and shaft, here set to 90◦; andlength is the length of the arrowhead (in inches on a printout). Finally,code=3 means that the arrow should have a head at both ends. Note thatthe x-coordinates of the stripcharts are simply the group numbers.

The indication of averages and the connecting lines are done with lines,where type="b" (both) means that both points and lines are printed,leaving gaps in the lines to make room for the symbols. pch=4 is a cross,and cex=2 requests that the symbols be drawn in double size.

7.1 One-way analysis of variance 135

N2O+O2,24h N2O+O2,op O2,24h

200

250

300

350

Figure 7.1. “Red cell folate” data with x̄ ± 1 SEM.

It is debatable whether you should draw the plot using 1 SEM as is donehere or whether perhaps it is better to draw proper confidence intervals forthe means (approximately 2 SEM), or maybe even SD instead of SEM. Thelatter point has to do with whether the plot is to be used in a descriptiveor an analytical manner. Standard errors of the mean are not useful fordescribing the distributions in the groups; they only say how preciselythe mean is determined. On the other hand, SDs do not enable the readerto see at a glance which groups are significantly different.

In many fields it appears to have become the tradition to use 1 SEM“because they are the smallest”; that is, it makes differences look moredramatic. Probably, the best thing to do is to follow the traditions in therelevant field and “calibrate your eyeballs” accordingly.

One word of warning, though: At small group sizes, the rule of thumb thatthe confidence interval is the mean ± 2 SEM becomes badly misleading.At a group size of 2, it actually has to be 12.7 SEM! That is a correctionheavily dependent on data having the normal distribution. If you havesuch small groups, it may be advisable to use a pooled SD for the entiredata set rather than the group-specific SDs. This does, of course, require

136 7. Analysis of variance and the Kruskal–Wallis test

that you can reasonably assume that the true standard deviation actuallyis the same in all groups.

7.1.4 Bartlett’s test

Testing whether the distribution of a variable has the same variance inall groups can be done using Bartlett’s test, although like the F test forcomparing two variances, it is rather nonrobust against departures fromthe assumption of normal distributions. As in var.test, it is assumedthat the data are from independent groups. The procedure is performedas follows:

> bartlett.test(folate~ventilation)

Bartlett test of homogeneity of variances

data: folate by ventilationBartlett’s K-squared = 2.0951, df = 2, p-value = 0.3508

That is, in this case, nothing in the data contradicts the assumption ofequal variances in the three groups.

7.2 Kruskal–Wallis test

A nonparametric counterpart of a one-way analysis of variance is theKruskal–Wallis test. As in the Wilcoxon two-sample test (see Section 5.5),data are replaced with their ranks without regard to the grouping, onlythis time the test is based on the between-group sum of squares calcu-lated from the average ranks. Again, the distribution of the test statisticcan be worked out based on the idea that, under the hypothesis of irrel-evant grouping, the problem reduces to a combinatorial one of samplingthe within-group ranks from a fixed set of numbers.

You can make R calculate the Kruskal–Wallis test as follows:

> kruskal.test(folate~ventilation)

Kruskal-Wallis rank sum test

data: folate by ventilationKruskal-Wallis chi-squared = 4.1852, df = 2, p-value = 0.1234

It is seen that there is no significant difference using this test. This shouldnot be too surprising in view of the fact that the F test in the one-way anal-ysis of variance was only borderline significant. Also, the Kruskal–Wallis

7.3 Two-way analysis of variance 137

test is less efficient than its parametric counterpart if the assumptionshold, although it does not invariably give a larger p-value.

7.3 Two-way analysis of variance

One-way analysis of variance deals with one-way classifications of data.It is also possible to analyze data that are cross-classified according toseveral criteria. When a cross-classified design is balanced, then you canalmost read the entire statistical analysis from a single analysis of vari-ance table, and that table generally consists of items that are simple tocompute, which was very important before the computer era. Balanced-ness is a concept that is hard to define exactly; for a two-way classification,a sufficient condition is that the cell counts be equal, but there are otherbalanced designs.

Here we restrict ourselves to the case of a single observation per cell.This typically arises from having multiple measurements on the sameexperimental unit and in this sense generalizes the paired t test.

Let xij denote the observation in row i and column j of the m×n table. Thisis similar to the notation used for one-way analysis of variance, but noticethat there is now a connection between observations with the same j, sothat it makes sense to look at both row averages x̄i· and column averagesx̄·j.

Consequently, it now makes sense to look at both variation between rows

SSDR = n ∑i(x̄i· − x̄..)2

and variation between columns

SSDC = m ∑j(x̄·j − x̄..)2

Subtracting these two from the total variation leaves the residual variation,which works out as

SSDres = ∑i

∑j(xij − x̄i· − x̄·j + x̄..)2

This corresponds to a statistical model in which it is assumed that theobservations are composed of a general level, a row effect, and a columneffect plus a noise term:

Xij = µ + αi + β j + eij eij ∼ N(0, σ2)

138 7. Analysis of variance and the Kruskal–Wallis test

The parameters of this model are not uniquely defined unless we imposesome restriction on the parameters. If we impose ∑ αi = 0 and ∑ β j = 0,then the estimates of αi, β j, and µ turn out to be x̄i· − x̄.., x̄·j − x̄.., and x̄…

Dividing the sums of squares by their respective degrees of freedom m−1for SSDR, n − 1 for SSDC , and (m − 1)(n − 1) for SSDres, we get a set ofmean squares. F tests for no row and column effect can be carried out bydividing the respective mean squares by the residual mean square.

It is important to notice that this works out so nicely only because of thebalanced design. If you have a table with “holes” in it, the analysis is con-siderably more complicated. The simple formulas for the sum of squaresare no longer valid and, in particular, the order independence is lost, sothat there is no longer a single SSDC but ones with and without adjustingfor row effects.

To perform a two-way ANOVA, it is necessary to have data in one vec-tor, with the two classifying factors parallel to it. We consider an exampleconcerning heart rate after administration of enalaprilate (Altman, 1991,p. 327). Data are found in this form in the heart.rate data set:

> attach(heart.rate)> heart.rate

hr subj time1 96 1 02 110 2 03 89 3 04 95 4 05 128 5 06 100 6 07 72 7 08 79 8 09 100 9 010 92 1 3011 106 2 3012 86 3 3013 78 4 3014 124 5 3015 98 6 3016 68 7 3017 75 8 3018 106 9 3019 86 1 6020 108 2 6021 85 3 6022 78 4 6023 118 5 6024 100 6 6025 67 7 6026 74 8 6027 104 9 60

7.3 Two-way analysis of variance 139

28 92 1 12029 114 2 12030 83 3 12031 83 4 12032 118 5 12033 94 6 12034 71 7 12035 74 8 12036 102 9 120

If you look inside the heart.rate.R file in the data directory of theISwR package, you will see that the actual definition of the data frame is

heart.rate <- data.frame(hr = c(96,110,89,95,128,100,72,79,100,92,106,86,78,124,98,68,75,106,86,108,85,78,118,100,67,74,104,92,114,83,83,118,94,71,74,102),

subj=gl(9,1,36),time=gl(4,9,36,labels=c(0,30,60,120)))

The gl (generate levels) function is specially designed for generating pat-terned factors for balanced experimental designs. It has three arguments:the number of levels, the block length (how many times each level shouldrepeat), and the total length of the result. The two patterns in the dataframe are thus

> gl(9,1,36)[1] 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1 2 3 4[32] 5 6 7 8 9Levels: 1 2 3 4 5 6 7 8 9> gl(4,9,36,labels=c(0,30,60,120))[1] 0 0 0 0 0 0 0 0 0 30 30 30 30 30 30[16] 30 30 30 60 60 60 60 60 60 60 60 60 120 120 120[31] 120 120 120 120 120 120Levels: 0 30 60 120

Once the variables have been defined, the two-way analysis of variance isspecified simply by

> anova(lm(hr~subj+time))Analysis of Variance Table

Response: hrDf Sum Sq Mean Sq F value Pr(>F)

subj 8 8966.6 1120.8 90.6391 4.863e-16 ***time 3 151.0 50.3 4.0696 0.01802 *Residuals 24 296.8 12.4—Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

140 7. Analysis of variance and the Kruskal–Wallis test

100

110

120

130

time

mean o

f h

0 30 60 120

subj

529613487

Figure 7.2. Interaction plot of heart-rate data.

Interchanging subj and time in the model formula (hr~time+subj)yields exactly the same analysis except for the order of the rows of theANOVA table. This is because we are dealing with a balanced design (acomplete two-way table with no missing values). In unbalanced cases, thefactor order will matter.

7.3.1 Graphics for repeated measurements

At least for your own use, it is useful to plot a “spaghettigram” of the data;that is, a plot where data from the same subject are connected with lines.To this end, you can use the function interaction.plot, which graphsthe values against one factor while connecting data for the other factorwith line segments to form traces.

> interaction.plot(time, subj, hr)

In fact there is a fourth argument, which specifies what should be done incase, there is more than one observation per cell. By default, the mean istaken, which is the reason why the y-axis in Figure 7.2 reads “mean of hr”.

7.4 The Friedman test 141

If you prefer to have the values plotted according to the times of measure-ment (which are not equidistant in this example), you could instead write(resulting plot not shown)

> interaction.plot(ordered(time),subj,hr)

7.4 The Friedman test

A nonparametric counterpart of two-way analysis of variance exists forthe case with one observation per cell. Friedman’s test is based on rankingobservations within each row assuming that if there is no column effect thenall orderings should be equally likely. A test statistic based on the columnsum of squares can be calculated and normalized to give a χ2-distributedtest statistic.

In the case of two columns, the Friedman test is equivalent to the sign test,in which one uses the binomial distribution to test for equal probabili-ties of positive and negative differences within pairs. This is a rather lesssensitive test than the Wilcoxon signed-rank test discussed in Section 5.2.

Practical application of the Friedman test is as follows:

> friedman.test(hr~time|subj,data=heart.rate)

Friedman rank sum test

data: hr and time and subjFriedman chi-squared = 8.5059, df = 3, p-value = 0.03664

Notice that the blocking factor is specified in a model formula using thevertical bar, which may be read as “time within subj”. It is seen that thetest is not quite as strongly significant as the parametric counterpart. Thisis unsurprising since the latter test is more powerful when its assumptionsare met.

7.5 The ANOVA table in regression analysis

We have seen the use of analysis of variance tables in grouped and cross-classified experimental designs. However, their use is not restricted tothese designs but applies to the whole class of linear models (more on thisin Chapter 12).

142 7. Analysis of variance and the Kruskal–Wallis test

The variation between and within groups for a one-way analysis ofvariance generalizes to model variation and residual variation

SSDmodel = ∑i(ŷi − ȳ.)2

SSDres = ∑i(yi − ŷi)2

which partition the total variation ∑i(yi − ȳ.)2. This applies only when themodel contains an intercept; see Section 12.2. The role of the group meansin the one-way classification is taken over by the fitted values ŷi in themore general linear model.

An F test for significance of the model is available in direct analogy withSection 7.1. In simple linear regression, this test is equivalent to testingthat the regression coefficient is zero.

The analysis of variance table corresponding to a regression analysiscan be extracted with the function anova, just as for one- and two-wayanalyses of variance. For the thuesen example, it will look like this:

> attach(thuesen)> lm.velo <- lm(short.velocity~blood.glucose)> anova(lm.velo)Analysis of Variance Table

Response: short.velocityDf Sum Sq Mean Sq F value Pr(>F)

blood.glucose 1 0.20727 0.20727 4.414 0.0479 *Residuals 21 0.98610 0.04696—Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Notice that the F test gives the same p-value as the t test for a zero slopefrom Section 6.1. It is the same F test that gets printed at the end of thesummary output:

…Residual standard error: 0.2167 on 21 degrees of freedomMultiple R-Squared: 0.1737, Adjusted R-squared: 0.1343F-statistic: 4.414 on 1 and 21 DF, p-value: 0.0479

The remaining elements of the three output lines above may also be de-rived from the ANOVA table. “Residual standard error” is the square rootof “Residual mean squares”, namely 0.2167 =

√0.04696. R2 is the propor-

tion of the total sum of squares explained by the regression line, 0.1737 =0.2073/(0.2073 + 0.9861); and, finally, the adjusted R2 is the relative im-provement of the residual variance, 0.1343 = (v − 0.04696)/v, wherev = (0.2073 + 0.9861)/22 = 0.05425 is the variance of short.velocityif the glucose values are not taken into account.

7.6 Exercises 143

7.6 Exercises

7.1 The zelazo data are in the form of a list of vectors, one for each ofthe four groups. Convert the data to a form suitable for the use of lm, andcalculate the relevant test. Consider t tests comparing selected subgroupsor obtained by combining groups.

7.2 In the lung data, do the three measurement methods give systemat-ically different results? If so, which ones appear to be different?

7.3 Repeat the previous exercises using the zelazo and lung data withthe relevant nonparametric tests.

7.4 The igf1 variable in the juul data set is arguably skewed and hasdifferent variances across Tanner groups. Try to compensate for this us-ing logarithmic and square-root transformations, and use the Welch test.However, the analysis is still problematic — why?

8Tabular data

This chapter describes a series of functions designed to analyze tabulardata. Specifically, we look at the functions prop.test, binom.test,chisq.test, and fisher.test.

8.1 Single proportions

Tests of single proportions are generally based on the binomial distribu-tion (see Section 3.3) with size parameter N and probability parameter p.For large sample sizes, this can be well approximated by a normal distri-bution with mean N p and variance N p(1 − p). As a rule of thumb, theapproximation is satisfactory when the expected numbers of “successes”and “failures” are both larger than 5.

Denoting the observed number of “successes” by x, the test for thehypothesis that p = p0 can be based on

u =x − N p0√

N p0(1 − p0)

which has an approximate normal distribution with mean zero and stan-dard deviation 1, or on u2, which has an approximate χ2 distribution with1 degree of freedom.

P. Dalgaard, Introductory Statistics with R,DOI: 10.1007/978-0-387-79054-1_8, © Springer Science+Business Media, LLC 2008

146 8. Tabular data

The normal approximation can be somewhat improved by the Yates correc-tion, which shrinks the observed value by half a unit towards the expectedvalue when calculating u.

We consider an example (Altman, 1991, p. 230) where 39 of 215 randomlychosen patients are observed to have asthma and one wants to test thehypothesis that the probability of a “random patient” having asthma is0.15. This can be done using prop.test:

> prop.test(39,215,.15)

1-sample proportions test with continuity correction

data: 39 out of 215, null probability 0.15X-squared = 1.425, df = 1, p-value = 0.2326alternative hypothesis: true p is not equal to 0.1595 percent confidence interval:0.1335937 0.2408799sample estimates:

p0.1813953

The three arguments to prop.test are the number of positive outcomes,the total number, and the (theoretical) probability parameter that youwant to test for. The latter is 0.5 by default, which makes sense for sym-metrical problems, but this is not the case here. The amount 15% is a bitsynthetic since it is rarely the case that one has a specific a priori value totest for. It is usually more interesting to compute a confidence interval forthe probability parameter, such as is given in the last part of the output.Notice that we have a slightly unfortunate double usage of the symbol pas the probability parameter of the binomial distribution and as the testprobability or p-value.

You can also use binom.test to obtain a test in the binomial distribution.In that way, you get an exact test probability, so it is generally preferableto using prop.test, but prop.test can do more than testing singleproportions. The procedure to obtain the p-value is to calculate the pointprobabilities for all the possible values of x and sum those that are lessthan or equal to the point probability of the observed x.

> binom.test(39,215,.15)

Exact binomial test

data: 39 and 215number of successes = 39, number of trials = 215, p-value = 0.2135alternative hypothesis: true probability … not equal to 0.1595 percent confidence interval:0.1322842 0.2395223sample estimates:

8.2 Two independent proportions 147

probability of success0.1813953

The “exact” confidence intervals at the 0.05 level are actually constructedfrom the two one-sided tests at the 0.025 level. Finding an exact confi-dence interval using two-sided tests is not a well-defined problem (seeExercise 8.5).

8.2 Two independent proportions

The function prop.test can also be used to compare two or more pro-portions. For that purpose, the arguments should be given as two vectors,where the first contains the number of positive outcomes and the secondthe total number for each group.

The theory is similar to that for a single proportion. Consider the dif-ference in the two proportions d = x1/N1 − x2/N2, which will beapproximately normally distributed with mean zero and variance Vp(d) =(1/N1 + 1/N2) × p(1 − p) if the counts are binomially distributed withthe same p parameter. So to test the hypothesis that p1 = p2, plug thecommon estimate p̂ = (x1 + x2)/(n1 + n2) into the variance formula andlook at u = d/

√Vp̂(d), which approximately follows a standard normal

distribution, or look at u2, which is approximately χ2(1)-distributed. AYates-type correction is possible, but we skip the details.

For illustration, we use an example originally due to Lewitt and Machin(Altman, 1991, p. 232):

> lewitt.machin.success <- c(9,4)> lewitt.machin.total <- c(12,13)> prop.test(lewitt.machin.success,lewitt.machin.total)

2-sample test for equality of proportions with continuitycorrection

data: lewitt.machin.success out of lewitt.machin.totalX-squared = 3.2793, df = 1, p-value = 0.07016alternative hypothesis: two.sided95 percent confidence interval:0.01151032 0.87310506sample estimates:

prop 1 prop 20.7500000 0.3076923

148 8. Tabular data

The confidence interval given is for the difference in proportions. The the-ory behind its calculation is similar to that of the test, but there are sometechnical complications, and a different approximation is used.

You can also perform the test without the Yates continuity correction.This is done by adding the argument correct=F. The continuity cor-rection makes the confidence interval somewhat wider than it wouldotherwise be, but notice that it nevertheless does not contain zero. Thus,the confidence interval is contradicting the test, which says that thereis no significant difference between the two groups with a two-sidedtest. The explanation lies in the different approximations, which becomesimportant for tables as sparse as the present one.

If you want to be sure that at least the p-value is correct, you can useFisher’s exact test. We illustrate this using the same data as in the preced-ing section. The test works by making the calculations in the conditionaldistribution of the 2 × 2 table given both the row and column marginals.This can be difficult to envision, but think of it like this: Take 13 whiteballs and 12 black balls (success and failure, respectively), and sample theballs without replacement into two groups of sizes 12 and 13. The num-ber of white balls in the first group obviously defines the whole table, andthe point is that its distribution can be found as a purely combinatorialproblem. The distribution is known as the hypergeometric distribution.

The relevant function is fisher.test, which requires that data be givenin matrix form. This is obtained as follows:

> matrix(c(9,4,3,9),2)[,1] [,2]

[1,] 9 3[2,] 4 9

> lewitt.machin <- matrix(c(9,4,3,9),2)> fisher.test(lewitt.machin)

Fisher’s Exact Test for Count Data

data: lewitt.machinp-value = 0.04718alternative hypothesis: true odds ratio is not equal to 195 percent confidence interval:0.9006803 57.2549701

sample estimates:odds ratio6.180528

Notice that the second column of the table needs to be the number ofnegative outcomes, not the total number of observations.

8.3 k proportions, test for trend 149

Notice also that the confidence interval is for the odds ratio; that is, for theestimate of (p1/(1 − p1))/(p2/(1 − p2)). One can show that if the ps arenot identical, then the conditional distribution of the table depends onlyon the odds ratio, so it is the natural measure of association in connectionwith the Fisher test. The exact distribution of the test can be worked outalso when the odds ratio differs from 1, but there is the same complicationas with binom.test that a two-sided 95% confidence interval must bepasted together from two one-sided 97.5% intervals. This leads to the op-posite inconsistency as with prop.test: The test is (barely) significant,but the confidence interval for the odds ratio includes 1.

The standard χ2 test (see also Section 8.4) in chisq.test works withdata in matrix form, like fisher.test does. For a 2 × 2 table, the test isexactly equivalent to prop.test.

> chisq.test(lewitt.machin)

Pearson’s Chi-squared test with Yates’ continuitycorrection

data: lewitt.machinX-squared = 3.2793, df = 1, p-value = 0.07016

8.3 k proportions, test for trend

Sometimes you want to compare more than two proportions. In thatcase, the categories are often ordered so that you would expect to find adecreasing or increasing trend in the proportions with the group number.

The example used in this section concerns data from a group of womengiving birth where it was recorded whether the child was delivered bycaesarean section and what shoe size the mother used (Altman, 1991,p. 229).

The table looks like this:

> caesar.shoe<4 4 4.5 5 5.5 6+

Yes 5 7 6 7 8 10No 17 28 36 41 46 140

To compare k > 2 proportions, another test based on the normal approx-imation is available. It consists of the calculation of a weighted sum ofsquared deviations between the observed proportions in each group andthe overall proportion for all groups. The test statistic has an approximateχ2 distribution with k − 1 degrees of freedom.

150 8. Tabular data

To use prop.test on a table like caesar.shoe, we need to convert it toa vector of “successes” (which in this case is close to being the opposite)and a vector of “trials”. The two vectors can be computed like this:

> caesar.shoe.yes <- caesar.shoe["Yes",]> caesar.shoe.total <- margin.table(caesar.shoe,2)> caesar.shoe.yes<4 4 4.5 5 5.5 6+5 7 6 7 8 10

> caesar.shoe.total<4 4 4.5 5 5.5 6+22 35 42 48 54 150

Thereafter it is easy to perform the test:

> prop.test(caesar.shoe.yes,caesar.shoe.total)6-sample test for equality of proportions withoutcontinuity correction

data: caesar.shoe.yes out of caesar.shoe.totalX-squared = 9.2874, df = 5, p-value = 0.09814alternative hypothesis: two.sidedsample estimates:

prop 1 prop 2 prop 3 prop 4 prop 5 prop 60.22727273 0.20000000 0.14285714 0.14583333 0.14814815 0.06666667

Warning message:In prop.test(caesar.shoe.yes, caesar.shoe.total) :Chi-squared approximation may be incorrect

It is seen that the test comes out nonsignificant, but the subdivision is re-ally unreasonably fine in view of the small number of caesarean sections.Notice, by the way, the warning about the χ2 approximation being dubi-ous, which is prompted by some cells having an expected count less than5.

You can test for a trend in the proportions using prop.trend.test. Ittakes three arguments: x, n, and score. The first two of these are exactlyas in prop.test, whereas the last one is the score given to the groups,by default simply 1, 2, . . . , k. The basis of the test is essentially a weightedlinear regression of the proportions on the group scores, where we test fora zero slope, which becomes a χ2 test on 1 degree of freedom.

> prop.trend.test(caesar.shoe.yes,caesar.shoe.total)

Chi-squared Test for Trend in Proportions

data: caesar.shoe.yes out of caesar.shoe.total ,using scores: 1 2 3 4 5 6X-squared = 8.0237, df = 1, p-value = 0.004617

8.4 r × c tables 151

So if we assume that the effect of shoe size is linear in the group score,then we can see a significant difference. This kind of assumption shouldnot be thought of as something that must hold for the test to be valid.Rather, it indicates the rough type of alternative to which the test shouldbe sensitive.

The effect of using a trend test can be viewed as an approximate subdivi-sion of the test for equal proportions (χ2 = 9.29) into a contribution fromthe linear effect (χ2 = 8.02) on 1 degree of freedom and a contributionfrom deviations from the linear trend (χ2 = 1.27) on 4 degrees of freedom.So you could say that the test for equal proportions is being diluted orwastes degrees of freedom on testing for deviations in a direction we arenot really interested in.

8.4 r × c tables

For the analysis of tables with more than two classes on both sides, youcan use chisq.test or fisher.test, although you should note thatthe latter can be very computationally demanding if the cell counts arelarge and there are more than two rows or columns. We have already seenchisq.test in a simple example, but with larger tables, some additionalfeatures are of interest.

An r × c table looks like this:n11 n12 · · · n1c n1·n21 n22 · · · n2c n2·

……

nr1 nr2 · · · nrc nr·n·1 n·2 · · · n·c n··

Such a table can arise from several different sampling plans, and thenotion of “no relation between rows and columns” is correspondingly dif-ferent. The total in each row might be fixed in advance, and you would beinterested in testing whether the distribution over columns is the samefor each row, or vice versa if the column totals were fixed. It might alsobe the case that only the total number is chosen and the individuals aregrouped randomly according to the row and column criteria. In the lattercase, you would be interested in testing the hypothesis of statistical inde-pendence, that the probability of an individual falling into the ijth cell isthe product pi·p·j of the marginal probabilities. However, the analysis ofthe table turns out to be the same in all cases.

152 8. Tabular data

If there is no relation between rows and columns, then you would expectto have the following cell values:

Eij =ni· × n·j

n··This can be interpreted as distributing each row total according to the pro-portions in each column (or vice versa) or as distributing the grand totalaccording to the products of the row and column proportions.

The test statistic

X2 = ∑(O − E)2

has an approximate χ2 distribution with (r − 1) × (c − 1) degrees of free-dom. Here the sum is over the entire table and the ij indices have beenomitted. O denotes the observed values and E the expected values asdescribed above.

We consider the table with caffeine consumption and marital status fromSection 4.5 and compute the χ2 test:

> caff.marital <- matrix(c(652,1537,598,242,36,46,38,21,218+ ,327,106,67),+ nrow=3,byrow=T)> colnames(caff.marital) <- c("0","1-150","151-300",">300")> rownames(caff.marital) <- c("Married","Prev.married","Single")> caff.marital

0 1-150 151-300 >300Married 652 1537 598 242Prev.married 36 46 38 21Single 218 327 106 67> chisq.test(caff.marital)

Pearson’s Chi-squared test

data: caff.maritalX-squared = 51.6556, df = 6, p-value = 2.187e-09

The test is highly significant, so we can safely conclude that the data con-tradict the hypothesis of independence. However, you would generallyalso like to know the nature of the deviations. To that end, you can look atsome extra components of the return value of chisq.test.

Notice that chisq.test (just like lm) actually returns more informationthan what is commonly printed:

8.5 Exercises 153

> chisq.test(caff.marital)$expected0 1-150 151-300 >300

Married 705.83179 1488.01183 578.06533 257.09105Prev.married 32.85648 69.26698 26.90895 11.96759Single 167.31173 352.72119 137.02572 60.94136> chisq.test(caff.marital)$observed

0 1-150 151-300 >300Married 652 1537 598 242Prev.married 36 46 38 21Single 218 327 106 67

These two tables may then be scrutinized to see where the differences lie.It is often useful to look at a table of the contributions from each cell to thetotal χ2. Such a table cannot be directly extracted, but it is easy to calculate:

> E <- chisq.test(caff.marital)$expected> O <- chisq.test(caff.marital)$observed> (O-E)^2/E

0 1-150 151-300 >300Married 4.1055981 1.612783 0.6874502 0.8858331Prev.married 0.3007537 7.815444 4.5713926 6.8171090Single 15.3563704 1.875645 7.0249243 0.6023355

There are some large contributions, particularly from too many “abstain-ing” singles, and the distribution among previously married is shiftedin the direction of a larger intake — insofar as they consume caffeine atall. Still, it is not easy to find a simple description of the deviation fromindependence in these data.

You can also use chisq.test directly on raw (untabulated) data, hereusing the juul data set from Section 4.5:

> attach(juul)> chisq.test(tanner,sex)

Pearson’s Chi-squared test

data: tanner and sexX-squared = 28.8672, df = 4, p-value = 8.318e-06

It may not really be relevant to test for independence between these par-ticular variables. The definition of Tanner stages is gender-dependent bynature.

8.5 Exercises

8.1 Reconsider the situation of Exercise 3.3, where 10 consecutive pa-tients had operations without complications and the expected rate was

154 8. Tabular data

20%. Calculate the relevant one-sided test in the binomial distribution.How large a sample (still with zero complications) would be necessary toobtain statistical significance?

8.2 In 747 cases of “Rocky Mountain spotted fever” from the west-ern United States, 210 patients died. Out of 661 cases from the easternUnited States, 122 died. Is the difference statistically significant? (See alsoExercise 13.4.)

8.3 Two drugs for the treatment of peptic ulcer were compared (Camp-bell and Machin, 1993, p. 72). The results were as follows:

Healed Not Healed TotalPirenzepine 23 7 30Trithiozine 18 13 31Total 41 20 61

Compute the χ2 test and Fisher’s exact test and discuss the difference.Find an approximate 95% confidence interval for the difference in healingprobability.

8.4 (From “Mathematics 5” exam, University of Copenhagen, Summer1969.) From September 20, 1968, to February 1, 1969, an instructor con-sumed 254 eggs. Every day, he recorded how many eggs broke duringboiling so that the white ran out and how many cracked so that the whitedid not run out. Additionally, he recorded whether the eggs were size Aor size B. From February 4, 1969, until April 10, 1969, he consumed 130eggs, but this time he used a “piercer” to create a small hole in the egg toprevent breaking and cracking. The results were as follows:

Period Size Total Broken CrackedSept. 20–Feb. 1 A 54 4 8Sept. 20–Feb. 1 B 200 15 28Feb. 4–Apr. 10 A 60 4 9Feb. 4–Apr. 10 B 70 1 7

Investigate whether or not the piercer seems to have had an effect.

8.5 Make a plot of the two-sided p-value for testing that the probabilityparameter is x when the observations are 3 successes in 15 trials for xvarying from 0 to 1 in steps of 0.001. Explain what makes the definition ofa two-sided confidence interval difficult.

9Power and the computation ofsample size

A statistical test will not be able to detect a true difference if the samplesize is too small compared with the magnitude of the difference. Whendesigning experiments, the experimenter should try to ensure that a suf-ficient amount of data are collected to be reasonably sure that a differenceof a specified size will be detected. R has methods for doing these calcu-lations in the simple cases of comparing means using one- or two-samplet tests and comparing two proportions.

9.1 The principles of power calculations

This section outlines the theory of power calculations and sample-sizechoice. If you are practically inclined and just need to find the necessarysample size in a particular situation, you can safely skim this section andmove quickly to subsequent sections that contain the actual R calls.

The basic idea of a hypothesis test should be clear by now. A test statistic isdefined, and its value is used to decide whether or not you can accept the(null) hypothesis. Acceptance and rejection regions are set up so that theprobability of getting a test statistic that falls into the rejection region is aspecified significance level (α) if the null hypothesis is true. In the presentcontext, it is useful to stick to this formulation (as opposed to the use ofp-values), as rigid as it might be.

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156 9. Power and the computation of sample size

Since data are sampled at random, there is always a risk of reaching awrong conclusion, and things can go wrong in two ways:

• The hypothesis is correct, but the test rejects it (type I error).

• The hypothesis is wrong, but the test accepts it (type II error).

The risk of a type I error is the significance level. The risk of a type IIerror will depend on the size and nature of the deviation you are tryingto detect. If there is very little difference, then you do not have much ofa chance of detecting it. For this reason, some statisticians disapprove ofterms like “acceptance region” because you can never prove that there isno difference — you can only fail to prove that there is one.

The probability of rejecting a false hypothesis is called the power of thetest, and methods exist for calculating or approximating the power in themost important practical situations. It is inconvenient to talk further aboutthese matters in the abstract, so let us move on to some concrete examples.

9.1.1 Power of one-sample and paired t tests

Consider the case of the comparison of a sample mean to a given value.For example, in a matched trial we wish to test whether the difference be-tween treatment A and treatment B is zero using a paired t test (describedin Chapter 5).

We call the true difference δ. Even if the null hypothesis is not true, we canstill work out the distribution of the test statistic, provided the other modelassumptions hold. It is called the noncentral t distribution and depends ona noncentrality parameter as well as the usual degrees of freedom. For thepaired t test, the noncentrality parameter ν is a function of δ, the standarddeviation of differences σ, and the sample size n and equals

ν =δ

σ/√

That is, it is simply the true difference divided by the standard error of themean.

The cumulative noncentral t distribution is available in R simply byadding an ncp argument to the pt function. Figure 9.1 shows a plot ofpt with ncp=3 and df=25. A vertical line indicates the upper end of theacceptance region for a two-sided test at the 0.05 significance level. Theplot was created as follows:

> curve(pt(x,25,ncp=3), from=0, to=6)> abline(v=qt(.975,25))

9.1 The principles of power calculations 157

0 1 2 3 4 5 6

0.0

0.2

0.4

0.6

0.8

1.0

pt(

x, 2

5, ncp

= 3

)

Figure 9.1. The cumulative noncentral t distribution with ν = 3 and 25 degrees offreedom. The vertical line marks the upper significance limit for a two-sided testat the 0.05 level.

The plot shows the main part of the distribution falling in the rejectionregion. The probability of getting a value in the acceptance region can beseen from the graph as the intersection between the curve and the verticalline. (Almost! See Exercise 9.4.) This value is easily calculated as

> pt(qt(.975,25),25,ncp=3)[1] 0.1779891

or roughly 0.18. The power of the test is the opposite, the probability ofgetting a significant result. In this case it is 0.82, and it is of course desirableto have the power as close to 1 as possible.

Notice that the power (traditionally denoted β) depends on four quanti-ties: δ, σ, n, and α. If we fix any three of these, we can adjust the fourth toachieve a given power. This can be used to determine the necessary sam-ple size for an experiment: You need to specify a desired power (β = 0.80and β = 0.90 are common choices), the significance level (usually given byconvention as α = 0.05), a guess of the standard deviation, and δ, whichis known as the “minimal relevant difference” (MIREDIF) or “smallestmeaningful difference” (SMD). This gives an equation that you can solve

158 9. Power and the computation of sample size

for n. The result will generally be a fractional number, which should ofcourse be rounded up.

You can also work on the opposite problem and answer the followingquestion: Given a feasible sample size, how large a difference should youreasonably be able to detect?

Sometimes a shortcut is made by expressing δ relative to the standarddeviation, in which case you would simply set σ to 1.

9.1.2 Power of two-sample t test

Procedures for two-sample t tests are essentially the same as for the one-sample case, except for the calculation of the noncentrality parameter,which is calculated as

ν =δ

σ√

1/n1 + 1/n2

It is generally assumed that the variance is the same in the two groups;that is, using the Welch procedure is not considered. In sample-size calcu-lations, one usually assumes that the group sizes are the same since thatgives the optimal power for a given total number of observations.

9.1.3 Approximate methods

For hand calculations, the power calculations can be considerably simpli-fied by assuming that the standard deviation is known, so that the t testis replaced by a test in the standard normal distribution. The practical ad-vantage is that the approximate formula for the power is easily invertedto give an explicit formula for n. For the one- and two-sample cases, thisworks out as

n =(

Φα/2 + Φβδ/σ

)2one-sample

n = 2 ×(

Φα/2 + Φβδ/σ

)2two-sample, each group

with the Φx denoting quantiles on the normal distribution. This is for two-sided tests. For one-sided tests, use α instead of α/2.

These formulas are often found in textbooks, and some computer pro-grams implement them rather than the more accurate method describedearlier. They do have the advantage of more clearly displaying theoreticalproperties such as the proportionality of δ and 1/

√n for a given power.

9.2 Two-sample problems 159

However, they become numerically unsatisfactory when the degrees offreedom falls below 20 or so.

9.1.4 Power of comparisons of proportions

Suppose you wish to compare the morbidity between two populationsand have to decide the number of persons to sample from each pop-ulation. That is, you plan to perform a comparison of two binomialdistributions as in Section 8.2 using prop.test or chisq.test.

For binomial comparisons, exact power calculations become unwieldy,so we rely on normal approximations to the binomial distribution. Thepower will depend on the probabilities in both groups, not just their dif-ference. As for the t test, the group sizes are assumed to be equal. Thetheoretical derivation of the power proceeds along the same lines as beforeby calculating the distribution of p̂1 − p̂2 when p1 6= p2 and the probabil-ity that it falls outside the range of values compatible with the hypothesisp1 = p2. Assuming equal numbers in the two groups, this leads to thesample-size formula

n =

(Φα/2

√2p(1 − p) + Φβ

√p1(1 − p1) + p2(1 − p2)

|p2 − p1|

)2in which p = (p1 + p2)/2.

Since the method is only approximate, the results are not reliable unlessthe expected number in each of the four cells in the 2 × 2 table is greaterthan 5.

9.2 Two-sample problems

The following example is from Altman (1991, p. 457) and concerns theinfluence of milk on growth. Two groups are to be given different diets,and their growth will be measured. We wish to compute the sample sizethat with a power of 90%, using a two-sided test at the 1% level, can finda difference of 0.5 cm in a distribution with a standard deviation of 2 cm.This is done as follows:

> power.t.test(delta=0.5, sd=2, sig.level = 0.01, power=0.9)

Two-sample t test power calculation

n = 477.8021delta = 0.5

160 9. Power and the computation of sample size

sd = 2sig.level = 0.01

power = 0.9alternative = two.sided

NOTE: n is number in *each* group

delta stands for the “true difference”, and sd is the standard deviation.As is seen, the calculation may return a fractional number of experimentalunits. This would, of course, in practice be rounded up to 478. In the orig-inal reference, a method employing nomograms (a graphical technique) isused and the value obtained is 450. The difference is probably due to diffi-culty in reading the value off the nomogram scale. To know which poweryou would actually obtain with 450 in each group, you would enter

> power.t.test(n=450, delta=0.5, sd=2, sig.level = 0.01)

Two-sample t test power calculation

n = 450delta = 0.5

sd = 2sig.level = 0.01

power = 0.8784433alternative = two.sided

NOTE: n is number in *each* group

The system is that exactly four out of five arguments (power, sig.level,delta, sd, and n) are given, and the function computes the missing one(defaults exist to set sd=1 and sig.level=0.05 — if you wish to havethose calculated, explicitly pass them as NULL). In addition, there are twooptional arguments: alternative, which can be used to specify one-sided tests; and type, which can be used to specify that you want tohandle a one-sample problem. An example of the former is

> power.t.test(delta=0.5, sd=2, sig.level = 0.01, power=0.9,+ alt="one.sided")

Two-sample t test power calculation

n = 417.898delta = 0.5

sd = 2sig.level = 0.01

power = 0.9alternative = one.sided

NOTE: n is number in *each* group

9.3 One-sample problems and paired tests 161

9.3 One-sample problems and paired tests

One-sample problems are handled by adding type="one.sample" inthe call to power.t.test. Similarly, paired tests are specified withtype="paired"; although these reduce to one-sample tests by formingdifferences, the printout will be slightly different.

One pitfall when planning a study with paired data is that the literaturesometimes gives the intra-individual variation as “standard deviation ofrepeated measurements on the same person” or similar. These may be cal-culated by measuring a number of persons several times and computinga common standard deviation within persons. This needs to be multipliedby√

2 to get the standard deviation of differences, which power.t.testrequires for paired data. If, for instance, it is known that the standard de-viation within persons is about 10, and you want to use a paired test at the5% level to detect a difference of 10 with a power of 85%, then you shouldenter

> power.t.test(delta=10, sd=10*sqrt(2), power=0.85, type="paired")

Paired t test power calculation

n = 19.96892delta = 10

sd = 14.14214sig.level = 0.05

power = 0.85alternative = two.sided

NOTE: n is number of *pairs*, sd is std.dev. of

*differences* within pairs

Notice that sig.level=0.05 was taken as the default.

9.4 Comparison of proportions

To calculate sample sizes and related quantities for comparisons ofproportions, you should use power.prop.test. This is based on ap-proximations with the normal distribution, so do not trust the results ifany of the expected cell counts drop below 5.

The use of power.prop.test is analogous to power.t.test, althoughdelta and sd are replaced by the hypothesized probabilities in the twogroups, p1 and p2. Currently, it is not possible to specify that one wantsto consider a one-sample problem.

162 9. Power and the computation of sample size

An example is given in Altman (1991, p. 459) in which two groups areadministered or not administered nicotine chewing gum and the binaryoutcome is smoking cessation. The stipulated values are p1 = 0.15 andp2 = 0.30. We want a power of 85%, and the significance level is thetraditional 5%. Inserting these values yields

> power.prop.test(power=.85,p1=.15,p2=.30)

Two-sample comparison of proportions power calculation

n = 137.6040p1 = 0.15p2 = 0.3

sig.level = 0.05power = 0.85

alternative = two.sided

NOTE: n is number in *each* group

9.5 Exercises

9.1 The ashina trial was designed to have 80% power if the truetreatment difference was 15% and the standard deviation of differenceswithin a person was 20%. Comment on the sample size chosen. (Thepower calculation was originally done using the approximative formula.The imbalance between the group sizes is due to the use of an openrandomization procedure.)

9.2 In a trial comparing a binary outcome between two groups, find therequired number of patients to find an increase in the success rate from60% to 75% with a power of 90%. What happens if we reduce the powerrequirement to 80%?

9.3 Plot the density of the noncentral t distribution for ncp=3 and df=25and compare it with the distribution of t + 3, where t has a central tdistribution with df=25.

9.4 In two-sided tests, there is also a risk of falling into the rejectionregion on the opposite side of the true value. The power calculationsin R only take this into account if you set strict=TRUE. Discuss theconsequences.

9.5 It is occasionally suggested to choose n to “make the true differ-ence significant”. What power would result from choosing n by such aprocedure?

10Advanced data handling

In the preceding text, we have covered a basic set of elementary statisticalprocedures. In the chapters that follow, we begin to discuss more elaboratestatistical modelling.

This is also a natural point to discuss some data handling techniques thatare useful in the practical analysis of data but were too advanced to coverin the first two chapters of the book.

10.1 Recoding variables

This section describes some techniques that are used to construct derivedvariables: grouping quantitative data, combining and renaming factorlevels, and handling date values.

10.1.1 The cut function

You may need to convert a quantitative variable to a grouping factor. Forinstance, you may wish to present your data in terms of age in 5-yeargroups, but age is in the data set as a quantitative variable, recorded aswhole years or perhaps to a finer resolution. This is what the cut function

P. Dalgaard, Introductory Statistics with R,DOI: 10.1007/978-0-387-79054-1_10, © Springer Science+Business Media, LLC 2008

164 10. Advanced data handling

is for. The basic principles are quite simple, although there are some finepoints to be aware of.

The function has two basic arguments: a numeric vector and a vector ofbreakpoints. The latter defines a set of intervals into which the variable isgrouped. You have to specify both ends of all intervals; that is, the totalnumber of break points must be one more than the number of intervals.It is a common mistake to believe that the outer breakpoints can be omit-ted but the result for a value outside all intervals is set to NA. The outerbreakpoints can be chosen as -Inf and Inf, though.

The intervals are left-open, right-closed by default. That is, they includethe breakpoint at the right end of each interval. The lowest breakpointis not included unless you set include.lowest=TRUE, making the firstinterval closed at both ends.

In (e.g.) epidemiology, you are more likely to want groupings like “40–49 years of age”. This opposite convention can be obtained by settingright=FALSE.

Of course, as you switch to left-closed, right-open intervals, the issue oflosing the extreme interval endpoint shifts to the other end of the scale.In that case, include.lowest actually includes the highest value! In theexample below, the difference lies in the inclusion of two subjects whowere exactly 16 years old.

> age <- subset(juul, age >= 10 & age <= 16)$age> range(age)[1] 10.01 16.00> agegr <- cut(age, seq(10,16,2), right=F, include.lowest=T)> length(age)[1] 502> table(agegr)agegr[10,12) [12,14) [14,16]

190 168 144> agegr2 <- cut(age, seq(10,16,2), right=F)> table(agegr2)agegr2[10,12) [12,14) [14,16)

190 168 142

It is sometimes desired to split data into roughly equal-sized groups. Thiscan be achieved by using breakpoints computed by quantile, which wasdescribed in Section 4.1. For instance, you could do

> q <- quantile(age, c(0, .25, .50, .75, 1))> q

0% 25% 50% 75% 100%10.0100 11.3825 12.6400 14.2275 16.0000

10.1 Recoding variables 165

> ageQ <- cut(age, q, include.lowest=T)> table(ageQ)ageQ[10,11.4] (11.4,12.6] (12.6,14.2] (14.2,16]

126 125 125 126

The level names resulting from cut turn out rather ugly at times. Fortu-nately they are easily changed. You can modify each of the factors createdabove as follows:

> levels(ageQ) <- c("1st", "2nd", "3rd", "4th")> levels(agegr) <- c("10-11", "12-13", "14-15")

Frank Harrell’s Hmisc package contains the cut2 function, which simpli-fies some of these matters.

10.1.2 Manipulating factor levels

In Section 1.2.8, we used levels(f)<- …. to change the level set of afactor. Some related tasks will be discussed in this section.

First, notice that the conversion from numeric input and renaming oflevels can be done in one operation:

> pain <- c(0,3,2,2,1)> fpain <- factor(pain,levels=0:3,+ labels=c("none","mild","medium","severe"))

Beware the slightly confusing distinction between levels and labels.The latter end up being the levels of the result, whereas the former refersto the coding of the input vector (pain in this case). That is, levels refersto the input and labels to the output.

If you do not specify a levels argument, the levels will be the sorted,unique values represented in the vector. This is not always desirable whendealing with text variables since the sorting is alphabetical. Consider, forinstance,

> text.pain <- c("none","severe", "medium", "medium", "mild")> factor(text.pain)[1] none severe medium medium mildLevels: medium mild none severe

Another reason for specifying levels is that the default levels, obvi-ously, do not include values that are not present in data. This may or maynot be a problem, but it has consequences for later analyses; for instance,whether tables contain zero entries or whether barplots leave space for theempty columns.

166 10. Advanced data handling

The factor function works on factors as if they were character vectors,so you can reorder the levels as follows

> ftpain <- factor(text.pain)> ftpain2 <- factor(ftpain,+ levels=c("none", "mild", "medium", "severe"))

Another typical task is to combine two or more levels. This is often donewhen groups would otherwise be too small for valid statistical analysis.Say you wish to combine the levels "medium" and "mild" into a sin-gle "intermediate" level. For this purpose, the assignment form oflevels allows the right-hand side to be a list:

> ftpain3 <- ftpain2> levels(ftpain3) <- list(+ none="none",+ intermediate=c("mild","medium"),+ severe="severe")> ftpain3[1] none severe intermediate intermediate[5] intermediateLevels: none intermediate severe

However, it is often easier just to change the level names and give thesame name to several groups:

> ftpain4 <- ftpain2> levels(ftpain4) <- c("none","intermediate","intermediate","severe")> ftpain4[1] none severe intermediate intermediate[5] intermediateLevels: none intermediate severe

The latter method is not quite as general as the former, though. It givesless control over the final ordering of levels.

10.1.3 Working with dates

In epidemiology and survival data, you often deal with time in the formof dates in calendar format. Different formats are used in different placesof the world, and the files you have to read were not necessarily writtenin the same region as the one you are currently in. The "Date" class andassociated conversion routines exist to help you deal with the complexity.

As an example, consider the Estonian stroke study, a preprocessed versionof which is contained in the data frame stroke. The raw data files can befound in the rawdata directory of the ISwR package and read using thefollowing code:

10.1 Recoding variables 167

> stroke <- read.csv2(+ system.file("rawdata","stroke.csv", package="ISwR"),+ na.strings=".")> names(stroke) <- tolower(names(stroke))> head(stroke)sex died dstr age dgn coma diab minf han

1 1 7.01.1991 2.01.1991 76 INF 0 0 1 02 1 <NA> 3.01.1991 58 INF 0 0 0 03 1 2.06.1991 8.01.1991 74 INF 0 0 1 14 0 13.01.1991 11.01.1991 77 ICH 0 1 0 15 0 23.01.1996 13.01.1991 76 INF 0 1 0 16 1 13.01.1991 13.01.1991 48 ICH 1 0 0 1

(You can of course also just substitute the full path to stroke.csv insteadof using the system.file construction.)

In this data set, the two date variables died and dstr (date of stroke) ap-pear as factor variables, which is the standard behaviour of read.table.To convert them to class "Date", we use the function as.Date. This isstraightforward but requires some attention to the date format. The for-mat used here is (day, month, year) separated by a period (dot character),with year given as four digits. This is not a standard format, so we needto specify it explicitly.

> stroke <- transform(stroke,+ died = as.Date(died, format="%d.%m.%Y"),+ dstr = as.Date(dstr, format="%d.%m.%Y"))

Notice the use of “percent-codes” to represent specific parts of the date:%d indicates the day of the month, %m means the month as a number, and%Y means that a four-digit year is used (notice the uppercase Y). The fullset of codes is documented on the help page for strptime.

Internally, dates are represented as the number of days before or after agiven point in time, known as the epoch. Specifically, the epoch is January1, 1970, although this is an implementation detail that should not be reliedupon.

It is possible to perform arithmetic on dates; that is, they behave mostlylike numeric vectors:

> summary(stroke$died)Min. 1st Qu. Median Mean 3rd Qu.

"1991-01-07" "1992-03-14" "1993-01-23" "1993-02-15" "1993-11-04"Max.

"1996-02-22"> summary(stroke$dstr)

Min. 1st Qu. Median Mean 3rd Qu."1991-01-02" "1991-11-08" "1992-08-12" "1992-07-27" "1993-04-30"

Max."1993-12-31"

168 10. Advanced data handling

> summary(stroke$died – stroke$dstr)Min. 1st Qu. Median Mean 3rd Qu. Max. NA’s0.0 8.0 28.0 225.7 268.5 1836.0 338.0

> head(stroke$died – stroke$dstr)Time differences in days[1] 5 NA 145 2 1836 0

Notice that means and quantiles are displayed in date format (even if theyare nonintegers). The count of NA values is not displayed for date variableseven though the date of death is unknown for quite a few patients; this isa bit unfortunate, but it would conflict with a convention that numericalsummaries have the same class as the object that is summarized (so youwould get the count displayed as a date!).

The vector of differences between the two dates is actually an object ofclass "difftime". Such objects can have different units — when basedon dates, it will always be "days", but for other kinds of time vari-ables it can be "hours" or "seconds". Accordingly, it is somewhat badpractice just to treat the vector of differences as a numeric variable. Therecommended procedure is to use as.numeric with an explicit unitsargument.

In the data file, NA for a death date means that the patient did not die be-fore the end of the study on January 1, 1996. Six patients were recorded ashaving died after this date, but since there may well be unrecorded deathsamong the remaining patients, we have to discard these death dates andjust record the patients as alive at the end of the study.

We shall transform the data so that all patients have an end date plus anindicator of what happened at the end date: died or survived.

> stroke <- transform(stroke,+ end = pmin(died, as.Date("1996-1-1"), na.rm = T),+ dead = !is.na(died) & died < as.Date("1996-1-1"))> head(stroke)sex died dstr age dgn coma diab minf han

1 1 1991-01-07 1991-01-02 76 INF 0 0 1 02 1 <NA> 1991-01-03 58 INF 0 0 0 03 1 1991-06-02 1991-01-08 74 INF 0 0 1 14 0 1991-01-13 1991-01-11 77 ICH 0 1 0 15 0 1996-01-23 1991-01-13 76 INF 0 1 0 16 1 1991-01-13 1991-01-13 48 ICH 1 0 0 1

end dead1 1991-01-07 TRUE2 1996-01-01 FALSE3 1991-06-02 TRUE4 1991-01-13 TRUE5 1996-01-01 FALSE6 1991-01-13 TRUE

10.1 Recoding variables 169

The pmin function calculates the minimum, but unlike the min function,which returns a single number, it does so in parallel across multiple vec-tors. The na.rm argument allows NA values to be ignored, so the resultis that wherever died is missing or later than 1996-01-01, the end datebecomes 1996-01-01 and the actual date of death otherwise.

The expression for dead is straightforward, although you should checkthat missing values are treated correctly. (They are. The & operator handlesmissingness such that if one argument is FALSE the result is FALSE, evenif the other is NA.)

Finally, to obtain the observation time for all individuals, we can do

> stroke <- transform(stroke,+ obstime = as.numeric(end – dstr, units="days")/365.25)

in which we pragmatically convert to “epidemiological years” of averagelength. (This cannot be done just by setting units="years". Objects ofclass "difftime" can only have units of "weeks" or less.)

Notice that we performed the transformations in three separate calls totransform. This was not just for the flow of the presentation; each ofthe last two calls refers to variables that were not defined previously. Thetransform function does not allow references to variables defined in thesame call (we could have used within, though; see Section 2.1.8).

Further time classes

R also has classes that represent time to a granularity finer than 1 day.The "POSIXct" class (calendar time according to the POSIX standards)is similar to "Date" except that it counts seconds rather than days, and"POSIXlt" (local time) represents date and time using a structure thatconsists of fields for various components: year, month, day of month,hours, minutes, seconds, and more. Working with such objects involves,by and large, the same issues as for the "Date" class, although with acouple of extra twists related to time zones and Daylight Savings Time.We shall not go deeper into this area here.

10.1.4 Recoding multiple variables

In the previous sections, we had some cases where essentially the sametransformation had to be applied to several variables. The solution inthose cases was simply to repeat the operation, but it can happen that adata set contains many similar variables that all need to be recoded (ques-tionnaire data may, for instance, have dozens of items rated on the same

170 10. Advanced data handling

five-point scale). In such cases, you can make use of the fact that dataframes are fundamentally lists and that lapply and indexing work onthem. For instance, in dealing with the raw stroke data, we could havedone the date handling as follows:

> rawstroke <- read.csv2(+ system.file("rawdata","stroke.csv", package="ISwR"),+ na.strings=".")> ix <- c("DSTR", "DIED")> rawstroke[ix] <- lapply(rawstroke[ix],+ as.Date, format="%d.%m.%Y")> head(rawstroke)SEX DIED DSTR AGE DGN COMA DIAB MINF HAN

Similarly, the four binary variables could be converted to “No/Yes”factors in a single operation.

> ix <- 6:9> rawstroke[ix] <- lapply(rawstroke[ix],+ factor, levels=0:1, labels=c("No","Yes"))

10.2 Conditional calculations

The ifelse function lets you apply different calculations to differentparts of data. For illustration, we use a subset of the stroke data dis-cussed in Section 10.1.3, but we use the “cooked” version contained in theISwR package.

> strokesub <- ISwR::stroke[1:10,2:3]> strokesub

died dstr1 1991-01-07 1991-01-022 <NA> 1991-01-033 1991-06-02 1991-01-084 1991-01-13 1991-01-115 <NA> 1991-01-136 1991-01-13 1991-01-137 1993-12-01 1991-01-148 1991-12-12 1991-01-149 <NA> 1991-01-1510 1993-11-10 1991-01-15

10.3 Combining and restructuring data frames 171

To compute the time on study and the event/censoring indicator neededfor survival models, we can do as follows:

> strokesub <- transform(strokesub,+ event = !is.na(died))> strokesub <- transform(strokesub,+ obstime = ifelse(event, died-dstr, as.Date("1996-1-1") – dstr))> strokesub

died dstr event obstime1 1991-01-07 1991-01-02 TRUE 52 <NA> 1991-01-03 FALSE 18243 1991-06-02 1991-01-08 TRUE 1454 1991-01-13 1991-01-11 TRUE 25 <NA> 1991-01-13 FALSE 18146 1991-01-13 1991-01-13 TRUE 07 1993-12-01 1991-01-14 TRUE 10528 1991-12-12 1991-01-14 TRUE 3329 <NA> 1991-01-15 FALSE 181210 1993-11-10 1991-01-15 TRUE 1030

The way ifelse works is that it takes three arguments: test, yes, andno. All three are vectors of the same length (if not, they will be made so byrecycling). The answer is “stitched together” of pieces of yes and no inthe sense that the yes element is selected wherever test is TRUE and theno element where it is FALSE. When the condition is NA, so is the result.

Notice that both alternatives are computed (exceptions are made for thecases where the condition is all TRUE or all FALSE). This is not usuallya problem in terms of speed, but it does mean that ifelse is not theright tool to use if you want to avoid, for example, taking the logarithmof negative values. Also notice that ifelse discards attributes, includingthe class, so that obstime is not of class "difftime" even though boththe yes and the no part are. This sometimes makes using ifelse moretrouble than it is worth, and it can be preferable simply to use explicitsubsetting operations.

10.3 Combining and restructuring data frames

In this section, we discuss ways of joining data frames either “vertically”(adding records) or “horizontally” (adding variables). We also look atthe issue of converting data with repeated measurements of the samevariables between the “long” and the “wide” formats.

172 10. Advanced data handling

10.3.1 Appending frames

Sometimes data are received from multiple sources and you need to com-bine them to form one bigger data set. In this subsection, we consider thecase where data are combined by “vertical stacking”; that is, you start outwith data frames which refer to separate rows of the result — typicallydifferent subjects. It is required that the data frames contain the same vari-ables, although not necessarily in the same order (this is unlike some otherstatistical systems, which will simply insert missing values for variablesthat are absent in a data set).

To simulate such a situation, suppose that the juul data set had beencollected separately for boys and girls. In that case, the data frames mightnot contain the variable sex, since this is the same for everyone in thesame data frame, and variables that only make sense for one gender mayalso have been omitted for the other group.

> juulgrl <- subset(juul, sex==2, select=-c(testvol,sex))> juulboy <- subset(juul, sex==1, select=-c(menarche,sex))

Notice the use of the select argument to subset. The processing of thisargument replaces column names by column numbers, and the resultingexpression is used to index the data frame. The net effect of the negativeindices is to remove, for example, testvol and sex from juulgrl.

To put the data frames back together, you must first add in the missingvariables

> juulgrl$sex <- factor("F")> juulgrl$testvol <- NA> juulboy$sex <- factor("M")> juulboy$menarche <- NA

and then it is just a matter of using the rbind method for data frames:

> juulall <- rbind(juulboy, juulgrl)> names(juulall)[1] "age" "igf1" "tanner" "testvol" "sex"[6] "menarche"

Notice that rbind uses the column names (so that it does not concatenateunrelated variables even though the order of columns differs in the twodata frames) and that the order of variables in the first data frame “wins”:The result has the variables in the same order as juulboy. Notice alsothat rbind is being smart about factor levels:

> levels(juulall$sex)[1] "M" "F"

10.3 Combining and restructuring data frames 173

10.3.2 Merging data frames

Just as you may have different groups of subjects collected in separatedata sets, you may also have different sorts of data on the same patientscollected separately. For example, you could have one data frame withregistry data, one with clinical biochemistry data, and one with question-naire data. It may work to use cbind to stick the data frames togetherside-by-side, but it could be dangerous: What if the data are not completein all data frames or out of sequence? You typically have to work with aunique subject identification code to avoid mistakes of this sort.

The merge function deals with these issues. It works by matching on oneor several variables from each data frame. By default, this is the set ofvariables that have the same name in both frames (typically, there is avariable called something like ID, which holds the subject identification).Assuming that this default works and that the two data frames are calledrespectively dfx and dfy, the merged frame is computed simply as

merge(dfx, dfy)

However, there may be variables of the same name in both frames. In suchcases, you can add a by argument, which contains the variable name ornames to match on as in

merge(dfx, dfy, by="ID")

Any other variables that appear in both frames will have .x or .y ap-pended to their name in the result. It is recommended to use this format inany case as a safeguard and for readability and explicitness. If the match-ing variable(s) have different names in the two data frames, you can useby.x and by.y.

Matching is not necessarily one-to-one. One of the data sets might for in-stance hold tabular material corresponding to the study population. Thecommon example is mortality tables. In such cases, there is generally amany-to-one relationship between the data frames. More than one subjectin the study population will belong to the table entry for 40–49 year-olds,and the rows of the table will have to be duplicated accordingly duringthe merge.

To illustrate these concepts, we use the data set nickel. This describes acohort of nickel smelting workers in South Wales. The data set ewratescontains a table of the population mortality by year and age group in five-year intervals.

> head(nickel)id icd exposure dob age1st agein ageout

1 3 0 5 1889.019 17.4808 45.2273 92.9808

174 10. Advanced data handling

2 4 162 5 1885.978 23.1864 48.2684 63.27123 6 163 10 1881.255 25.2452 52.9917 54.16444 8 527 9 1886.340 24.7206 47.9067 69.67945 9 150 0 1879.500 29.9575 54.7465 76.84426 10 163 2 1889.915 21.2877 44.3314 62.5413> head(ewrates)year age lung nasal other

1 1931 10 1 0 12692 1931 15 2 0 22013 1931 20 6 0 31164 1931 25 14 0 30245 1931 30 30 1 31886 1931 35 68 1 4165

Suppose we wish to merge these two data sets according to the valuesat entry into the study population. This age is contained in agein, andthe date of entry is computed as dob + agein. You can compute groupcodes corresponding to ewrates as follows:

> nickel <- transform(nickel,+ agr = trunc(agein/5)*5,+ ygr = trunc((dob+agein-1)/5)*5+1)

The trunc function rounds values towards zero. Notice that the agegroups start on values that are evenly divisible by 5, whereas the yeargroups end on such values; this is why the expression for ygr subtracts 1and adds it back after truncation. (Actually this does not matter becauseall enrollment dates were April 1 of 1934, 1939, 1944, or 1949.) Notice alsothat we do not use the same variable names as in ewrates. We could havedone so, but the names age and year would be unintuitive in the contextof the nickel data.

With the age and year groups defined, it is an easy matter to perform themerge. We just need to account for the fact that we have used differentvariable names in the two data frames.

> mrg <- merge(nickel, ewrates,+ by.x=c("agr","ygr"), by.y=c("age","year"))> head(mrg,10)

agr ygr id icd exposure dob age1st agein ageout1 20 1931 273 154 0 1909.500 14.6913 24.7465 55.93022 20 1931 213 162 0 1910.129 14.2018 24.1177 63.04933 20 1931 546 0 0 1909.500 14.4945 24.7465 72.50004 20 1931 574 491 0 1909.729 14.0356 24.5177 70.65925 20 1931 110 0 0 1909.247 14.0302 24.9999 72.75346 20 1931 325 434 0 1910.500 14.0737 23.7465 43.03437 25 1931 56 502 2 1904.500 18.2917 29.7465 51.58478 25 1931 690 420 0 1906.500 17.2206 27.7465 55.12199 25 1931 443 420 0 1905.326 14.5562 28.9204 65.761610 25 1931 137 465 0 1905.386 19.0808 28.8601 74.2794

lung nasal other

10.3 Combining and restructuring data frames 175

1 6 0 31162 6 0 31163 6 0 31164 6 0 31165 6 0 31166 6 0 31167 14 0 30248 14 0 30249 14 0 302410 14 0 3024

We have only described the main function of merge. There are also op-tions to include rows that only exist in one of the two frames (all, all.x,all.y), and it may also be useful to know that the pseudo-variablerow.names will allow matching on row names.

We have discussed the cases of one-to-one and many-to-one matching.Many-to-many is possible but rarely useful. What happens in that caseis that the “Cartesian product” is formed by generating all combinationsof rows from the two frames within each matching set. The extreme caseof many-to-many matching occurs if the by set is empty, which gives aresult with as many rows as the product of the row counts. This sometimessurprises people who expect that the row number will act as an implicitID.

10.3.3 Reshaping data frames

Longitudinal data come in two different forms: a “wide” format, wherethere is a separate column for each time point but only one record percase; and a “long” format, where there are multiple records for each case,one for each time point. The long format is more general since it does notneed to assume that the cases are recorded at the same set of times, butwhen applicable it may be easier to work with data in the wide format,and some statistical functions expect it that way. Other functions expect tofind data in the long format. Either way, there is a need to convert fromone format to another. This is what the reshape function does.

Consider the following data from a randomized study of bone metabolismdata during Tamoxifen treatment after breast cancer. The concentrationof alkaline phosphatase is recorded at baseline and 3, 6, 9, 12, 18, and24 months after treatment start.

> head(alkfos)grp c0 c3 c6 c9 c12 c18 c24

1 1 142 140 159 162 152 175 1482 1 120 126 120 146 134 119 1163 1 175 161 168 164 213 194 221

176 10. Advanced data handling

4 1 234 203 174 197 289 174 1895 1 94 107 146 124 128 98 1146 1 128 97 113 203 NA NA NA

In the simplest uses of reshape, the function will assume that the vari-able names encode the information necessary for reshaping to the longformat. By default, it assumes that variable names are separated from timeof measurement by a "." (dot), so we might oblige by modifying the nameformat.

> a2 <- alkfos> names(a2) <- sub("c", "c.", names(a2))> names(a2)[1] "grp" "c.0" "c.3" "c.6" "c.9" "c.12" "c.18" "c.24"

The sub function does substitutions within character strings, in this casereplacing the string "c" with "c.". Alternatively, the original name for-mat (c0, . . . , c24) can be handled by adding sep="" to the reshapecall.

Once we have the variable naming in place, the only things we need tospecify are the direction of the reshape and the set of variables to be con-sidered time-varying. As a convenience feature, the latter can be specifiedby index rather than by name.

> a.long <- reshape(a2, varying=2:8, direction="long")> head(a.long)

grp time c id1.0 1 0 142 12.0 1 0 120 23.0 1 0 175 34.0 1 0 234 45.0 1 0 94 56.0 1 0 128 6> tail(a.long)

grp time c id38.24 2 24 95 3839.24 2 24 NA 3940.24 2 24 192 4041.24 2 24 94 4142.24 2 24 194 4243.24 2 24 129 43

Notice that the sort order of the result is that id varies within time. Thisis the most convenient format to generate technically, but if you prefer theopposite sort order, just use

> o <- with(a.long, order(id, time))> head(a.long[o,], 10)

grp time c id

10.3 Combining and restructuring data frames 177

1.0 1 0 142 11.3 1 3 140 11.6 1 6 159 11.9 1 9 162 11.12 1 12 152 11.18 1 18 175 11.24 1 24 148 12.0 1 0 120 22.3 1 3 126 22.6 1 6 120 2

To demonstrate the reverse procedure, we use the same data, in the longformat. Actually, this is a bit too easy because reshape has insertedenough information in its output to let you convert to the wide format justby saying reshape(a.long). To simulate the situation where the orig-inal data are given in the long format, we remove the "reshapeLong"attribute, which holds these data. Furthermore, we remove the records forwhich we have missing data by using na.omit.

> a.long2 <- na.omit(a.long)> attr(a.long2, "reshapeLong") <- NULL

To convert a.long2 to the wide format, use

> a.wide2 <- reshape(a.long2, direction="wide", v.names="c",+ idvar="id", timevar="time")> head(a.wide2)

grp id c.0 c.3 c.6 c.9 c.12 c.18 c.241.0 1 1 142 140 159 162 152 175 1482.0 1 2 120 126 120 146 134 119 1163.0 1 3 175 161 168 164 213 194 2214.0 1 4 234 203 174 197 289 174 1895.0 1 5 94 107 146 124 128 98 1146.0 1 6 128 97 113 203 NA NA NA

Notice that NA values are filled in for patient no. 6, for whom only the firstfour observations are available.

The arguments idvar and timevar specify the names of the variablesthat contain the ID and the time for each observation. It is not strictlynecessary to specify them if they have their default names, but it is goodpractice to do so. The argument v.names specifies the time-varying vari-ables; notice that if it were omitted, then the grp variable would also betreated as time-varying.

178 10. Advanced data handling

10.4 Per-group and per-case procedures

A specific data management task involves operations within subsets ofa data frame, particularly those where there are multiple records foreach individual. Examples include calculation of cumulative dosage in apharmacokinetic experiment and various methods of normalization andstandardization.

A nice general approach to such tasks is first to split the data into a list ofgroups, operate on each group, and then put the pieces back together.

Consider the task of normalizing the values of alkaline phosphatase ina.long to their baseline values. The split function can be used togenerate a list of the individual time courses:

> l <- split(a.long$c, a.long$id)> l[1:3]$‘1‘[1] 142 140 159 162 152 175 148

$‘2‘[1] 120 126 120 146 134 119 116

$‘3‘[1] 175 161 168 164 213 194 221

Next, we apply a function to each element of the list and collect the resultsusing lapply.

> l2 <- lapply(l, function(x) x / x[1])

Finally, we put the pieces back together using unsplit, which is thereverse operation of split. Notice that a.long has id varying withintime, so this is not just a matter of concatenating the elements of l2. Thedata for the first patient are now

> a.long$c.adj <- unsplit(l2, a.long$id)> subset(a.long, id==1)

grp time c id c.adj1.0 1 0 142 1 1.00000001.3 1 3 140 1 0.98591551.6 1 6 159 1 1.11971831.9 1 9 162 1 1.14084511.12 1 12 152 1 1.07042251.18 1 18 175 1 1.23239441.24 1 24 148 1 1.0422535

In fact, there is a function that formalizes this sort of split-modify-unsplitoperation. It is called ave because the default use is to replace data with

10.5 Time splitting 179

group averages, but it can also be used for more general transformations.The following is an alternative way of doing the same computation asabove:

> a.long$c.adj <- ave(a.long$c, a.long$id,+ FUN = function(x) x / x[1])

In the preceding code, we worked on the single vector a.long$c.Alternatively, we can split the entire data frame and use code like

> l <- split(a.long, a.long$id)> l2 <- lapply(l, transform, c.adj = c / c[1])> a.long2 <- unsplit(l2, a.long$id)

Notice how the last argument to lapply is passed on to transform, sothat you effectively call transform(x, c.adj = c / c[1]) for eachdata frame x in the list l. This procedure is somewhat less efficient thanthe first one because there is more copying of data, but it generalizes tomore complex transformations.

10.5 Time splitting

This section is rather advanced, and the beginner may want to skip it onthe first read. Understanding the contents is not crucial for the later partsof the book. On the other hand, apart from solving the particular problem,this is also a rather nice first example of the use of ad hoc programming inR and also of the “lateral thinking” that is sometimes required.

The merge operation of the nickel and ewrates data in Section 10.3.2does not really make sense statistically: We merged in the mortality tablecorresponding to the age at the time of entry into the study population.However, the data set is about cancer, a slow disease, and an exposurethat perhaps leads to an increased risk 20 or more years later. If the subjectstypically die around age 50, the population mortality for people of age 30is hardly relevant.

A sensible statistical analysis needs to consider the population mortalityduring the entire follow-up period. One way to handle this issue is to splitthe individuals into multiple “sub-individuals”.

In the data set, the first six observations are (after the merge in Sec-tion 10.3.2)

> head(nickel)id icd exposure dob age1st agein ageout agr ygr

1 3 0 5 1889.019 17.4808 45.2273 92.9808 45 1931

180 10. Advanced data handling

2 4 162 5 1885.978 23.1864 48.2684 63.2712 45 19313 6 163 10 1881.255 25.2452 52.9917 54.1644 50 19314 8 527 9 1886.340 24.7206 47.9067 69.6794 45 19315 9 150 0 1879.500 29.9575 54.7465 76.8442 50 19316 10 163 2 1889.915 21.2877 44.3314 62.5413 40 1931

Consider the individual with id == 4; this person entered the study atthe age of 48.2684 and died (from lung cancer) at the age of 63.2712 (apolo-gies for the excess precision). The time-splitting method treats this subjectas four separate subjects, one entering the study at age 48.2684 and leav-ing at age 50 (on his 50th birthday) and the others covering the intervals50–55, 55–60, and 60–63.2712. The first three are censored observations, asthe subject did not die.

If we merge these data with the population tables, then we can computethe expected number of deaths in a given age interval and compare thatwith the actual number of deaths.

Taking advantage of the vectorized nature of computations in R, thenice way of doing this is to loop over age intervals, “trimming” everyobservation period to each interval.

To trim the observation periods to ages between (say) 60 and 65, the entryand exit times should be adjusted to the interval if they fall outside of it,cases that are unobserved during the interval should be removed, and ifthe subject did not die inside the interval, icd should be set to 0.

The easiest procedure is to “shoot first and ask later”. The adjusted entryand exit times are

> entry <- pmax(nickel$agein, 60)> exit <- pmin(nickel$ageout, 65)

or rather they would be if there were always a suitable overlap betweenthe observation period and the target age interval. However, there are peo-ple leaving the study population before age 60 (by death or otherwise) andpeople entering the study after age 65. In either case, what goes wrong isthat entry >= exit, and we can check for such cases by calculating

> valid <- (entry < exit)> entry <- entry[valid]> exit <- exit[valid]

The censoring indicator for valid cases is

> cens <- (nickel$ageout[valid] > 65)

(We might have used cens <- (exit == 65), but it is a good rule toavoid testing floating point data for equality.)

10.5 Time splitting 181

The trimmed data set can then be obtained as

> nickel60 <- nickel[valid,]> nickel60$icd[cens] <- 0> nickel60$agein <- entry> nickel60$ageout <- exit> nickel60$agr <- 60> nickel60$ygr <- with(nickel60, trunc((dob+agein-1)/5)*5+1)

and the first lines of the result are

> head(nickel60)id icd exposure dob age1st agein ageout agr ygr

1 3 0 5 1889.019 17.4808 60 65.0000 60 19462 4 162 5 1885.978 23.1864 60 63.2712 60 19414 8 0 9 1886.340 24.7206 60 65.0000 60 19465 9 0 0 1879.500 29.9575 60 65.0000 60 19366 10 163 2 1889.915 21.2877 60 62.5413 60 19467 15 334 0 1890.500 23.2836 60 62.0000 60 1946

A couple of fine points: If someone dies exactly at age 65, they are countedas dying inside the age interval. Conversely, we do not include peopledying exactly at age 60; they belong in the interval 55–60 (for purposeslike those of Chapter 15, one should avoid observation intervals of lengthzero). It was also necessary to recompute ygr since this was based on theoriginal agein.

To get the fully expanded data set, you could repeat the above for each ageinterval (20–25, . . . , 95–100) and append the resulting 16 data frames withrbind. However, this gets rather long-winded, and there is a substantialrisk of copy-paste errors. Instead, you can do a little programming. First,wrap up the procedure for one group as a function:

> trim <- function(start)+ {+ end <- start + 5+ entry <- pmax(nickel$agein, start)+ exit <- pmin(nickel$ageout, end)+ valid <- (entry < exit)+ cens <- (nickel$ageout[valid] > end)+ result <- nickel[valid,]+ result$icd[cens] <- 0+ result$agein <- entry[valid]+ result$ageout <- exit[valid]+ result$agr <- start+ result$ygr <- with(result, trunc((dob+agein-1)/5)*5+1)+ result+ }

(In practice, you should not type all this at the command line but use ascript window or an editor; see Section 2.1.3.)

182 10. Advanced data handling

This is typical ad hoc programming. The function is far from generalsince it relies on knowing various names, and it also hardcodes the in-terval length as 5. However, more generality is not required for a one-offcalculation. The important thing for the purpose at hand is to make thedependence on start explicit so that we can loop over it.

With this definition, trim(60) is equivalent to the nickel60 wecomputed earlier:

> head(trim(60))id icd exposure dob age1st agein ageout agr ygr

To get results for all intervals, do the following:

> nickel.expand <- do.call("rbind", lapply(seq(20,95,5), trim))> head(nickel.expand)

id icd exposure dob age1st agein ageout agr ygr84 110 0 0 1909.247 14.0302 24.9999 25 20 1931156 213 0 0 1910.129 14.2018 24.1177 25 20 1931197 273 0 0 1909.500 14.6913 24.7465 25 20 1931236 325 0 0 1910.500 14.0737 23.7465 25 20 1931384 546 0 0 1909.500 14.4945 24.7465 25 20 1931400 574 0 0 1909.729 14.0356 24.5177 25 20 1931

The do.call construct works by creating a call to rbind with a givenargument list, which in this case is the return value from lapply, whichin turn has applied the trim function to each of the values 20, 25, . . . 95.That is, the whole thing is equivalent to

rbind(trim(20), trim(25), ……, trim(95))

Displaying the result for a single subject yields, for example,

> subset(nickel.expand, id==4)id icd exposure dob age1st agein ageout agr ygr

2 4 0 5 1885.978 23.1864 48.2684 50.0000 45 19312100 4 0 5 1885.978 23.1864 50.0000 55.0000 50 19312102 4 0 5 1885.978 23.1864 55.0000 60.0000 55 19362104 4 162 5 1885.978 23.1864 60.0000 63.2712 60 1941

(The strange row names occur because multiple data frames with thesame row names are being rbind-ed together and data frames must haveunique row names.)

10.6 Exercises 183

A weakness of the ygr computation is that since ygr refers to the calen-dar time group at agein, it may be off by up to 5 years. However, lungcancer death rates by age do not change that quickly, so we leave it at this.A more careful procedure, and in fact the common practice in epidemiol-ogy, is to split on both age and calendar time. The Epi package containsgeneralized time-splitters splitLexis and cutLexis, which are usefulfor this purpose and also for handling the related case of splitting timebased on individual events (e.g., childbirth).

As a final step, we can merge in the mortality table as we did inSection 10.3.2.

> nickel.expand <- merge(nickel.expand, ewrates,+ by.x=c("agr","ygr"), by.y=c("age","year"))> head(nickel.expand)agr ygr id icd exposure dob age1st agein ageout lung

1 20 1931 325 0 0 1910.500 14.0737 23.7465 25 62 20 1931 273 0 0 1909.500 14.6913 24.7465 25 63 20 1931 110 0 0 1909.247 14.0302 24.9999 25 64 20 1931 574 0 0 1909.729 14.0356 24.5177 25 65 20 1931 213 0 0 1910.129 14.2018 24.1177 25 66 20 1931 546 0 0 1909.500 14.4945 24.7465 25 6nasal other

1 0 31162 0 31163 0 31164 0 31165 0 31166 0 3116

For later use, the expanded data set is made available “precooked” in theISwR package under the name nickel.expand. We return to the dataset in connection with the analysis of rates in Chapter 15.

10.6 Exercises

10.1 Create a factor in which the blood.glucose variable in thethuesen data is divided into the intervals (4, 7], (7, 9], (9, 12], and (12, 20].Change the level names to “low”, “intermediate”, “high”, and “veryhigh”.

10.2 In the bcmort data set, the four-level factor cohort can be consid-ered the product of two two-level factors, say period and area. Howcan you generate them?

184 10. Advanced data handling

10.3 Convert the ashina data to the long format. Consider how toencode whether the vas measurement is from the first or the secondmeasurement session.

10.4 Split the stroke data according to obsmonths into time intervals0–0.5, 0.5–2, 2–12, and 12+ months after stroke.

11Multiple regression

This chapter discusses the case of regression analysis with multiple pre-dictors. There is not really much new here since model specification andoutput do not differ a lot from what has been described for regressionanalysis and analysis of variance. The news is mainly the model searchaspect, namely among a set of potential descriptive variables to look for asubset that describes the response sufficiently well.

The basic model for multiple regression analysis is

y = β0 + β1 x1 + · · · + βk xk + e

where x1, . . . xk are explanatory variables (also called predictors) and theparameters β1, . . . , βk can be estimated using the method of least squares(see Section 6.1). A closed-form expression for the estimates can be derivedusing matrix calculus, but we do not go into the details of that here.

11.1 Plotting multivariate data

As an example in this chapter, we use a study concerning lung function inpatients with cystic fibrosis in Altman (1991, p. 338). The data are in thecystfibr data frame in the ISwR package.

P. Dalgaard, Introductory Statistics with R,DOI: 10.1007/978-0-387-79054-1_11, © Springer Science+Business Media, LLC 2008

186 11. Multiple regression

age

0.0 0.6 20 50 20 40 100 200 60 120 200

0.0

0.6

sex

height

weight

bmp

fev1

150

300

100

200

frc

tlc

110

10 20

120

200

110 150 65 80 95 150 300 450 80 110

pemax

Figure 11.1. Pairwise plots for cystic fibrosis data.

You can obtain pairwise scatterplots between all the variables in the dataset. This is done using the function pairs. To get Figure 11.1, you simplywrite

> par(mex=0.5)> pairs(cystfibr, gap=0, cex.labels=0.9)

The arguments gap and cex.labels control the visual appearance byremoving the space between subplots and decreasing the font size. Themex graphics parameter reduces the interline distance in the margins.

A similar plot is obtained by simply saying plot(cystfibr) since theplot function is generic and behaves differently depending on the classof its arguments (see Section 2.3.2). Here the argument is a data frame anda pairs plot is a fairly reasonable thing to get when asking for a plot of an

11.2 Model specification and output 187

entire data frame (although you might equally reasonably have expecteda histogram or a barchart of each variable instead).

The individual plots do get rather small, probably not suitable for di-rect publication, but such plots are quite an effective way of obtainingan overview of multidimensional issues. For example, the close relationsamong age, height, and weight appear clearly on the plot.

In order to be able to refer directly to the variables in cystfibr, we addit to the search path (a harmless warning about masking of tlc ensues atthis point):

> attach(cystfibr)

Because this data set contains common variable names such as age,height, and weight, it is a good idea to ensure that you do not haveidentically named variables in the workspace at this point. In particular,such names were used in the introductory session.

11.2 Model specification and output

Specification of a multiple regression analysis is done by setting up amodel formula with + between the explanatory variables:

lm(pemax~age+sex+height+weight+bmp+fev1+rv+frc+tlc)

which is meant to be read as “pemax is described using a model thatis additive in age, sex, and so forth.” (pemax is the maximal expira-tory pressure. See Appendix B for a description of the other variables incystfibr.)

As usual, there is not much output from lm itself, but with the aid ofsummary you can obtain some more interesting output:

> summary(lm(pemax~age+sex+height+weight+bmp+fev1+rv+frc+tlc))

Call:lm(formula = pemax ~ age + sex + height + weight + bmp + fev1 +

rv + frc + tlc)

Residuals:Min 1Q Median 3Q Max

-37.338 -11.532 1.081 13.386 33.405

Coefficients:Estimate Std. Error t value Pr(>|t|)

(Intercept) 176.0582 225.8912 0.779 0.448

188 11. Multiple regression

age -2.5420 4.8017 -0.529 0.604sex -3.7368 15.4598 -0.242 0.812height -0.4463 0.9034 -0.494 0.628weight 2.9928 2.0080 1.490 0.157bmp -1.7449 1.1552 -1.510 0.152fev1 1.0807 1.0809 1.000 0.333rv 0.1970 0.1962 1.004 0.331frc -0.3084 0.4924 -0.626 0.540tlc 0.1886 0.4997 0.377 0.711

Residual standard error: 25.47 on 15 degrees of freedomMultiple R-squared: 0.6373, Adjusted R-squared: 0.4197F-statistic: 2.929 on 9 and 15 DF, p-value: 0.03195

The layout should be well known by now. Notice that there is not onesingle significant t value, but the joint F test is nevertheless significant,so there must be an effect somewhere. The reason is that the t tests onlysay something about what happens if you remove one variable and leavein all the others. You cannot see whether a variable would be statisticallysignificant in a reduced model; all you can see is that no variable must beincluded.

Note further that there is quite a large difference between the unadjustedand the adjusted R2, which is due to the large number of variables relativeto the number of degrees of freedom for the variance. Recall that the for-mer is the change in residual sum of squares relative to an empty model,whereas the latter is the similar change in residual variance:

> 1-25.5^2/var(pemax)[1] 0.4183949

The 25.5 comes from “residual standard error” in the summary output.

The ANOVA table for a multiple regression analysis is obtained usinganova and gives a rather different picture:

> anova(lm(pemax~age+sex+height+weight+bmp+fev1+rv+frc+tlc))Analysis of Variance Table

Response: pemaxDf Sum Sq Mean Sq F value Pr(>F)

age 1 10098.5 10098.5 15.5661 0.001296 **sex 1 955.4 955.4 1.4727 0.243680height 1 155.0 155.0 0.2389 0.632089weight 1 632.3 632.3 0.9747 0.339170bmp 1 2862.2 2862.2 4.4119 0.053010 .fev1 1 1549.1 1549.1 2.3878 0.143120rv 1 561.9 561.9 0.8662 0.366757frc 1 194.6 194.6 0.2999 0.592007tlc 1 92.4 92.4 0.1424 0.711160Residuals 15 9731.2 648.7

11.2 Model specification and output 189

—Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Note that, except for the very last line (“tlc”), there is practically nocorrespondence between these F tests and the t tests from summary. Inparticular, the effect of age is now significant. That is because these testsare successive; they correspond to (reading upward from the bottom) astepwise removal of terms from the model until finally only age is left.During the process, bmp came close to the magical 5% limit, but in view ofthe number of tests, this is hardly noteworthy.

The probability that one out of eight independent tests gives a p-value of0.053 or below is actually just over 35%! The tests in the ANOVA table arenot completely independent, but the approximation should be good.

The ANOVA table indicates that there is no significant improvement ofthe model once age is included. It is possible to perform a joint test forwhether all the other variables can be removed by adding up the sums ofsquares contributions and using the sum for an F test; that is,

> 955.4+155.0+632.3+2862.2+1549.1+561.9+194.6+92.4[1] 7002.9> 7002.9/8[1] 875.3625> 875.36/648.7[1] 1.349407> 1-pf(1.349407,8,15)[1] 0.2935148

This corresponds to collapsing the eight lines of the table so that it wouldlook like this:

Df Sum Sq Mean Sq F Pr(>F)age 1 10098.5 10098.5 15.566 0.00130others 8 7002.9 875.4 1.349 0.29351Residual 15 9731.2 648.7

(Note that this is “cheat output”, in which we have manually inserted thenumbers computed above.)

A procedure leading directly to the result is

> m1<-lm(pemax~age+sex+height+weight+bmp+fev1+rv+frc+tlc)> m2<-lm(pemax~age)> anova(m1,m2)Analysis of Variance Table

Model 1: pemax ~ age + sex + height + weight + bmp + fev1 + rv +frc + tlc

Model 2: pemax ~ age

190 11. Multiple regression

Res.Df RSS Df Sum of Sq F Pr(>F)1 15 9731.22 23 16734.2 -8 -7002.9 1.3493 0.2936

which gives the appropriate F test with no manual computation.

Notice, however, that you need to be careful to ensure that the two modelsare actually nested. R does not check this, although it does verify thatthe number of response observations is the same to safeguard against themore obvious mistakes. (When there are missing values in the descriptivevariables, it’s easy for the smaller model to contain more data points.)

From the ANOVA table, we can thus see that it is allowable to remove allvariables except age. However, that this particular variable is left in themodel is primarily due to the fact that it was mentioned first in the modelspecification, as we see below.

11.3 Model search

R has the step() function for performing model searches by the Akaikeinformation criterion. Since that is well beyond the scope of this book, weuse simple manual variants of backwards elimination.

In the following, we go through a practical model reduction for the exam-ple data. Notice that the output has been slightly edited to take up lessspace.

> summary(lm(pemax~age+sex+height+weight+bmp+fev1+rv+frc+tlc))…

Estimate Std. Error t value Pr(>|t|)(Intercept) 176.0582 225.8912 0.779 0.448age -2.5420 4.8017 -0.529 0.604sex -3.7368 15.4598 -0.242 0.812height -0.4463 0.9034 -0.494 0.628weight 2.9928 2.0080 1.490 0.157bmp -1.7449 1.1552 -1.510 0.152fev1 1.0807 1.0809 1.000 0.333rv 0.1970 0.1962 1.004 0.331frc -0.3084 0.4924 -0.626 0.540tlc 0.1886 0.4997 0.377 0.711…

One advantage of doing model reductions by hand is that you may im-pose some logical structure on the process. In the present case, it may, forinstance, be natural to try to remove other lung function indicators first.

> summary(lm(pemax~age+sex+height+weight+bmp+fev1+rv+frc))

11.3 Model search 191

…Estimate Std. Error t value Pr(>|t|)

(Intercept) 221.8055 185.4350 1.196 0.2491age -3.1346 4.4144 -0.710 0.4879sex -4.6933 14.8363 -0.316 0.7558height -0.5428 0.8428 -0.644 0.5286weight 3.3157 1.7672 1.876 0.0790 .bmp -1.9403 1.0047 -1.931 0.0714 .fev1 1.0183 1.0392 0.980 0.3417rv 0.1857 0.1887 0.984 0.3396frc -0.2605 0.4628 -0.563 0.5813…> summary(lm(pemax~age+sex+height+weight+bmp+fev1+rv))…

Estimate Std. Error t value Pr(>|t|)(Intercept) 166.71822 154.31294 1.080 0.2951age -1.81783 3.66773 -0.496 0.6265sex 0.10239 11.89990 0.009 0.9932height -0.40981 0.79257 -0.517 0.6118weight 2.87386 1.55120 1.853 0.0814 .bmp -1.94971 0.98415 -1.981 0.0640 .fev1 1.41526 0.74788 1.892 0.0756 .rv 0.09567 0.09798 0.976 0.3425…> summary(lm(pemax~age+sex+height+weight+bmp+fev1))…

Estimate Std. Error t value Pr(>|t|)(Intercept) 260.6313 120.5215 2.163 0.0443 *age -2.9062 3.4898 -0.833 0.4159sex -1.2115 11.8083 -0.103 0.9194height -0.6067 0.7655 -0.793 0.4384weight 3.3463 1.4719 2.273 0.0355 *bmp -2.3042 0.9136 -2.522 0.0213 *fev1 1.0274 0.6329 1.623 0.1219…> summary(lm(pemax~age+sex+height+weight+bmp))…

Estimate Std. Error t value Pr(>|t|)(Intercept) 280.4482 124.9556 2.244 0.0369 *age -3.0750 3.6352 -0.846 0.4081sex -11.5281 10.3720 -1.111 0.2802height -0.6853 0.7962 -0.861 0.4001weight 3.5546 1.5281 2.326 0.0312 *bmp -1.9613 0.9263 -2.117 0.0476 *…

As is seen, there was no obstacle to removing the four lung functionvariables. Next we try to reduce among the variables that describe thepatient’s state of physical development or size. Initially, we avoid remov-ing weight and bmp since they appear to be close to the 5% significancelimit.

192 11. Multiple regression

> summary(lm(pemax~age+height+weight+bmp))…

Estimate Std. Error t value Pr(>|t|)(Intercept) 274.5307 125.5745 2.186 0.0409 *age -3.0832 3.6566 -0.843 0.4091height -0.6985 0.8008 -0.872 0.3934weight 3.6338 1.5354 2.367 0.0282 *bmp -1.9621 0.9317 -2.106 0.0480 *…> summary(lm(pemax~height+weight+bmp))…

Estimate Std. Error t value Pr(>|t|)(Intercept) 245.3936 119.8927 2.047 0.0534 .height -0.8264 0.7808 -1.058 0.3019weight 2.7717 1.1377 2.436 0.0238 *bmp -1.4876 0.7375 -2.017 0.0566 ….> summary(lm(pemax~weight+bmp))…

Estimate Std. Error t value Pr(>|t|)(Intercept) 124.8297 37.4786 3.331 0.003033 **weight 1.6403 0.3900 4.206 0.000365 ***bmp -1.0054 0.5814 -1.729 0.097797 ….> summary(lm(pemax~weight))…

Estimate Std. Error t value Pr(>|t|)(Intercept) 63.5456 12.7016 5.003 4.63e-05 ***weight 1.1867 0.3009 3.944 0.000646 ***…

Notice that, once age and height were removed, bmp was no longer sig-nificant. In the original reference (Altman, 1991), weight, fev1, and bmpall ended up with p-values below 5%. However, far from all eliminationprocedures lead to that result.

It is also a good idea to pay close attention to the age, weight, andheight variables, which are heavily correlated since we are dealing withchildren and adolescents.

> summary(lm(pemax~age+weight+height))…

Estimate Std. Error t value Pr(>|t|)(Intercept) 64.65555 82.40935 0.785 0.441age 1.56755 3.14363 0.499 0.623weight 0.86949 0.85922 1.012 0.323height -0.07608 0.80278 -0.095 0.925…> summary(lm(pemax~age+height))…

Estimate Std. Error t value Pr(>|t|)(Intercept) 17.8600 68.2493 0.262 0.796

11.4 Exercises 193

age 2.7178 2.9325 0.927 0.364height 0.3397 0.6900 0.492 0.627…> summary(lm(pemax~age))…

Estimate Std. Error t value Pr(>|t|)(Intercept) 50.408 16.657 3.026 0.00601 **age 4.055 1.088 3.726 0.00111 **…> summary(lm(pemax~height))…

Estimate Std. Error t value Pr(>|t|)(Intercept) -33.2757 40.0445 -0.831 0.41453height 0.9319 0.2596 3.590 0.00155 **…

As it turns out, there is really no reason to prefer one of the three variablesover the two others. The fact that an elimination method ends up with amodel containing only weight is essentially a coincidence. You can easilybe misled by model search procedures that end up with one highly sig-nificant variable — it is far from certain that the same variable would bechosen if you were to repeat the analysis on a new, similar data set.

What you may reasonably conclude is that there is probably a connectionwith the patient’s physical development or size, which may be describedin terms of age, height, or weight. Which description to use is arbitrary. Ifyou want to choose one over the others, a decision cannot be based on thedata, although possibly on theoretical considerations and/or results fromprevious investigations.

11.4 Exercises

11.1 The secher data are best analyzed after log-transforming birthweight as well as the abdominal and biparietal diameters. Fit a predictionequation for birth weight. How much is gained by using both diameters ina prediction equation? The sum of the two regression coefficients is almostexactly 3 — can this be given a nice interpretation?

11.2 The tlc data set contains a variable also called tlc. This is not ingeneral a good idea; explain why. Describe tlc using the other variablesin the data set and discuss the validity of the model.

11.3 The analyses of cystfibr involve sex, which is a binary variable.How would you interpret the results for this variable?

11.4 Consider the juul2 data set and select the group of those over 25years old. Perform a regression analysis of

√igf1 on age, and extend

194 11. Multiple regression

the model by including height and weight. Generate the analysis ofvariance table for the extended model. What is the surprise, and why doesit happen?

11.5 Analyze and interpret the effect of explanatory variables on the milkintake in the kfm data set using a multiple regression model. Notice thatsex is a factor here; what does that imply for the analyses?

12Linear models

Many data sets are inherently too complex to be handled adequately bystandard procedures and thus require the formulation of ad hoc models.The class of linear models provides a flexible framework into which many— although not all — of these cases can be fitted.

You may have noticed that the lm function is applied to data classified intogroups (Chapter 7) as well as to (multiple) linear regression (Chapters 6and 11) problems, even though the theory for these procedures appearsto be quite different. However, they are, in fact, special cases of the samegeneral model.

The basic point is that a multiple regression model can describe a wide va-riety of situations if you choose the explanatory variables suitably. Thereis no requirement that the explanatory variables should follow a normaldistribution, or any continuous distribution for that matter. One simpleexample (which we use without comment in Chapter 11) is that a group-ing into two categories can be coded as a 0/1 variable and used in aregression analysis. The regression coefficient in that case corresponds toa difference between two groups rather than the slope of an actual line. Toencode a grouping with more than two categories, you can use multiple0/1 variables.

Generating these dummy variables becomes tedious, but it can be auto-mated by the use of model formulas. Among other things, such formulasprovide a convenient abstraction by treating classification variables (fac-tors) and continuous variables symmetrically. You will need to learn

P. Dalgaard, Introductory Statistics with R,DOI: 10.1007/978-0-387-79054-1_12, © Springer Science+Business Media, LLC 2008

196 12. Linear models

exactly what model formulas do in order to become able to express yourown modelling ideas.

This chapter contains a collection of models and their handling by lm,mainly in the form of relatively minor extensions and modifications ofmethods described earlier. It is meant only to give you a feel for the scopeof possibilities and does not pretend to be complete.

12.1 Polynomial regression

One basic observation showing that multiple regression analysis can domore than meets the eye is that you can include second-order and higherpowers of a variable in the model along with the original linear term. Thatis, you can have a model like

y = α + β1 x + β2 x2 + · · · + βk x

k + e

This obviously describes a nonlinear relation between y and x, but thatdoes not matter; the model is still a linear model. What does matter is thatthe relation between the parameters and the expected observations is lin-ear. It also does not matter that there is a deterministic relation betweenthe regression variables x, x2, x3, . . . , as long as there is no linear relationbetween them. However, fitting high-degree polynomials can be diffi-cult because near-collinearity between terms makes the fit numericallyunstable.

We return to the cystic fibrosis data set for an example. The plot of pemaxand height in Figure 11.1 may suggest that the relation is not quite linear.One way to test this is to try to add a term that is the square of the height.

> attach(cystfibr)> summary(lm(pemax~height+I(height^2)))…

Estimate Std. Error t value Pr(>|t|)(Intercept) 615.36248 240.95580 2.554 0.0181 *height -8.08324 3.32052 -2.434 0.0235 *I(height^2) 0.03064 0.01126 2.721 0.0125 *…

Notice that the computed height2 in the model formula needs to be “pro-tected” by I(…). This technique is often used to prevent any specialinterpretation of operators in a model formula. Such an interpretation willnot take place inside a function call, and I is the identity function thatreturns its argument unaltered.

12.1 Polynomial regression 197

We find a significant deviation from linearity. However, considering theprocess that led to doing this particular analysis, the p-values have tobe taken with more than a grain of salt. This is getting dangerouslyclose to “data dredging”, fishing expeditions in data. Consider it morean illustration of a technique than an exemplary data analysis.

To draw a plot of the fitted curve with prediction and confidence bands,we can use predict. To avoid problems caused by data not being sortedby height, we use newdata, which allows the prediction of values for achosen set of predictors. Here we choose a set of heights between 110 and180 cm in steps of 2 cm:

> pred.frame <- data.frame(height=seq(110,180,2))> lm.pemax.hq <- lm(pemax~height+I(height^2))> predict(lm.pemax.hq,interval="pred",newdata=pred.frame)

fit lwr upr1 96.90026 37.94461 155.85592 94.33611 36.82985 151.84243 92.01705 35.73077 148.3033…34 141.68922 88.70229 194.676135 147.21294 93.51117 200.914736 152.98174 98.36718 207.5963

Based on these predicted data, Figure 12.1 is obtained as follows:

> pp <- predict(lm.pemax.hq,newdata=pred.frame,interval="pred")> pc <- predict(lm.pemax.hq,newdata=pred.frame,interval="conf")> plot(height,pemax,ylim=c(0,200))> matlines(pred.frame$height,pp,lty=c(1,2,2),col="black")> matlines(pred.frame$height,pc,lty=c(1,3,3),col="black")

It is seen that the fitted curve is slightly decreasing for small heights. Thisis probably an artifact caused by the choice of a second-order polynomialto fit data. More likely, the reality is that pemax is relatively constant upto about 150 cm, after which it increases quickly with height. Note alsothat there seems to be a discrepancy between the prediction limits and theactual distribution of data for the smaller heights. The standard deviationmight be larger for larger heights, but it is not impossible to obtain a sim-ilar distribution of points by coincidence, and there is also an issue withpotential overfitting to the observed data. It is really not advisable to con-struct prediction intervals based on data as limited as these unless you aresure that the model is correct.

198 12. Linear models

110 120 130 140 150 160 170 180

050

100

150

200

height

pem

Figure 12.1. Quadratic regression with confidence and prediction limits.

12.2 Regression through the origin

It sometimes makes sense to assume that a regression line passes through(0, 0) — that the intercept of the regression line is zero. This can be spec-ified in the model formula by adding the term -1 (“minus intercept”) tothe right-hand side: y ~ x – 1.

The logic of the notation can be seen by writing the linear regressionmodel as y = α × 1 + β × x + e. The intercept corresponds to having anextra descriptive variable, which is the constant 1. Removing this variableyields regression through the origin.

This is a simulated example of a linear relationship through the origin(y = 2x + e):

> x <- runif(20)> y <- 2*x+rnorm(20,0,0.3)> summary(lm(y~x))

Call:lm(formula = y ~ x)

12.2 Regression through the origin 199

Residuals:Min 1Q Median 3Q Max

-0.50769 -0.08766 0.03802 0.14512 0.26358

Coefficients:Estimate Std. Error t value Pr(>|t|)

(Intercept) -0.14896 0.08812 -1.69 0.108x 2.39772 0.15420 15.55 7.05e-12 ***—Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.2115 on 18 degrees of freedomMultiple R-squared: 0.9307, Adjusted R-squared: 0.9269F-statistic: 241.8 on 1 and 18 DF, p-value: 7.047e-12

> summary(lm(y~x-1))

Call:lm(formula = y ~ x – 1)

Residuals:Min 1Q Median 3Q Max

-0.62178 -0.16855 -0.04019 0.12044 0.27346

Coefficients:Estimate Std. Error t value Pr(>|t|)

x 2.17778 0.08669 25.12 4.87e-16 ***—Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.2216 on 19 degrees of freedomMultiple R-squared: 0.9708, Adjusted R-squared: 0.9692F-statistic: 631.1 on 1 and 19 DF, p-value: 4.873e-16

In the first analysis, the intercept is not significant, which is, of course,not surprising. In the second analysis we force the intercept to be zero,resulting in a slope estimate with a substantially improved accuracy.

Comparison of the R2-values in the two analyses shows something thatoccasionally causes confusion: R2 is much larger in the model with no in-tercept! This does not, however, mean that the relation is “more linear”when the intercept is not included or that more of the variation is ex-plained. What is happening is that the definition of R2 itself is changing.It is most easily seen from the ANOVA tables in the two cases:

> anova(lm(y~x))Analysis of Variance Table

Response: yDf Sum Sq Mean Sq F value Pr(>F)

x 1 10.8134 10.8134 241.80 7.047e-12 ***Residuals 18 0.8050 0.0447

200 12. Linear models

—Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1> anova(lm(y~x-1))Analysis of Variance Table

Response: yDf Sum Sq Mean Sq F value Pr(>F)

x 1 30.9804 30.9804 631.06 4.873e-16 ***Residuals 19 0.9328 0.0491—Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Notice that the total sum of squares and the total number of degrees offreedom is not the same in the two analyses. In the model with an inter-cept, there are 19 DF in all and the total sum of squares is ∑(yi − ȳ)2, whilethe model without an intercept has a total of 20 DF and the total sum ofsquares is defined as ∑ y2i . Unless ȳ is close to zero, the latter “total SS”will be much larger than the former, so if the residual variance is similar,R2 will be much closer to 1.

The reason for defining the total sum of squares like this for models with-out intercepts is that it has to correspond to the residual sum of squaresin a minimal model. The minimal model has to be a submodel of the re-gression model; otherwise the ANOVA table simply does not make sense.In an ordinary regression analysis, the minimal model is y = α + e, butwhen the regression model does not include α, the only sensible minimalmodel is y = 0 + e.

12.3 Design matrices and dummy variables

The function model.matrix gives the design matrix for a given model. Itcan look like this:

> model.matrix(pemax~height+weight)(Intercept) height weight

1 1 109 13.12 1 112 12.93 1 124 14.14 1 125 16.2…24 1 175 51.125 1 179 71.5attr(,"assign")[1] 0 1 2

(The cystfibr data set was attached previously.)

12.3 Design matrices and dummy variables 201

You should not worry about the "assign" attribute at this stage, but thethree columns are important. If you add them together, weighted by thecorresponding regression coefficients, you get exactly the fitted values.Notice that the intercept enters as the coefficient to a column of ones.

If the same is attempted for a model containing a factor, the followinghappens. (We return to the anesthetic ventilation example from p. 129.)

> attach(red.cell.folate)> model.matrix(folate~ventilation)

(Intercept) ventilationN2O+O2,op ventilationO2,24h1 1 0 02 1 0 0…16 1 1 017 1 1 018 1 0 119 1 0 120 1 0 121 1 0 122 1 0 1attr(,"assign")[1] 0 1 1attr(,"contrasts")attr(,"contrasts")$ventilation[1] "contr.treatment"

The two columns of zeros and ones are sometimes called dummy variables.They are interpreted exactly as above: Multiplying them by the respectiveregression coefficients and adding the results yields the fitted value. No-tice that, for example, the second column is 1 for observations in group 2and 0 otherwise; that is, the corresponding regression coefficient describessomething that is added to the intercept for observations in that particulargroup. Both columns have zeros for observations from the first group, themean value of which is described by the intercept (β0) alone. The regres-sion coefficient β1 thus describes the difference in means between groups 1and 2, and β2 between groups 1 and 3.

You may be confused by the use of the term “regression coefficients” eventhough no regression lines are present in models like that above. The pointis that you formally rewrite a model for groups as a multiple regressionmodel, so that you can use the same software. As is seen, there is a uniquecorrespondence between the formal regression coefficients and the groupmeans.

You can define dummy variables in several different ways to describe agrouping. This particular scheme is called treatment contrasts because ifthe first group is “no treatment” then the coefficients immediately givethe treatment effects for each of the other groups. We do not discuss other

202 12. Linear models

choices here; see Venables and Ripley (2002) for a much deeper discussion.Note only that contrast type can be set on a per-term basis and that this iswhat is reflected in the "contrasts" attribute of the design matrix.

For completeness, the "assign" attribute indicates which columns be-long together. When, for instance, you request an analysis of varianceusing anova, the sum of squares for ventilation will have 2 degrees offreedom, corresponding to the removal of both columns simultaneously.

Removing the intercept from a model containing a factor term will notcorrespond to a model in which a particular group has mean zero sincesuch models are usually nonsensical. Instead, R generates a simpler set ofdummy variables, which are indicator variables of the levels of the factor.This corresponds to the same model as when the intercept is included (thefitted values are identical), but the regression coefficients have a differentinterpretation.

12.4 Linearity over groups

Sometimes data are grouped according to a division of a continuous scale(e.g., by age group), or an experiment was designed to take several mea-surements at each of a fixed set of x-values. In both cases it is relevantto compare the results of a linear regression with those of an analysis ofvariance.

In the case of grouped x-values, you might take a central value as rep-resentative for everyone in a given group, for instance formally lettingeveryone in a “20–29-year” category be 25 years old. If individual x-values are available, they may of course be used in a linear regression,but it makes the analysis a little more complicated, so we discuss only thesituation where that is not the case.

We thus have two alternative models for the same data. Both belong to theclass of linear models that lm is capable of handling. The linear regressionmodel is a submodel of the model for one-way analysis of variance becausethe former can be obtained by placing restrictions on the parameters ofthe latter (namely that the true group means lie on a straight line).

It is possible to test whether or not a model reduction is allowable by com-paring the reduction in the amount of variation explained to the residualvariation in the larger model, resulting in an F test.

In the following example on trypsin concentrations in age groups (Alt-man, 1991, p. 212), data are given as the mean and SD within each of sixgroups. This is a kind of data that R is not quite prepared to handle, and it

12.4 Linearity over groups 203

has therefore been necessary to create “fake” data giving the same meansand SDs. These can be obtained via

> attach(fake.trypsin)

The actual results of the analysis of variance depend only on the meansand SDs and are therefore independent of the faking. Readers inter-ested in how to perform the actual faking should take a look at the filefake.trypsin.R in the rawdata directory of the ISwR package.

The fake.trypsin data frame contains three variables, as seen by

> summary(fake.trypsin)trypsin grp grpf

Min. :-39.96 Min. :1.000 1: 321st Qu.:119.52 1st Qu.:2.000 2:137Median :167.59 Median :2.000 3: 38Mean :168.68 Mean :2.583 4: 443rd Qu.:213.98 3rd Qu.:3.000 5: 16Max. :390.13 Max. :6.000 6: 4

Notice that there are both grp, which is a numeric vector, and grpf,which is a factor with six levels.

Performing a one-way analysis of variance on the fake data gives thefollowing ANOVA table:

> anova(lm(trypsin~grpf))Analysis of Variance Table

Response: trypsinDf Sum Sq Mean Sq F value Pr(>F)

grpf 5 224103 44821 13.508 9.592e-12 ***Residuals 265 879272 3318

If you had used grp instead of grpf in the model formula, you wouldhave obtained a linear regression on the group number instead. In somecircumstances, that would have been a serious error, but here it actuallymakes sense. The midpoints of the age intervals are equidistant, so themodel is equivalent to assuming a linear development with age (the in-terpretation of the regression coefficient requires some care, though). TheANOVA table looks as follows:

> anova(lm(trypsin~grp))Analysis of Variance Table

Response: trypsinDf Sum Sq Mean Sq F value Pr(>F)

grp 1 206698 206698 62.009 8.451e-14 ***Residuals 269 896677 3333

204 12. Linear models

Notice that the residual mean squares did not change very much, indicat-ing that the two models describe the data nearly equally well. If you wantto have a formal test of the simple linear model against the model wherethere is a separate mean for each group, it can be done easily as follows:

> model1 <- lm(trypsin~grp)> model2 <- lm(trypsin~grpf)> anova(model1,model2)Analysis of Variance Table

Model 1: trypsin ~ grpModel 2: trypsin ~ grpfRes.Df RSS Df Sum of Sq F Pr(>F)

1 269 8966772 265 879272 4 17405 1.3114 0.2661

So we see that the model reduction has a nonsignificant p-value and hencethat model2 does not fit data significantly better than model1.

This technique works only when one model is a submodel of the other,which is the case here since the linear model is defined by a restriction onthe group means.

Another way to achieve the same result is to add the two models togetherformally as follows:

> anova(lm(trypsin~grp+grpf))Analysis of Variance Table

Response: trypsinDf Sum Sq Mean Sq F value Pr(>F)

grp 1 206698 206698 62.2959 7.833e-14 ***grpf 4 17405 4351 1.3114 0.2661Residuals 265 879272 3318

This model is exactly the same as when only grpf was included. How-ever, the ANOVA table now contains a subdivision of the model sum ofsquares in which the grpf line describes the change incurred by expand-ing the model from one to five parameters. The ANOVA table in Altman(1991, p. 213) is different, erroneously.

The plot in Figure 12.2 is made like this:

> xbar.trypsin <- tapply(trypsin,grpf,mean)> stripchart(trypsin~grp, method="jitter",+ jitter=.1, vertical=T, pch=20)> lines(1:6,xbar.trypsin,type="b",pch=4,cex=2,lty=2)> abline(lm(trypsin~grp))

The graphical techniques used here are essentially identical to those usedfor Figure 7.1, so we do not go into further details.

12.4 Linearity over groups 205

1 2 3 4 5 6

0100

200

300

400

Figure 12.2. “Fake” data for the trypsin example with fitted line and empiricalmeans.

Notice that the fakeness of the data is exposed by a point showing a neg-ative trypsin concentration! The original data are unavailable but wouldlikely show a distribution skewed slightly upwards.

Actually, it is possible to analyze the data in R without generating fakedata. A weighted regression analysis of the group means, with weightsequal to the number of observations in each group, will yield the first twolines of the ANOVA table, and the last one can be computed from the SDs.The details are as follows:

> n <- c(32,137, 38,44,16,4)> tryp.mean <- c(128,152,194,207,215,218)> tryp.sd <-c(50.9,58.5,49.3,66.3,60,14)> gr<-1:6> anova(lm(tryp.mean~gr+factor(gr),weights=n))Analysis of Variance Table

Response: tryp.meanDf Sum Sq Mean Sq F value Pr(>F)

gr 1 206698 206698factor(gr) 4 17405 4351Residuals 0 0

206 12. Linear models

Notice that the “Residuals” line is zero and that the F tests are not cal-culated. Omitting the factor(gr) term will cause that line to go intoResiduals and be treated as an estimate of the error variation, but thatis not what you want since it does not include the information aboutthe variation within groups. Instead, you need to fill in the missinginformation computed from the group standard deviations and sizes.The following gives the residual sum of squares and the correspondingdegrees of freedom and mean squares:

> sum(tryp.sd^2*(n-1))[1] 879271.9> sum(n-1)[1] 265> sum(tryp.sd^2*(n-1))/sum(n-1)[1] 3318.007

There is no simple way of updating the ANOVA table with an externalvariance estimate, but it is easy enough to do the computations directly:

> 206698/3318.007 # F statistic for gr[1] 62.29583> 1-pf(206698/3318.007,1,265) # p-value[1] 7.838175e-14> 4351/3318.007 # F statistic for factor(gr)[1] 1.311329> 1-pf(4351/3318.007,4,265) # p-value[1] 0.2660733

12.5 Interactions

A basic assumption in a multiple regression model is that terms act ad-ditively on the response. However, this does not mean that linear modelscannot describe nonadditivity. You can add special interaction terms thatspecify that the effect of one term is modified according to the level ofanother. In the model formulas in R, such terms are generated usingthe colon operator; for example, a:b. Usually, you will also include theterms a and b, and R allows the notation a*b for a+b+a:b. Higher-orderinteractions among three or more variables are also possible.

The exact definition of the interaction terms and the interpretation of theirassociated regression coefficients can be elusive. Some peculiar thingshappen if an interaction term is present but one or more of the maineffects are missing. The full details are probably best revealed throughexperimentation. However, depending on the nature of the terms a and bas factors or numeric variables, the overall effect of including interactionterms can be described as follows:

12.6 Two-way ANOVA with replication 207

• Interaction between two factors. This is conceptually the simplest case.The model with interaction corresponds to having different levelsfor all possible combinations of levels of the two factors.

• Interaction between a factor and a numeric variable. In this case, themodel with interaction contains linear effects of the continuous vari-able but with different slopes within each group defined by thefactor.

• Interaction between two continuous variables. This gives a slightly pecu-liar model containing a new regression variable that is the productof the two. The interpretation is that you have a linear effect of vary-ing one variable while keeping the other constant, but with a slopethat changes as you vary the other variable.

12.6 Two-way ANOVA with replication

The coking data set comes from Johnson (1994, Section 13.1). The timerequired to make coke from coal is analyzed in a 2×3 experiment varyingthe oven temperature and the oven width. There were three replicationsat each combination.

> attach(coking)> anova(lm(time~width*temp))Analysis of Variance Table

Response: timeDf Sum Sq Mean Sq F value Pr(>F)

width 2 123.143 61.572 222.102 3.312e-10 ***temp 1 17.209 17.209 62.076 4.394e-06 ***width:temp 2 5.701 2.851 10.283 0.002504 **Residuals 12 3.327 0.277—Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

We see that the interaction term is significant. If we take a look at the cellmeans, we can get an idea of why this happens:

> tapply(time,list(width,temp),mean)1600 1900

4 3.066667 2.3000008 7.166667 5.53333312 10.800000 7.333333

The difference between high and low temperatures increases with ovenwidth, making an additive model inadequate. When this is the case, theindividual tests for the two factors make no sense. If the interaction had

208 12. Linear models

not been significant, then we would have been able to perform separate Ftests for the two factors.

12.7 Analysis of covariance

As the example in this section, we use a data set concerning growth condi-tions of Tetrahymena cells, collected by Per Hellung-Larsen. Data are fromtwo groups of cell cultures where glucose was either added or not addedto the growth medium. For each culture, the average cell diameter (µ) andcell concentration (count per ml) were recorded. The cell concentrationwas set at the beginning of the experiment, and there is no systematic dif-ference in cell concentration between the two glucose groups. However, itis expected that the cell diameter is affected by the presence of glucose inthe medium.

Data are in the data frame hellung, which can be loaded and viewed likethis:

> hellungglucose conc diameter

1 1 631000 21.22 1 592000 21.53 1 563000 21.34 1 475000 21.0…49 2 14000 24.450 2 13000 24.351 2 11000 24.2

The coding of glucose is such that 1 and 2 mean yes and no, respectively.There are no missing values.

Summarizing the data frame yields

> summary(hellung)glucose conc diameter

Min. :1.000 Min. : 11000 Min. :19.201st Qu.:1.000 1st Qu.: 27500 1st Qu.:21.40Median :1.000 Median : 69000 Median :23.30Mean :1.373 Mean :164325 Mean :23.003rd Qu.:2.000 3rd Qu.:243000 3rd Qu.:24.35Max. :2.000 Max. :631000 Max. :26.30

Notice that the distribution of the concentrations is strongly right-skewedwith a mean more than twice as big as the median. Note also that glucoseis regarded as a numeric vector by summary, even though it has only twodifferent values.

12.7 Analysis of covariance 209

It will be more convenient to have glucose as a factor, so it is recoded asshown below. Recall that to change a variable inside a data frame, you use$-notation (p. 21) to specify the component you want to change:

> hellung$glucose <- factor(hellung$glucose, labels=c("Yes","No"))> summary(hellung)glucose conc diameterYes:32 Min. : 11000 Min. :19.20No :19 1st Qu.: 27500 1st Qu.:21.40

Median : 69000 Median :23.30Mean :164325 Mean :23.003rd Qu.:243000 3rd Qu.:24.35Max. :631000 Max. :26.30

It is convenient to be able to refer to the variables of hellung without thehellung$ prefix, so we put hellung in the search path.

> attach(hellung)

12.7.1 Graphical description

First, we plot the raw data (Figure 12.3):

> plot(conc,diameter,pch=as.numeric(glucose))

By calculating as.numeric(glucose), we convert the factor glucoseto the underlying codes, 1 and 2. The specification of pch thus implies thatgroup 1 (“Yes”) is drawn using plotting character 1 (circles) and group 2with plotting character 2 (triangles).

To get different plotting symbols, you must first create a vector containingthe symbol numbers and give that as the pch argument. The followingform yields open and filled circles: c(1,16)[glucose]. It looks a bitcryptic at first, but it is really just a consequence of R’s way of indexing.For indexing purposes, a factor like glucose behaves as a vector of 1sand 2s, so you get the first element of c(1,16), namely 1, whenever anobservation is from group 1; when the observation is from group 2, yousimilarly get 16.

The explanatory text is inserted with legend like this:

> legend(locator(n=1),legend=c("glucose","no glucose"),pch=1:2)

Notice that both the function and one of its arguments are named legend.

The function locator returns the coordinates of a point on a plot. Itworks so that the function awaits a click with a mouse button and then re-turns the cursor position. You may want to call locator() directly from

210 12. Linear models

0e+00 1e+05 2e+05 3e+05 4e+05 5e+05 6e+05

conc

dia

mete

glucoseno glucose

Figure 12.3. Plot of diameter versus concentration for Tetrahymena data.

the command line to see the effect. Notice that if you do not specify a valuefor n, then you need to right-click when you are done selecting points.

The plot shows a clear inverse and nonlinear relation between concentra-tion and cell diameter. Further, it is seen that the cultures without glucoseare systematically below cultures with added glucose.

You get a much nicer plot (Figure 12.4) by using a logarithmic x-axis:

> plot(conc,diameter,pch=as.numeric(glucose),log="x")

Now the relation suddenly looks linear!

You could also try a log-log plot (shown in Figure 12.5 with regressionlines as described below):

> plot(conc,diameter,pch=as.numeric(glucose),log="xy")

As is seen, this really does not change much, but it was neverthelessdecided to analyze data with both diameter and concentration log-transformed because a power-law relation was expected (y = αxβ, whichgives a straight line on a log-log plot).

12.7 Analysis of covariance 211

1e+04 2e+04 5e+04 1e+05 2e+05 5e+05

conc

dia

mete

Figure 12.4. Tetrahymena data with logarithmic x-axis.

When adding regression lines to a log plot or log-log plot, you shouldnotice that abline interprets them as lines in the coordinate system ob-tained after taking (base-10) logarithms. Thus, you can add a line foreach group with abline applied to the results of a regression analysisof log10(diameter) on log10(conc). First, however, it is convenientto define data frames corresponding to the two glucose groups:

> tethym.gluc <- hellung[glucose=="Yes",]> tethym.nogluc <- hellung[glucose=="No",]

Notice that you have to use the names, not the numbers, of the factorlevels.

Since we only need the two data frames for adding lines to the figure,it would be cumbersome to add them in turn to the search path withattach, do the plotting, and then use detach to remove them. It is eas-ier to use the data argument to lm; this allows you to explicitly specifythe data frame in which to look for variables. The two regression lines aredrawn with

> lm.nogluc <- lm(log10(diameter)~ log10(conc),data=tethym.nogluc)> lm.gluc <- lm(log10(diameter)~ log10(conc),data=tethym.gluc)

212 12. Linear models

1e+04 2e+04 5e+04 1e+05 2e+05 5e+05

conc

dia

mete

Figure 12.5. Tetrahymena data, log-log plot with regression lines.

> abline(lm.nogluc)> abline(lm.gluc)

after which the plot looks like Figure 12.5. It is seen that the lines fit thedata quite well and that they are almost, but not perfectly, parallel. Thequestion is whether the difference in slope is statistically significant. Thisis the topic of the next section.

12.7.2 Comparison of regression lines

Corresponding to the two lines from before, we have the followingregression analyses:

> summary(lm(log10(diameter)~ log10(conc), data=tethym.gluc))

Call:lm(formula = log10(diameter) ~ log10(conc), data = tethym.gluc)

Residuals:Min 1Q Median 3Q Max

-0.0267219 -0.0043361 0.0006891 0.0035489 0.0176077

12.7 Analysis of covariance 213

Coefficients:Estimate Std. Error t value Pr(>|t|)

(Intercept) 1.63134 0.01345 121.29 <2e-16 ***log10(conc) -0.05320 0.00272 -19.56 <2e-16 ***—Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.008779 on 30 degrees of freedomMultiple R-squared: 0.9273, Adjusted R-squared: 0.9248F-statistic: 382.5 on 1 and 30 DF, p-value: < 2.2e-16

> summary(lm(log10(diameter)~ log10(conc), data=tethym.nogluc))

Call:lm(formula = log10(diameter) ~ log10(conc), data = tethym.nogluc)

Residuals:Min 1Q Median 3Q Max

-2.192e-02 -4.977e-03 5.598e-05 5.597e-03 1.663e-02

Coefficients:Estimate Std. Error t value Pr(>|t|)

(Intercept) 1.634761 0.020209 80.89 < 2e-16 ***log10(conc) -0.059677 0.004125 -14.47 5.48e-11 ***—Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.009532 on 17 degrees of freedomMultiple R-squared: 0.9249, Adjusted R-squared: 0.9205F-statistic: 209.3 on 1 and 17 DF, p-value: 5.482e-11

Notice that you can use arithmetic expressions in the model formula[here log10(…)]. There are limitations, though, because, for example,z~x+y means a model where z is described by an additive model in x andy, which is not the same as a regression analysis on the sum of the two.The latter may be specified using z~I(x+y) (I for “identity”).

A quick assessment of the significance of the difference between theslopes of the two lines can be obtained as follows: The difference be-tween the slope estimates is 0.0065, and the standard error of that is√

0.00412 + 0.00272 = 0.0049. Since t = 0.0065/0.0049 = 1.3, it wouldseem that we are allowed to assume that the slopes are the same.

It is, however, preferable to fit a model to the entire data set and test thehypothesis of equal slopes in that model. One reason that this approachis preferable is that it can be generalized to more complicated models.Another reason is that even though there is nothing seriously wrong withthe simple test for equal slopes, that procedure gives you little informationon how to proceed. If the slopes are the same, you would naturally want

214 12. Linear models

to find an estimate of the common slope and the distance between theparallel lines.

First, we set up a model that allows the relation between concentrationand cell diameter to have different slopes and intercepts in the two glucosegroups:

> summary(lm(log10(diameter)~log10(conc)*glucose))

Call:lm(formula = log10(diameter) ~ log10(conc) * glucose)

Residuals:Min 1Q Median 3Q Max

-2.672e-02 -4.888e-03 5.598e-05 3.767e-03 1.761e-02

Coefficients:Estimate Std. Error t value Pr(>|t|)

(Intercept) 1.631344 0.013879 117.543 <2e-16 ***log10(conc) -0.053196 0.002807 -18.954 <2e-16 ***glucoseNo 0.003418 0.023695 0.144 0.886log10(conc):glucoseNo -0.006480 0.004821 -1.344 0.185—Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.009059 on 47 degrees of freedomMultiple R-squared: 0.9361, Adjusted R-squared: 0.9321F-statistic: 229.6 on 3 and 47 DF, p-value: < 2.2e-16

These regression coefficients should be read as follows. The expectedvalue of the log cell diameter for an observation with cell concentrationC is obtained as the sum of the following four quantities:

1. The intercept, 1.6313

2. −0.0532 × log10 C

3. 0.0034, but only for a culture without glucose

4. −0.0065 × log10 C, but only for cultures without glucose

Accordingly, for cell cultures with glucose, we have the linear relation

log10 D = 1.6313 − 0.0532 × log10 C

and for cultures without glucose we have

log10 D = (1.6313 + 0.0034) − (0.0532 + 0.0065) × log10 C

Put differently, the first two coefficients in the joint model can be inter-preted as the estimates for intercept and slope in group 1, whereas thelatter two are the differences between group 1 and group 2 in interceptand slope, respectively. Comparison with the separate regression analyses

12.7 Analysis of covariance 215

shows that slopes and intercepts are the same as in the joint analysis. Thestandard errors differ a little from the separate analyses because a pooledvariance estimate is now used. Notice that the rough test of difference inslope outlined above is essentially the t test for the last coefficient.

Notice also that the glucose and log10(conc).glucose terms indi-cate items to be added for cultures without glucose. This is because thefactor levels are ordered yes = 1 and no = 2, and the base level is the firstgroup.

Fitting an additive model, we get

> summary(lm(log10(diameter)~log10(conc)+glucose))…Coefficients:

Estimate Std. Error t value Pr(>|t|)(Intercept) 1.642132 0.011417 143.83 < 2e-16 ***log10(conc) -0.055393 0.002301 -24.07 < 2e-16 ***glucoseNo -0.028238 0.002647 -10.67 2.93e-14 ***…

Here the interpretation of the coefficients is that the estimated relation forcultures with glucose is

log10 D = 1.6421 − 0.0554 × log10 C

and for cultures without glucose it is

log10 D = (1.6421 − 0.0282) − 0.0554 × log10 C

That is, the lines for the two cultures are parallel, but the log diame-ters for cultures without glucose are 0.0282 below those with glucose. Onthe original (nonlogarithmic) scale, this means that the former are 6.3%lower (a constant absolute difference on a logarithmic scale correspondsto constant relative differences on the original scale and 10−0.0282 = 0.937).

The joint analysis presumes that the variance around the regression lineis the same in the two groups. This assumption should really have beentested before embarking on the analysis above. A formal test can beperformed with var.test, which conveniently allows a pair of linearmodels as arguments instead of a model formula or two group vectors:

> var.test(lm.gluc,lm.nogluc)

F test to compare two variances

data: lm.gluc and lm.noglucF = 0.8482, num df = 30, denom df = 17, p-value = 0.6731alternative hypothesis: true ratio of variances is not equal to 195 percent confidence interval:0.3389901 1.9129940sample estimates:

216 12. Linear models

ratio of variances0.8481674

When there are more than two groups, Bartlett’s test can be used. It, too,allows linear models to be compared. The reservations about robustnessagainst nonnormality apply here, too.

It is seen that it is possible to assume that the lines have the same slope andthat they have the same intercept, but — as we will see below — not bothat once. The hypothesis of a common intercept is silly anyway unless theslopes are also identical: The intercept is by definition the y-value at x = 0,which because of the log scale corresponds to a cell concentration of 1.That is far outside the region the data cover, and it is a completely arbitrarypoint that will change if the concentrations are measured in different units.

The ANOVA table for the model is

> anova(lm(log10(diameter)~ log10(conc)*glucose))Analysis of Variance Table

Response: log10(diameter)Df Sum Sq Mean Sq F value Pr(>F)

log10(conc) 1 0.046890 0.046890 571.436 < 2.2e-16 ***glucose 1 0.009494 0.009494 115.698 2.89e-14 ***log10(conc):glucose 1 0.000148 0.000148 1.807 0.1853Residuals 47 0.003857 0.000082—Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

The model formula a*b, where in the present case a is log10(conc)and b is glucose, is a short form for a + b + a:b, which is read “effectof a plus effect of b plus interaction”. The F test in the penultimate lineof the ANOVA table is a test for the hypothesis that the last term (a:b)can be omitted, reducing the model to be additive in log10(conc) andglucose, which corresponds to the parallel regression lines. The F testone line earlier indicates whether you can subsequently remove glucoseand the one in the first line to whether you can additionally removelog10(conc), leaving an empty model.

Alternatively, you can read the table from top to bottom as adding termsdescribing more and more of the total sum of squares. To those familiarwith the SAS system, this kind of ANOVA table is known as type I sumsof squares.

The p-value for log10(conc):glucose can be recognized as that of thet test for the coefficient labelled log10(conc).glucose in the previousoutput. The F statistic is exactly the square of t as well. However, this istrue only because there are just two groups. Had there been three or more,there would have been several regression coefficients and the F test would

12.7 Analysis of covariance 217

have tested them all against zero simultaneously, just like when all groupsare tested equal in a one-way analysis of variance.

Note that the test for removing log10(conc) does not make sense be-cause you would have to remove glucose first, which is “forbidden”when glucose has a highly significant effect. It makes perfectly goodsense to test log10(conc) without removing glucose — which corre-sponds to testing that the two parallel regression lines can be assumedhorizontal — but that test is not found in the ANOVA table. You canget the right test by changing the order of terms in the model formula;compare, for instance, these two regression analyses:

> anova(lm(log10(diameter)~glucose+log10(conc)))Analysis of Variance Table

Response: log10(diameter)Df Sum Sq Mean Sq F value Pr(>F)

glucose 1 0.008033 0.008033 96.278 4.696e-13 ***log10(conc) 1 0.048351 0.048351 579.494 < 2.2e-16 ***Residuals 48 0.004005 0.000083—Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1> anova(lm(log10(diameter)~log10(conc)+ glucose))Analysis of Variance Table

Response: log10(diameter)Df Sum Sq Mean Sq F value Pr(>F)

log10(conc) 1 0.046890 0.046890 561.99 < 2.2e-16 ***glucose 1 0.009494 0.009494 113.78 2.932e-14 ***Residuals 48 0.004005 0.000083—Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

They both describe exactly the same model, as is indicated by the resid-ual sum of squares being identical. The partitioning of the sum of squaresis not the same, though — and the difference may be much more dra-matic than it is here. The difference is whether log10(conc) is added toa model already containing glucose or vice versa. Since the second F testin both tables is highly significant, no model reduction is possible and theF test in the line above it is irrelevant.

If you go back and look at the regression coefficients in the model withparallel regression lines, you will see that the squares of the t tests are579.49 and 113.8, precisely the last F test in the two tables above.

It is informative to compare the covariance analysis above with thesimpler analysis in which the effect of cell concentration is ignored:

> t.test(log10(diameter)~glucose)

218 12. Linear models

Welch Two Sample t-test

data: log10(diameter) by glucoset = 2.7037, df = 36.31, p-value = 0.01037alternative hypothesis: true difference in means is not equal to 095 percent confidence interval:0.006492194 0.045424241sample estimates:mean in group Yes mean in group No

1.370046 1.344088

Notice that the p-value is much less extreme. It is still significant in thiscase, but in smaller data sets the statistical significance could easily disap-pear completely. The difference in means between the two groups is 0.026,which is comparable to the 0.028 that was the glucose effect in the analysisof covariance. However, the confidence interval goes from 0.006 to 0.045,whereas the analysis of covariance had 0.023 to 0.034 [0.0282 ± t.975(48) ×0.0026], which is almost four times as narrow, obviously a substantial gainin efficiency.

12.8 Diagnostics

Regression diagnostics are used to evaluate the model assumptions and in-vestigate whether or not there are observations with a large influenceon the analysis. A basic set of these is available via the plot method forlm objects. Four of them are displayed in a 2 × 2 layout (Figure 12.6) asfollows:

> attach(thuesen)> options(na.action="na.exclude")> lm.velo <- lm(short.velocity~blood.glucose)> opar <- par(mfrow=c(2,2), mex=0.6, mar=c(4,4,3,2)+.3)> plot(lm.velo, which=1:4)> par(opar)

The par commands set up for a 2×2 layout with compressed margin textsand go back to normal after plotting.

The top left panel shows residuals versus fitted values. The top right panelis a Q–Q normal distribution plot of standardized residuals. Notice thatthere are residuals and standardized residuals; the latter have been cor-rected for differences in the SD of residuals depending on their position inthe design. (Residuals corresponding to extreme x-values generally havea lower SD due to overfitting.) The third plot is of the square root of theabsolute value of the standardized residuals; this reduces the skewnessof the distribution and makes it much easier to detect if there might be a

12.8 Diagnostics 219

1.20 1.30 1.40 1.50

−0

.40

.00

.20

Fitted values

Resi

dua

lsResiduals vs Fitted

−2 −1 0 1 2

−2

−1

Theoretical Quantiles

Sta

ndard

ized

resi

dua

Normal Q−Q

1.20 1.30 1.40 1.50

0.0

0.5

1.0

1.5

Fitted values

Sta

ndard

ize

d r

esi

als

Scale−Location13

2024

5 10 15 20

0.0

0.2

0.4

0.6

Obs. number

Cook’

s dis

tance

Cook’s distance13

Figure 12.6. Regression diagnostics.

trend in the dispersion. The fourth plot is of “Cook’s distance”, which is ameasure of the influence of each observation on the regression coefficients.We will return to Cook’s distance shortly. Actually, this is not the defaultset of plots; the default replaces the fourth plot by a plot that contains thetwo components that enter into the calculation of Cook’s distance, but thisis harder to explain at this level.

The plots for the thuesen data show observation no. 13 as extreme inseveral respects. It has the largest residual as well as a prominent spike inthe Cook’s distance plot. Observation no. 20 also has a large residual, butnot quite as conspicuous a Cook’s distance.

> opar <- par(mfrow=c(2,2), mex=0.6, mar=c(4,4,3,2)+.3)> plot(rstandard(lm.velo))> plot(rstudent(lm.velo))> plot(dffits(lm.velo),type="l")> matplot(dfbetas(lm.velo),type="l", col="black")> lines(sqrt(cooks.distance(lm.velo)), lwd=2)> par(opar)

It is also possible to obtain individual diagnostics; a selection is shownin Figure 12.7. The function rstandard gives the standardized residuals

220 12. Linear models

5 10 15 20

−2

−1

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ard

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.ve

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.50.0

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Figure 12.7. Further regression diagnostics.

discussed above. There is also rstudent, which gives leave-out-one residu-als, in which the fitted value is calculated omitting the current point; if themodel is correct, then these will follow a (Student’s) t distribution. (Unfor-tunately, some texts use “studentized residuals” for residuals divided bytheir standard deviation; i.e., what rstandard calculates in R.) It takes akeen eye to see the difference between the two types of residuals, but theextreme residuals tend to be a little further out in the case of rstudent.

The function dffits expresses how much an observation affects the as-sociated fitted value. As with the residuals, observations 13 and maybe20 seem to stick out. Notice that there is a gap in the line. This is dueto the missing observation 16 and the use of na.exclude. This looksa little awkward but has the advantage of making the x-axis match theobservation number.

The function dfbetas gives the change in the estimated parameters ifan observation is excluded relative to its standard error. It is a matrix, somatplot is useful to plot them all in one plot. Notice that observation 13affects both α (the solid line) and β by nearly one standard error.

The name dfbetas refers to its use in multiple regression analysis, whereyou write the model as y = β0 + β1 x1 + β2 x2 + · · · . This gets a little con-

12.8 Diagnostics 221

fusing in a simple regression analysis, where the intercept is otherwisecalled α.

Cook’s distance D calculated by cooks.distance is essentially a jointmeasure of the components of dfbetas. The exact procedure is to takethe unnormalized change in coefficients and use the norm defined by theestimated covariance matrix for β̂ and then divide by the number of co-efficients.

√D is on the same scale as dfbetas and was added to that

plot as a double-width line. (If you look inside the R functions for someof these quantities, you will find them apparently quite different from thedescriptions above, but they are in fact the same, only computationallymore efficient.)

Thus, the picture is that observation 13 seems to be influential. Let us lookat the analysis without this observation.

We use the subset argument to lm, which, like other indexing operations,can be used with negative numbers to remove observations.

> summary(lm(short.velocity~blood.glucose, subset=-13))

Call:lm(formula = short.velocity ~ blood.glucose, subset = -13)

Residuals:Min 1Q Median 3Q Max

-0.31346 -0.11136 -0.01247 0.06043 0.40794

Coefficients:Estimate Std. Error t value Pr(>|t|)

(Intercept) 1.18929 0.11061 10.752 9.22e-10 ***blood.glucose 0.01082 0.01029 1.052 0.305—Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.193 on 20 degrees of freedom(1 observation deleted due to missingness)

Multiple R-squared: 0.05241, Adjusted R-squared: 0.005026F-statistic: 1.106 on 1 and 20 DF, p-value: 0.3055

The relation practically vanished in thin air! The whole analysis actuallyhinges on a single observation. If the data and model are valid, then ofcourse the original p-value is correct, and perhaps you could also say thatthere will always be influential observations in small data sets, but somecaution in the interpretation does seem advisable.

The methods for finding influential observations and outliers are evenmore important in regression analysis with multiple descriptive variables.One of the big problems is how to present the quantities graphically in asensible way. This might be done using three-dimensional plots (the add-

222 12. Linear models

110 120 130 140 150 160 170 180

height

weig

Figure 12.8. Cook’s distance (colour coded) in pemax ~ height + weight.

on package scatterplot3d makes this possible), but you can get quitefar using colour coding.

Here, we see how to display the value of Cook’s distance (which is alwayspositive) graphically for a model where pemax is described using heightand weight, as in Figure 12.8:

> cookd <- cooks.distance(lm(pemax~height+weight))> cookd <- cookd/max(cookd)> cook.colors <- gray(1-sqrt(cookd))> plot(height,weight,bg=cook.colors,pch=21,cex=1.5)> points(height,weight,pch=1,cex=1.5)

The first line computes Cook’s distance and the second scales it to a valuebetween 0 and 1. Thereafter, a colour coding of the values in cookd ismade with the function gray. The latter interprets its argument as the de-gree of whiteness, so if you want a large distance represented as black,you need to subtract the value from 1. Furthermore, it is convenient totake the square root of cookd because it is a quadratic distance measure(which in practice shows up in the form of too many white or nearly whitepoints). Then a scatterplot of height versus weight is drawn with the cho-

12.8 Diagnostics 223

70 80 90 100 110 120 130

100

bpd

Figure 12.9. Studentized residuals in the Secher data, colour coded. Positive valuesare marked with upward-pointing triangles; negative ones point down.

sen colours. A filled plotting symbol in enlarged symbol size is used to getthe grayscale to stand out more clearly.

You can use similar techniques to describe other influence measures. Inthe case of signed measures, you might use different symbols for positiveand negative values. Here is an example on Studentized residuals in adata set describing birth weight as a function of abdominal and biparietaldiameters determined by ultrasonography of the fetus immediately beforebirth, also used in Exercise 11.1 (Figure 12.9):

> attach(secher)> rst <- rstudent(lm(log10(bwt)~log10(ad)+log10(bpd)))> range(rst)[1] -3.707509 3.674050> rst <- rst/3.71> plot(ad,bpd,log="xy",bg=gray(1-abs(rst)),+ pch=ifelse(rst>0,24,25), cex=1.5)

224 12. Linear models

12.9 Exercises

12.1 Set up an additive model for the ashina data (see Exercise 5.6) con-taining additive effects of subjects, period, and treatment. Compare theresults with those obtained from t tests.

12.2 Perform a two-way analysis of variance on the tb.dilute data.Modify the model to have a dose effect that is linear in log dose. Computea confidence interval for the slope. An alternative approach could be tocalculate a slope for each animal and perform a test based on them. Com-pute a confidence interval for the mean slope, and compare it with thepreceding result.

12.3 Consider the following definitions:

a <- gl(2, 2, 8)b <- gl(2, 4, 8)x <- 1:8y <- c(1:4,8:5)z <- rnorm(8)

Generate the model matrices for models z ~ a*b, z ~ a:b, etc. Dis-cuss the implications. Carry out the model fits, and notice which modelscontain singularities.

12.4 (Advanced) In the secretin experiment, you may expect to findinter-individual differences not only in the level of glucose but also inthe change induced by the injection of secretin. The factor time.combcombines time values at 30, 60, and 90 minutes. The factor time20pluscombines all values from 20 minutes onward. Discuss the differences andrelations among the following linear models:

attach(secretin)model1 <- lm(gluc ~ person * time)model2 <- lm(gluc ~ person + time)model3 <- lm(gluc ~ person * time20plus + time)model4 <- lm(gluc ~ person * time20plus + time.comb)

12.5 Analyze the blood pressure in the bp.obese data set as a functionof obesity and gender.

12.6 Analyze the vitcap2 data set using analysis of covariance. RevisitExercise 5.2 and compare the conclusions. Try using the drop1 functionwith test="F" instead of summary in this model.

12.7 In the juul data set make regression analyses for prepubescent chil-dren (Tanner stage 1) of

√igf1 versus age separately for boys and girls.

Compare the two regression lines.

12.9 Exercises 225

12.8 Try step on the kfm data and discuss the result. One observationappears to be influential on the diagnostic plot for this model — explainwhy. What happens if you reduce the model further?

12.9 For the juul data, fit a model for igf1 with interactions betweenage, sex, and Tanner stage for those under 25 years old. Explain the inter-pretation of this model. Hint: A plot of the fitted values against age shouldbe helpful. Use diagnostic plots to evaluate possible transformations of thedependent variable: untransformed, log, or square root.

13Logistic regression

Sometimes you wish to model binary outcomes, variables that can haveonly two possible values: diseased or nondiseased, and so forth. For in-stance, you want to describe the risk of getting a disease depending onvarious kinds of exposures. Chapter 8 discusses some simple techniquesbased on tabulation, but you might also want to model dose-response re-lationships (where the predictor is a continuous variable) or model theeffect of multiple variables simultaneously. It would be very attractive tobe able to use the same modelling techniques as for linear models.

However, it is not really attractive to use additive models for probabili-ties since they have a limited range and regression models could predictoff-scale values below zero or above 1. It makes better sense to modelthe probabilities on a transformed scale; this is what is done in logisticregression analysis.

A linear model for transformed probabilities can be set up as

logit p = β0 + β1 x1 + β2 x2 + . . . βk xk

in which logit p = log[p/(1 − p)] is the log odds. A constant additive ef-fect on the logit scale corresponds to a constant odds ratio. The choice ofthe logit function is not the only one possible, but it has some mathemat-ically convenient properties. Other choices do exist; the probit function(the quantile function of the normal distribution) or log(− log p), whichhas a connection to survival analysis models.

P. Dalgaard, Introductory Statistics with R,DOI: 10.1007/978-0-387-79054-1_13, © Springer Science+Business Media, LLC 2008

228 13. Logistic regression

One thing to notice about the logistic model is that there is no error term asin linear models. We are modelling the probability of an event directly, andthat in itself will determine the variability of the binary outcome. There isno variance parameter as in the normal distribution.

The parameters of the model can be estimated by the method of maximumlikelihood. This is a quite general technique, similar to the least-squaresmethod in that it finds a set of parameters that optimizes a goodness-of-fit criterion (in fact, the least-squares method itself is a slightly modifiedmaximum-likelihood procedure). The likelihood function L(β) is simply theprobability of the entire observed data set for varying parameters.

The deviance is the difference between the maximized value of −2 log Land the similar quantity under a “maximal model” that fits data perfectly.Changes in deviance caused by a model reduction will be approximatelyχ2-distributed with degrees of freedom equal to the change in the numberof parameters.

In this chapter, we see how to perform logistic regression analysis in R.There naturally is quite a large overlap with the material on linear modelssince the description of models is quite similar, but there are also somespecial issues concerning deviance tables and the specification of modelsfor pretabulated data.

13.1 Generalized linear models

Logistic regression analysis belongs to the class of generalized linear models.These models are characterized by their response distribution (here thebinomial distribution) and a link function, which transfers the mean valueto a scale in which the relation to background variables is described aslinear ans additive. In a logistic regression analysis, the link function islogit p = log[p/(1 − p)].

There are several other examples of generalized linear models; for in-stance, analysis of count data is often handled by the multiplicativePoisson model, where the link function is log λ, with λ the mean of thePoisson-distributed observation. All of these models can be handled usingthe same algorithm, which also allows the user some freedom to define hisor her own models by defining suitable link functions.

In R generalized linear models are handled by the glm function. Thisfunction is very similar to lm, which we have used many times for lin-ear normal models. The two functions use essentially the same modelformulas and extractor functions (summary, etc.), but glm also needs tohave specified which generalized linear model is desired. This is done via

13.2 Logistic regression on tabular data 229

the family argument. To specify a binomial model with logit link (i.e.,logistic regression analysis), you write family=binomial("logit").

13.2 Logistic regression on tabular data

In this section, we analyze the example concerning hypertension fromAltman (1991, p. 353). First, we need to enter data, which is done asfollows:

> no.yes <- c("No","Yes")> smoking <- gl(2,1,8,no.yes)> obesity <- gl(2,2,8,no.yes)> snoring <- gl(2,4,8,no.yes)> n.tot <- c(60,17,8,2,187,85,51,23)> n.hyp <- c(5,2,1,0,35,13,15,8)> data.frame(smoking,obesity,snoring,n.tot,n.hyp)smoking obesity snoring n.tot n.hyp

1 No No No 60 52 Yes No No 17 23 No Yes No 8 14 Yes Yes No 2 05 No No Yes 187 356 Yes No Yes 85 137 No Yes Yes 51 158 Yes Yes Yes 23 8

The gl function to “generate levels” was briefly introduced in Section 7.3.The first three arguments to gl are, respectively, the number of levels,the repeat count of each level, and the total length of the vector. A fourthargument can be used to specify the level names of the resulting factor.The result is apparent from the printout of the generated variables. Theywere put together in a data frame to get a nicer layout. Another way ofgenerating a regular pattern like this is to use expand.grid:

> expand.grid(smoking=no.yes, obesity=no.yes, snoring=no.yes)smoking obesity snoring

1 No No No2 Yes No No3 No Yes No4 Yes Yes No5 No No Yes6 Yes No Yes7 No Yes Yes8 Yes Yes Yes

R is able to fit logistic regression analyses for tabular data in two differentways. You have to specify the response as a matrix, where one column is

230 13. Logistic regression

the number of “diseased” and the other is the number of “healthy” (or“success” and “failure”, depending on context).

> hyp.tbl <- cbind(n.hyp,n.tot-n.hyp)> hyp.tbl

n.hyp[1,] 5 55[2,] 2 15[3,] 1 7[4,] 0 2[5,] 35 152[6,] 13 72[7,] 15 36[8,] 8 15

The cbind function (“c” for “column”) is used to bind variables together,columnwise, to form a matrix. Note that it would be a horrible mistake touse the total count for column 2 instead of the number of failures.

Then, you can specify the logistic regression model as

> glm(hyp.tbl~smoking+obesity+snoring,family=binomial("logit"))

Actually, "logit" is the default for binomial and the family argumentis the second argument to glm, so it suffices to write

> glm(hyp.tbl~smoking+obesity+snoring,binomial)

The other way to specify a logistic regression model is to give theproportion of diseased in each cell:

> prop.hyp <- n.hyp/n.tot> glm.hyp <- glm(prop.hyp~smoking+obesity+snoring,+ binomial,weights=n.tot)

It is necessary to give weights because R cannot see how manyobservations a proportion is based on.

As output, you get in either case (except for minor details)

Call: glm(formula = hyp.tbl ~ smoking + obesity + snoring, …

Coefficients:(Intercept) smokingYes obesityYes snoringYes

-2.37766 -0.06777 0.69531 0.87194

Degrees of Freedom: 7 Total (i.e. Null); 4 ResidualNull Deviance: 14.13Residual Deviance: 1.618 AIC: 34.54

13.2 Logistic regression on tabular data 231

which is in a minimal style similar to that used for printing lm objects.Also in the result of glm is some nonvisible information, which may beextracted with particular functions. You can, for instance, save the resultof a fit of a generalized linear model in a variable and obtain a table ofregression coefficients and so forth using summary:

> glm.hyp <- glm(hyp.tbl~smoking+obesity+snoring,binomial)> summary(glm.hyp)

Call:glm(formula = hyp.tbl ~ smoking + obesity + snoring, family …

Deviance Residuals:1 2 3 4 5 6

-0.04344 0.54145 -0.25476 -0.80051 0.19759 -0.466027 8

-0.21262 0.56231

Coefficients:Estimate Std. Error z value Pr(>|z|)

(Intercept) -2.37766 0.38018 -6.254 4e-10 ***smokingYes -0.06777 0.27812 -0.244 0.8075obesityYes 0.69531 0.28509 2.439 0.0147 *snoringYes 0.87194 0.39757 2.193 0.0283 *—Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for binomial family taken to be 1)

Null deviance: 14.1259 on 7 degrees of freedomResidual deviance: 1.6184 on 4 degrees of freedomAIC: 34.537

Number of Fisher Scoring iterations: 4

In the following, we go through the components of summary output forgeneralized linear models:

Call:glm(formula = hyp.tbl ~ smoking + obesity + snoring, family = …

As usual, we start off with a repeat of the model specification. Obviously,more interesting is when the output is not viewed in connection with thefunction call that generated it.

Deviance Residuals:1 2 3 4 5 6

-0.04344 0.54145 -0.25476 -0.80051 0.19759 -0.466027 8

-0.21262 0.56231

232 13. Logistic regression

This is the contribution of each cell of the table to the deviance of themodel (the deviance corresponds to the sum of squares in linear normalmodels), with a sign according to whether the observation is larger orsmaller than expected. They can be used to pinpoint cells that are par-ticularly poorly fitted, but you have to be wary of the interpretation insparse tables.

Coefficients:Estimate Std. Error z value Pr(>|z|)

(Dispersion parameter for binomial family taken to be 1)

This is the table of primary interest. Here, we get estimates of the re-gression coefficients, standard errors of same, and tests for whether eachregression coefficient can be assumed to be zero. The layout is nearlyidentical to the corresponding part of the lm output.

The note about the dispersion parameter is related to the fact that the bino-mial variance depends entirely on the mean. There is no scale parameterlike the variance in the normal distribution.

Null deviance: 14.1259 on 7 degrees of freedomResidual deviance: 1.6184 on 4 degrees of freedomAIC: 34.537

“Residual deviance” corresponds to the residual sum of squares in ordi-nary regression analyses which is used to estimate the standard deviationabout the regression line. In binomial models, however, the standard devi-ation of the observations is known, and you can therefore use the deviancein a test for model specification. The AIC (Akaike information criterion) isa measure of goodness of fit that takes the number of fitted parametersinto account.

R is reluctant to associate a p-value with the deviance. This is just as wellbecause no exact p-value can be found, only an approximation that is validfor large expected counts. In the present case, there are actually a coupleof places where the expected cell count is rather small.

The asymptotic distribution of the residual deviance is a χ2 distributionwith the stated degrees of freedom, so even though the approximationmay be poor, nothing in the data indicates that the model is wrong (the5% significance limit is at 9.49 and the value found here is 1.62).

13.2 Logistic regression on tabular data 233

The null deviance is the deviance of a model that contains only the in-tercept (that is, describes a fixed probability, here for hypertension, in allcells). What you would normally be interested in is the difference from theresidual deviance, here 14.13 − 1.62 = 12.51, which can be used for a jointtest for whether any effects are present in the model. In the present case, ap-value of approximately 0.6% is obtained.

Number of Fisher Scoring iterations: 4

This refers to the actual fitting procedure and is a purely technical item.There is no statistical information in it, but you should keep an eye onwhether the number of iterations becomes too large because that might bea sign that the model is too complex to fit based on the available data. Nor-mally, glm halts the fitting procedure if the number of iterations exceeds25, but it is possible to configure the limit.

The fitting procedure is iterative in that there is no explicit formula thatcan be used to compute the estimates, only a set of equations that theyshould satisfy. However, there is an approximate solution of the equationsif you supply an initial guess at the solution. This solution is then used asa starting point for an improved solution, and the procedure is repeateduntil the guesses are sufficiently stable.

A table of correlations between parameter estimates can be obtained viathe optional argument corr=T to summary (this also works for linearmodels). It looks like this:

Correlation of Coefficients:(Intercept) smokingYes obesityYes

smokingYes -0.1520obesityYes -0.1361 -9.499e-05snoringYes -0.8965 -6.707e-02 -0.07186

It is seen that the correlation between the estimates is fairly small, so that itmay be expected that removing a variable from the model does not changethe coefficients and p-values for other variables much. (The correlationsbetween the regression coefficients and intercept are not very informative;they mostly relate to whether the variable in question has many or fewobservations in the “Yes” category.)

The z test in the table of regression coefficients immediately shows thatthe model can be simplified by removing smoking. The result then looksas follows (abbreviated):

> glm.hyp <- glm(hyp.tbl~obesity+snoring,binomial)> summary(glm.hyp)

…

234 13. Logistic regression

Coefficients:Estimate Std. Error z value Pr(>|z|)

(Intercept) -2.3921 0.3757 -6.366 1.94e-10 ***obesityYes 0.6954 0.2851 2.440 0.0147 *snoringYes 0.8655 0.3967 2.182 0.0291 *

13.2.1 The analysis of deviance table

Deviance tables correspond to ANOVA tables for multiple regressionanalyses and are generated like these with the anova function:

> glm.hyp <- glm(hyp.tbl~smoking+obesity+snoring,binomial)> anova(glm.hyp, test="Chisq")Analysis of Deviance Table

Model: binomial, link: logit

Response: hyp.tbl

Terms added sequentially (first to last)

Df Deviance Resid. Df Resid. Dev P(>|Chi|)NULL 7 14.1259smoking 1 0.0022 6 14.1237 0.9627obesity 1 6.8274 5 7.2963 0.0090snoring 1 5.6779 4 1.6184 0.0172

Notice that the Deviance column gives differences between models asvariables are added to the model in turn. The deviances are approximatelyχ2-distributed with the stated degrees of freedom. It is necessary to addthe test="chisq" argument to get the approximate χ2 tests.

Since the snoring variable on the last line is significant, it may not beremoved from the model and we cannot use the table to justify modelreductions. If, however, the terms are rearranged so that smoking comeslast, we get a deviance-based test for removal of that variable:

> glm.hyp <- glm(hyp.tbl~snoring+obesity+smoking,binomial)> anova(glm.hyp, test="Chisq")…

Df Deviance Resid. Df Resid. Dev P(>|Chi|)NULL 7 14.1259snoring 1 6.7887 6 7.3372 0.0092obesity 1 5.6591 5 1.6781 0.0174smoking 1 0.0597 4 1.6184 0.8069

From this you can read that smoking is removable, whereas obesity isnot, after removal of smoking.

13.2 Logistic regression on tabular data 235

For good measure, you should also set up the analysis with the two re-maining explanatory variables interchanged, so that you get a test ofwhether snoring may be removed from a model that also containsobesity:

> glm.hyp <- glm(hyp.tbl~obesity+snoring,binomial)> anova(glm.hyp, test="Chisq")…

Df Deviance Resid. Df Resid. Dev P(>|Chi|)NULL 7 14.1259obesity 1 6.8260 6 7.2999 0.0090snoring 1 5.6218 5 1.6781 0.0177

An alternative method is to use drop1 to try removing one term at a time:

> drop1(glm.hyp, test="Chisq")Single term deletions

Model:hyp.tbl ~ obesity + snoring

Df Deviance AIC LRT Pr(Chi)<none> 1.678 32.597obesity 1 7.337 36.256 5.659 0.01737 *snoring 1 7.300 36.219 5.622 0.01774 *—Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Here LRT is the likelihood ratio test, another name for the deviancechange.

The information in the deviance tables is fundamentally the same as thatgiven by the z tests in the table of regression coefficients. The results maydiffer due to the use of different approximations, though. From theoreticalconsiderations, the deviance test is preferable, but in practice the differ-ence is often small because of the large-sample approximation χ2 ≈ z2for tests with a single degree of freedom. However, to test factors withmore than two categories, you have to use the deviance table because thez tests only relate to some of the possible group comparisons. Also, thesmall-sample situation requires special attention; see the next section.

13.2.2 Connection to test for trend

In Chapter 8, we considered tests for comparing relative frequencies usingprop.test and prop.trend.test, in particular the example of cae-sarean section versus shoe size. This example can also be analyzed as alogistic regression analysis on a “shoe score”, which — for want of a bet-ter idea — may be chosen as the group number. This gives essentially thesame analysis in the sense that the same models are involved.

236 13. Logistic regression

> caesar.shoe<4 4 4.5 5 5.5 6+

Yes 5 7 6 7 8 10No 17 28 36 41 46 140> shoe.score <- 1:6> shoe.score[1] 1 2 3 4 5 6

> summary(glm(t(caesar.shoe)~shoe.score,binomial))…Coefficients:

Estimate Std. Error z value Pr(>|z|)(Intercept) -0.87058 0.40506 -2.149 0.03161 *shoe.score -0.25971 0.09361 -2.774 0.00553 **—Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for binomial family taken to be 1)

Null deviance: 9.3442 on 5 degrees of freedomResidual deviance: 1.7845 on 4 degrees of freedomAIC: 27.616…

Notice that caesar.shoe had to be transposed with t(…), so that thematrix was “stood on its end” in order to be used as the response variableby glm.

You can also write the results in a deviance table

> anova(glm(t(caesar.shoe)~shoe.score,binomial))…

Df Deviance Resid. Df Resid. DevNULL 5 9.3442shoe.score 1 7.5597 4 1.7845

from the last line of which you see that there is no significant deviationfrom linearity (1.78 on 4 degrees of freedom), whereas shoe.score has asignificant contribution.

For comparison, the previous analyses using standard tests are repeated:

> caesar.shoe.yes <- caesar.shoe["Yes",]> caesar.shoe.no <- caesar.shoe["No",]> caesar.shoe.total <- caesar.shoe.yes+caesar.shoe.no> prop.trend.test(caesar.shoe.yes,caesar.shoe.total)

Chi-squared Test for Trend in Proportions…X-squared = 8.0237, df = 1, p-value = 0.004617

> prop.test(caesar.shoe.yes,caesar.shoe.total)

13.3 Likelihood profiling 237

6-sample test for equality of proportions withoutcontinuity correction

…X-squared = 9.2874, df = 5, p-value = 0.09814…Warning message:In prop.test(caesar.shoe.yes, caesar.shoe.total) :Chi-squared approximation may be incorrect

The 9.29 from prop.test corresponds to the 9.34 in residual deviancefrom a NULL model, whereas the 8.02 in the trend test corresponds tothe 7.56 in the test of significance of shoe.score. Thus, the tests do notgive exactly the same result but generally almost the same. Theoreticalconsiderations indicate that the specialized trend test is probably slightlybetter than the regression-based test. However, testing the linearity bysubtracting the two χ2 tests is definitely not as good as the real test forlinearity.

13.3 Likelihood profiling

The z tests in the summary output are based on the Wald approximation,which calculates what the approximate standard error of the parameterestimate would be if the true values of the parameters were equal to the es-timates. In large data sets, this is fine because the result is nearly the samefor all parameter values that fit the data reasonably well. In smaller datasets, however, the difference between the Wald tests and the likelihoodratio test can be considerable.

This also affects the calculation of confidence intervals since these arebased on inverting the tests, giving a set of parameter values that arenot rejected by a statistical test. As an alternative to the Wald-based±1.96 × s.e. technique, the MASS package allows you to compute inter-vals that are based on inverting the likelihood ratio test. In practice, thisworks like this

> confint(glm.hyp)Waiting for profiling to be done…

2.5 % 97.5 %(Intercept) -3.2102369 -1.718143obesityYes 0.1254382 1.246788snoringYes 0.1410865 1.715860

The standard type of result can be obtained using confint.default.The difference in this case is not very large, although visible in the linesrelating to snoring and the intercept:

238 13. Logistic regression

−4.0 −3.0 −2.0 −1.0

−4

−2

(Intercept)

tau

−0.5 0.0 0.5 1.0 1.5

−4

−2

obesityYes

tau

−0.5 0.5 1.0 1.5 2.0 2.5

−4

−2

snoringYes

tau

Figure 13.1. Profile plot for hypertension model.

> confint.default(glm.hyp)2.5 % 97.5 %

(Intercept) -3.12852108 -1.655631obesityYes 0.13670388 1.254134snoringYes 0.08801498 1.642902

The way this works is via likelihood profiling. For a set of trial values of theparameter, the likelihood is maximized over the other parameters in themodel. The result can be displayed in a profile plot as follows:

> library(MASS)> plot(profile(glm.hyp))

Notice that we need to load the MASS package at this point. (The functionwas used by confint earlier on, but without putting it on the searchpath.)

The plots require a little explanation. The quantity on the y-axis, labelledtau, is the signed square root of the likelihood ratio test.

τ(β) = sgn(β − β̂)√−2(`(β) − `(β̂))

13.4 Presentation as odds-ratio estimates 239

Here ` denotes the profile log-likelihood. The main idea is that when theprofile likelihood function is approximately quadratic, τ(β) is approxi-mately linear. Conversely, likelihood functions not well approximated bya quadratic show up as nonlinear profile plots.

One important thing to notice, though, is that although the profilingmethod will capture nonquadratic behaviour of the likelihood function,confidence intervals based on the likelihood ratio test will always belimited in accuracy by the approximation of the distribution of the test.

13.4 Presentation as odds-ratio estimates

In parts of the epidemiological literature, it has become traditional topresent logistic regression analyses in terms of odds ratios. In the caseof a quantitative covariate, this means odds ratio per unit change in thecovariate. That is, the antilogarithm (exp) of the regression coefficients isgiven instead of the coefficients themselves. Since standard errors makelittle sense after the transformation, it is also customary to give confidenceintervals instead. This can be obtained quite easily as follows:

> exp(cbind(OR=coef(glm.hyp), confint(glm.hyp)))Waiting for profiling to be done…

OR 2.5 % 97.5 %(Intercept) 0.09143963 0.04034706 0.1793989obesityYes 2.00454846 1.13364514 3.4791490snoringYes 2.37609483 1.15152424 5.5614585

The (Intercept) is really the odds of hypertension (for the not snoringnon-obese) and not an odds ratio.

13.5 Logistic regression using raw data

In this section, we again use Anders Juul’s data (see p. 85). For easy ref-erence, here is how to read data and convert the variables that describegroupings into factors (this time slightly simplified):

> juul$menarche <- factor(juul$menarche, labels=c("No","Yes"))> juul$tanner <- factor(juul$tanner)

In the following, we look at menarche as the response variable. This vari-able indicates for each girl whether or not she has had her first period. Itis coded 1 for “no” and 2 for “yes”. It is convenient to look at a subset ofdata consisting of 8–20-year-old girls. This can be extracted as follows:

240 13. Logistic regression

> juul.girl <- subset(juul,age>8 & age<20 &+ complete.cases(menarche))> attach(juul.girl)

For obvious reasons, no boys have a nonmissing menarche, so it is notnecessary to select on gender explicitly.

Then you can analyze menarche as a function of age like this:

> summary(glm(menarche~age,binomial))Call:glm(formula = menarche ~ age, family = binomial)

Deviance Residuals:Min 1Q Median 3Q Max

-2.32759 -0.18998 0.01253 0.12132 2.45922

Coefficients:Estimate Std. Error z value Pr(>|z|)

(Intercept) -20.0132 2.0284 -9.867 <2e-16 ***age 1.5173 0.1544 9.829 <2e-16 ***—Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for binomial family taken to be 1)

Null deviance: 719.39 on 518 degrees of freedomResidual deviance: 200.66 on 517 degrees of freedomAIC: 204.66

Number of Fisher Scoring iterations: 7

The response variable menarche is a factor with two levels, where the lastlevel is considered the event. It also works to use a variable that has thevalues 0 and 1 (but not, for instance, 1 and 2!).

Notice that from this model you can estimate the median menarcheal ageas the age where logit p = 0. A little thought (solve −20.0132 + 1.5173 ×age = 0) reveals that it is 20.0132/1.5173 = 13.19 years.

You should not pay too much attention to the deviance residuals in thiscase since they automatically become large in every case where the fittedprobability “goes against” the observations (which is bound to happen insome cases). The residual deviance is also difficult to interpret when thereis only one observation per cell.

A hint of a more complicated analysis is obtained by including the Tan-ner stage of puberty in the model. You should be warned that the exactinterpretation of such an analysis is quite tricky and qualitatively differentfrom the analysis of menarche as a function of age. It can be used for pre-diction purposes (although asking the girl whether she has had her first

13.6 Prediction 241

period would likely be much easier than determining her Tanner stage!),but the interpretation of the terms is not clear-cut.

> summary(glm(menarche~age+tanner,binomial))…Coefficients:

Estimate Std. Error z value Pr(>|z|)(Intercept) -13.7758 2.7630 -4.986 6.17e-07 ***age 0.8603 0.2311 3.723 0.000197 ***tanner2 -0.5211 1.4846 -0.351 0.725609tanner3 0.8264 1.2377 0.668 0.504313tanner4 2.5645 1.2172 2.107 0.035132 *tanner5 5.1897 1.4140 3.670 0.000242 ***…

Notice that there is no joint test for the effect of tanner. There are a cou-ple of significant z-values, so you would expect that the tanner variablehas some effect (which, of course, you would probably expect even in theabsence of data!). The formal test, however, must be obtained from thedeviances:

> drop1(glm(menarche~age+tanner,binomial),test="Chisq")…

Df Deviance AIC LRT Pr(Chi)<none> 106.599 118.599age 1 124.500 134.500 17.901 2.327e-05 ***tanner 4 161.881 165.881 55.282 2.835e-11 ***…

Clearly, both terms are highly significant.

13.6 Prediction

The predict function works for generalized linear models, too. Let usfirst consider the hypertension example, where data were given in tabularform:

> predict(glm.hyp)1 2 3 4 5 6

-2.3920763 -2.3920763 -1.6966575 -1.6966575 -1.5266180 -1.52661807 8

-0.8311991 -0.8311991

Recall that smoking was eliminated from the model, which is why theexpected values come in identical pairs.

These numbers are on the logit scale, which reveals the additive structure.Notice that 2.392 − 1.697 = 1.527 − 0.831 = 0.695 (except for roundoff er-

242 13. Logistic regression

ror), which is exactly the regression coefficient to obesity. Likewise, theregression coefficient to snoring is obtained by looking at the differences2.392 − 1.527 = 1.697 − 0.831 = 0.866.

To get predicted values on the response scale (i.e., probabilities), use thetype="response" argument to predict:

> predict(glm.hyp, type="response")1 2 3 4 5 6

0.08377892 0.08377892 0.15490233 0.15490233 0.17848906 0.178489067 8

0.30339158 0.30339158

These may also be obtained using fitted, although you then cannot usethe techniques for predicting on new data, etc.

In the analysis of menarche, the primary interest is probably in seeing aplot of the expected probabilities versus age (Figure 13.2). A crude plotcould be obtained using something like

plot(age, fitted(glm(menarche~age,binomial)))

(it will look better if a different plotting symbol in a smaller size, using thepch and cex arguments, is used) but here is a more ambitious plan:

> glm.menarche <- glm(menarche~age, binomial)> Age <- seq(8,20,.1)> newages <- data.frame(age=Age)> predicted.probability <- predict(glm.menarche,+ newages,type="resp")> plot(predicted.probability ~ Age, type="l")

This is Figure 13.2. Recall that seq generates equispaced vectors, here agesfrom 8 to 20 in steps of 0.1, so that connecting the points with lines willgive a nearly smooth curve.

13.7 Model checking

For tabular data it is obvious to try to compare observed and fittedproportions. In the hypertension example you get

> fitted(glm.hyp)1 2 3 4 5 6

0.08377892 0.08377892 0.15490233 0.15490233 0.17848906 0.178489067 8

0.30339158 0.30339158> prop.hyp

13.7 Model checking 243

8 10 12 14 16 18 20

0.0

0.2

0.4

0.6

0.8

1.0

Age

pre

dic

ted.p

robabili

Figure 13.2. Fitted probability of menarche having occurred.

[1] 0.08333333 0.11764706 0.12500000 0.00000000 0.18716578[6] 0.15294118 0.29411765 0.34782609

The problem with this is that you get no feeling for how well the rela-tive frequencies are determined. It can be better to look at observed andexpected counts instead. The former can be computed as

> fitted(glm.hyp)*n.tot1 2 3 4 5 6

5.0267351 1.4242416 1.2392186 0.3098047 33.3774535 15.17156987 8

15.4729705 6.9780063

and to get a nice print for the comparison, you can use

> data.frame(fit=fitted(glm.hyp)*n.tot,n.hyp,n.tot)fit n.hyp n.tot

1 5.0267351 5 602 1.4242416 2 173 1.2392186 1 84 0.3098047 0 25 33.3774535 35 1876 15.1715698 13 85

244 13. Logistic regression

8 10 12 14 16 18 20

0.0

0.2

0.4

0.6

0.8

1.0

Age

pre

dic

ted.p

robabili

Figure 13.3. Fitted probability for menarche having occurred and observedproportion in age groups.

7 15.4729705 15 518 6.9780063 8 23

Notice that the discrepancy in cell 4 between 15% expected and 0% ob-served really is that there are 0 hypertensives out of 2 in a cell where themodel yields an expectation of 0.3 hypertensives!

For complex models with continuous background variables, it becomesmore difficult to perform an adequate model check. It is especially ahindrance that nothing really corresponds to a residual plot when theobservations have only two different values.

Consider the example of the probability of menarche as a function of age.The problem here is whether the relation can really be assumed linear onthe logit scale. For this case, you might try subdividing the x-axis in anumber of intervals and see how the counts in each interval fit with theexpected probabilities. This is presented graphically in Figure 13.3. Noticethat the code adds points to Figure 13.2, which you are assumed not tohave deleted at this point.

> age.group <- cut(age,c(8,10,12,13,14,15,16,18,20))

13.7 Model checking 245

> tb <- table(age.group,menarche)> tb

menarcheage.group No Yes(8,10] 100 0(10,12] 97 4(12,13] 32 21(13,14] 22 20(14,15] 5 36(15,16] 0 31(16,18] 0 105(18,20] 0 46

> rel.freq <- prop.table(tb,1)[,2]> rel.freq

(8,10] (10,12] (12,13] (13,14] (14,15] (15,16]0.00000000 0.03960396 0.39622642 0.47619048 0.87804878 1.00000000

(16,18] (18,20]1.00000000 1.00000000> points(rel.freq ~ c(9,11,12.5,13.5,14.5,15.5,17,19),pch=5)

The technique used above probably requires some explanation. First, cutis used to define the factor age.group, which describes a groupinginto age intervals. Then a crosstable tb is formed from menarche andage.group. Using prop.table, the numbers are expressed relative tothe row total, and column 2 of the resulting table is extracted. This con-tains the relative proportion in each age group of girls for whom menarchehas occurred. Finally, a plot of expected probabilities is made, overlaid bythe observed proportions.

The plot looks reasonable on the whole, although the observed proportionamong 12–13-year-olds appears a bit high and the proportion among 13–14-year-olds is a bit too low.

But how do you evaluate whether the deviation is larger than what canbe expected from the statistical variation? One thing to try is to extend themodel with a factor that describes a division into intervals. It is not prac-tical to use the full division of age.group because there are cells whereeither none or all of the girls have had their menarche.

We therefore try a division into four groups, with cutpoints at 12, 13, and14 years, and add this factor to the model containing a linear age effect.

> age.gr <- cut(age,c(8,12,13,14,20))> summary(glm(menarche~age+age.gr,binomial))…Coefficients:

Estimate Std. Error z value Pr(>|z|)(Intercept) -21.5683 5.0645 -4.259 2.06e-05 ***age 1.6250 0.4416 3.680 0.000233 ***age.gr(12,13] 0.7296 0.7856 0.929 0.353024age.gr(13,14] -0.5219 1.1184 -0.467 0.640765

246 13. Logistic regression

8 10 12 14 16 18 20

0.0

0.2

0.4

0.6

0.8

1.0

Age

pre

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Figure 13.4. Logit-cubical fit of menarche data.

age.gr(14,20] 0.2751 1.6065 0.171 0.864053…

> anova(glm(menarche~age+age.gr,binomial))…

Df Deviance Resid. Df Resid. DevNULL 518 719.39age 1 518.73 517 200.66age.gr 3 8.06 514 192.61> 1-pchisq(8.058,3)[1] 0.04482811

That is, the addition of the grouping actually does give a significantlybetter deviance. The effect is not highly significant, but since the devia-tion concerns the ages where “much happens”, you should probably becautious about postulating a logit-linear age effect.

Another possibility is to try a polynomial regression model. Here youneed at least a third-degree polynomial to describe the apparent stagna-tion of the curve around 13 years of age. We do not look at this in greatdetail, but just show part of the output and in Figure 13.4 a graphicalpresentation of the model.

13.8 Exercises 247

> anova(glm(menarche~age+I(age^2)+I(age^3)+age.gr,binomial))…

Df Deviance Resid. Df Resid. DevNULL 518 719.39age 1 518.73 517 200.66I(age^2) 1 0.05 516 200.61I(age^3) 1 8.82 515 191.80age.gr 3 3.34 512 188.46Warning messages:1: In glm.fit(x = X, y = Y, weights = weights, …. :fitted probabilities numerically 0 or 1 occurred

2: In method(x = x[, varseq <= i, drop = FALSE], …. :fitted probabilities numerically 0 or 1 occurred

> glm.menarche <- glm(menarche~age+I(age^2)+I(age^3), binomial)Warning message:In glm.fit(x = X, y = Y, weights = weights, start = start, …. :fitted probabilities numerically 0 or 1 occurred

> predicted.probability <-+ predict(glm.menarche, newages, type="resp")> plot(predicted.probability ~ Age, type="l")> points(rel.freq~c(9,11,12.5,13.5,14.5,15.5,17,19), pch=5)

The warnings about fitted probabilities of 0 or 1 occur because the cubicterm makes the logit tend much faster to ±∞ than the linear model did.There are two occurrences for the anova call because two of the modelsinclude the cubic term.

The thing to note in the deviance table is that the cubic term gives a sub-stantial improvement of the deviance, but once that is included, the agegrouping gives no additional improvement. The plot should speak foritself.

13.8 Exercises

13.1 In the malaria data set, analyze the risk of malaria with age andlog-transformed antibody level as explanatory variables.

13.2 Fit a logistic regression model to the graft.vs.host data set, pre-dicting the gvhd response. Use different transformations of the indexvariable. Reduce the model using backwards elimination.

13.3 In the analyses of the malaria and graft.vs.host data, try us-ing the confint function to find improved confidence intervals for theregression coefficients.

13.4 Following up on Exercise 8.2 about “Rocky Mountain spottedfever”, splitting the data by age groups gives the table below. Does this

248 13. Logistic regression

confirm the earlier analysis?

Western Type Eastern TypeAge Group Total Fatal Total FatalUnder 15 108 13 310 4015–39 264 40 189 2140 or above 375 157 162 61

747 210 661 122

13.5 A probit regression is just like a logistic regression but uses a differ-ent link function. Try the analysis of the menarche variable in the juuldata set with this link. Does the fit improve?

14Survival analysis

The analysis of lifetimes is an important topic within biology andmedicine in particular but also in reliability analysis with engineering ap-plications. Such data are often highly nonnormally distributed, so that theuse of standard linear models is problematic.

Lifetime data are often censored: You do not know the exact lifetime, onlythat it is longer than a given value. For instance, in a cancer trial, somepeople are lost to follow-up or simply live beyond the study period. Itis an error to ignore the censoring in the statistical analysis, sometimeswith extreme consequences. Consider, for instance, the case where a newtreatment is introduced towards the end of the study period, so that nearlyall the observed lifetimes will be cut short.

14.1 Essential concepts

Let X be the true lifetime and T a censoring time. What you observe isthe minimum of X and T together with an indication of whether it is oneor the other. T can be a random variable or a fixed time depending oncontext, but if it is random, then it should generally be noninformative forthe methods we describe here to be applicable. Sometimes “dead fromother causes” is considered a censoring event for the mortality of a given

P. Dalgaard, Introductory Statistics with R,DOI: 10.1007/978-0-387-79054-1_14, © Springer Science+Business Media, LLC 2008

250 14. Survival analysis

disease, and in those cases it is particularly important to ensure that theseother causes are unassociated with the disease state.

The survival function S(t) measures the probability of being alive at a giventime. It is really just 1 minus the cumulative distribution function for X,1 − F(t).

The hazard function or force of mortality h(t) measures the (infinitesimal)risk of dying within a short interval of time t, given that the subject is aliveat time t. If the lifetime distribution has density f , then h(t) = f (t)/S(t).This is often considered a more fundamental quantity than (say) the meanor median of the survival distribution and is used as a basis for modelling.

14.2 Survival objects

We use the package survival, written by Terry Therneau and portedto R by Thomas Lumley. The package implements a large number of ad-vanced techniques. For the present purposes, we use only a small subsetof it.

To load survival, use

> library(survival)

(This may produce a harmless warning about masking the lung data setfrom the ISwR package.)

The routines in survival work with objects of class "Surv", which isa data structure that combines times and censoring information. Such ob-jects are constructed using the Surv function, which takes two arguments:an observation time and an event indicator. The latter can be coded as alogical variable, a 0/1 variable, or a 1/2 variable. The latter coding is notrecommended since Surv will assume 0/1 coding if all values are 1.

Actually, Surv can also be used with three arguments for dealing withdata that have a start time as well as an end time (“staggered entry”) andalso interval censored data (where you know that an event happened be-tween two dates, as happens, for instance, in repeated testing for a disease)can be handled.

We use the data set melanom collected by K. T. Drzewiecki and re-produced in Andersen et al. (1991). The data become accessible asfollows:

> attach(melanom)> names(melanom)

14.3 Kaplan–Meier estimates 251

[1] "no" "status" "days" "ulc" "thick" "sex"

The variable status is an indicator of the patient’s status by the end ofthe study: 1 means “dead from malignant melanoma”, 2 means “alive onJanuary 1, 1978”, and 3 means “dead from other causes”. The variabledays is the observation time in days, ulc indicates (1 for present and 2for absent) whether the tumor was ulcerated, thick is the thickness in1/100 mm, and sex contains the gender of the patient (1 for women and2 for men).

We want to create a Surv object in which we consider the values 2 and 3of the status variable as censorings. This is done as follows:

> Surv(days, status==1)[1] 10+ 30+ 35+ 99+ 185 204 210 232 232+ 279[11] 295 355+ 386 426 469 493+ 529 621 629 659[21] 667 718 752 779 793 817 826+ 833 858 869…[181] 3476+ 3523+ 3667+ 3695+ 3695+ 3776+ 3776+ 3830+ 3856+ 3872+[191] 3909+ 3968+ 4001+ 4103+ 4119+ 4124+ 4207+ 4310+ 4390+ 4479+[201] 4492+ 4668+ 4688+ 4926+ 5565+

Associated with the Surv objects is a print method that displays the ob-jects in the format above, with a ‘+’ marking censored observations. Forexample, 10+ means that the patient did not die from melanoma within10 days and was then unavailable for further study (in fact, he died fromother causes), whereas 185 means that the patient died from the disease alittle over half a year after his operation.

Notice that the second argument to Surv is a logical vector; status==1is TRUE for those who died of malignant melanoma and FALSE otherwise.

14.3 Kaplan–Meier estimates

The Kaplan–Meier estimator allows the computation of an estimated sur-vival function in the presence of right-censoring. It is also called theproduct-limit estimator because one way of describing the procedure isthat it multiplies together conditional survival curves for intervals inwhich there are either no censored observations or no deaths. This be-comes a step function where the estimated survival is reduced by a factor(1 − 1/Rt) if there is a death at time t and a population of Rt is still aliveand uncensored at that time.

Computing the Kaplan–Meier estimator for the survival function is donewith a function called survfit. In its simplest form, it takes just a single

252 14. Survival analysis

argument, namely a Surv object. It returns a survfit object. As de-scribed above, we consider “dead from other causes” a kind of censoringand do as follows:

> survfit(Surv(days,status==1))Call: survfit(formula = Surv(days, status == 1))

n events median 0.95LCL 0.95UCL205 57 Inf Inf Inf

As is seen, using survfit by itself is not very informative (just as theprinted output of a “bare” lm is not). You get a couple of summary statis-tics and an estimate of the median survival, and in this case the latter isnot even interesting because the estimate is infinite. The survival curvedoes not cross the 50% mark before all patients are censored.

To see the actual Kaplan–Meier estimate, use summary on the survfitobject. We first save the survfit object into a variable, here namedsurv.all because it contains the raw survival function for all patientswithout regard to patient characteristics.

> surv.all <- survfit(Surv(days,status==1))> summary(surv.all)Call: survfit(formula = Surv(days, status == 1))

time n.risk n.event survival std.err lower 95% CI upper 95% CI185 201 1 0.995 0.00496 0.985 1.000204 200 1 0.990 0.00700 0.976 1.000210 199 1 0.985 0.00855 0.968 1.000232 198 1 0.980 0.00985 0.961 1.000279 196 1 0.975 0.01100 0.954 0.997295 195 1 0.970 0.01202 0.947 0.994

…2565 63 1 0.689 0.03729 0.620 0.7662782 57 1 0.677 0.03854 0.605 0.7573042 52 1 0.664 0.03994 0.590 0.7473338 35 1 0.645 0.04307 0.566 0.735

This contains the values of the survival function at the event times. Thecensoring times are not displayed but are contained in the survfit objectand can be obtained by passing censored=T to summary (see the helppage for summary.survfit for such details).

The Kaplan–Meier estimate is the step function whose jump points aregiven in time and whose values right after a jump are given in survival.Additionally, both an estimate of the standard error of the curve and a(pointwise) confidence interval for the true curve are given.

14.3 Kaplan–Meier estimates 253

0 1000 2000 3000 4000 5000

0.0

0.2

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0.6

0.8

1.0

Figure 14.1. Kaplan–Meier plot for melanoma data (all observations).

Normally, you would be more interested in showing the Kaplan–Meierestimate graphically than numerically. To do this (Figure 14.1), you simplywrite

> plot(surv.all)

The markings on the curve indicate censoring times, and the bands giveapproximate confidence intervals. If you look closely, you will see that thebands are not symmetrical around the estimate. They are constructed as asymmetric interval on the log scale and transformed back to the originalscale.

It is often useful to plot two or more survival functions on the same plotso that they can be directly compared (Figure 14.2). To obtain survivalfunctions split by gender, do the following:

> surv.bysex <- survfit(Surv(days,status==1)~sex)> plot(surv.bysex)

That is, you use a model formula as in lm and glm, specifying that thesurvival object generated from day and status should be described bysex. Notice that there are no confidence intervals on the curves. These are

254 14. Survival analysis

0 1000 2000 3000 4000 5000

0.0

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Figure 14.2. Kaplan–Meier plots for melanoma data, grouped by gender.

turned off when there are two or more curves because the display easilybecomes confusing. They can be turned on again by passing conf.int=Tto plot, in which case it can be recommended to use separate colours forthe curves, as in

> plot(surv.bysex, conf.int=T, col=c("black","gray"))

Similarly, you can avoid plotting the confidence bands in the single-sample case by setting conf.int=F. If you want the bands but at a 99%confidence level, you should pass conf.int=0.99 to survfit. Noticethat the level of confidence is an argument to the fitting function (whichneeds it to compute the confidence limits), whereas the decision to plotthe bands is controlled by a similarly named argument to plot.

14.4 The log-rank test

The log-rank test is used to test whether two or more survival curves areidentical. It is based on looking at the population at each death time andcomputing the expected number of deaths in proportion to the number of

14.4 The log-rank test 255

individuals at risk in each group. This is then summed over all death timesand compared with the observed number of deaths by a procedure similar(but not identical) to the χ2 test. Notice that the interpretation of “ex-pected” and “observed” is slightly peculiar: If the difference in mortalityis sufficiently large, then you can easily “expect” the same individuals todie several times over the course of the trial. If the population is observedto extinction with no censoring, then the observed number of deaths willequal the group size by definition and the expected values will contain allthe random variation.

The log-rank test is formally nonparametric since the distribution of thetest statistic depends only on the assumption that the groups have thesame survival function. However, it can also be viewed as a model-basedtest under the assumption of proportional hazards (see Section 14.1). You canset up a semiparametric model in which the hazard itself is unspecifiedbut it is assumed that the hazards are proportional between groups. Test-ing that the proportionality factors are all unity then leads to a log-ranktest. The log-rank test will work best against this class of alternatives.

Computation of the log-rank test is done by the function survdiff.This actually implements a whole family of tests specified by a param-eter ρ, allowing various nonproportional hazards alternatives to the nullhypothesis, but the default value of ρ = 0 gives the log-rank test.

> survdiff(Surv(days,status==1)~sex)Call:survdiff(formula = Surv(days, status == 1) ~ sex)

N Observed Expected (O-E)^2/E (O-E)^2/Vsex=1 126 28 37.1 2.25 6.47sex=2 79 29 19.9 4.21 6.47

Chisq= 6.5 on 1 degrees of freedom, p= 0.011

The specification is using a model formula as for linear and generalizedlinear models. However, the test can deal only with grouped data, so ifyou specify multiple variables on the right-hand side it will work on thegrouping of data generated by all combinations of predictor variables. Italso makes no distinction between factors and numerical codes. The sameis true of survfit.

It is also possible to specify stratified analyses, in which the observed andexpected value calculations are carried out separately within a stratifica-tion of the data set. For instance, you can compute the log-rank test for agender effect stratified by ulceration as follows:

> survdiff(Surv(days,status==1)~sex+strata(ulc))Call:

256 14. Survival analysis

survdiff(formula = Surv(days, status == 1) ~ sex + strata(ulc))

N Observed Expected (O-E)^2/E (O-E)^2/Vsex=1 126 28 34.7 1.28 3.31sex=2 79 29 22.3 1.99 3.31

Chisq= 3.3 on 1 degrees of freedom, p= 0.0687

Notice that this makes the effect of sex appear less significant. A possibleexplanation might be that males seek treatment when the disease is ina more advanced state than women do, so that the gender difference isreduced when adjusted for a measure of disease progression.

14.5 The Cox proportional hazards model

The proportional hazards model allows the analysis of survival data byregression models similar to those of lm and glm. The scale on whichlinearity is assumed is the log-hazard scale. Models can be fitted via themaximization of Cox’s likelihood, which is not a true likelihood but it canbe shown that it may be used as one. It is calculated in a manner similarto that of the log-rank test, as the product of conditional likelihoods of theobserved death at each death time.

As a first example, consider a model with the single regressor sex:

> summary(coxph(Surv(days,status==1)~sex))Call:coxph(formula = Surv(days, status == 1) ~ sex)

n= 205coef exp(coef) se(coef) z p

sex 0.662 1.94 0.265 2.5 0.013

exp(coef) exp(-coef) lower .95 upper .95sex 1.94 0.516 1.15 3.26

Rsquare= 0.03 (max possible= 0.937 )Likelihood ratio test= 6.15 on 1 df, p=0.0131Wald test = 6.24 on 1 df, p=0.0125Score (logrank) test = 6.47 on 1 df, p=0.0110

The coef is the estimated logarithm of the hazard ratio between the twogroups, which for convenience is also given as the actual hazard ratioexp(coef). The line following that also gives the inverted ratio (swap-ping the groups) and confidence intervals for the hazard ratio. Finally,three overall tests for significant effects in the model are given. These areall equivalent in large samples but may differ somewhat in small-sample

14.5 The Cox proportional hazards model 257

cases. Notice that the Wald test is identical to the z test based on the es-timated coefficient divided by its standard error, whereas the score test isequivalent to the log-rank test (as long as the model involves only a simplegrouping).

A more elaborate example, involving a continuous covariate and astratification variable, is

> summary(coxph(Surv(days,status==1)~sex+log(thick)+strata(ulc)))Call:coxph(formula = Surv(days, status == 1) ~ sex + log(thick) +

strata(ulc))

n= 205

coef exp(coef) se(coef) z psex 0.36 1.43 0.270 1.33 0.1800log(thick) 0.56 1.75 0.178 3.14 0.0017

exp(coef) exp(-coef) lower .95 upper .95sex 1.43 0.698 0.844 2.43log(thick) 1.75 0.571 1.234 2.48

Rsquare= 0.063 (max possible= 0.9 )Likelihood ratio test= 13.3 on 2 df, p=0.00130Wald test = 12.9 on 2 df, p=0.00160Score (logrank) test = 13.0 on 2 df, p=0.00152

It is seen that the significance of the sex variable has been further reduced.

The Cox model assumes an underlying baseline hazard function with acorresponding survival curve. In a stratified analysis, there will be onesuch curve for each stratum. They can be extracted by using survfit onthe output of coxph and of course be plotted using the plot method forsurvfit objects (Figure 14.3):

> plot(survfit(coxph(Surv(days,status==1)~+ log(thick)+sex+strata(ulc))))

Be aware that the default for survfit is to generate curves for a pseudo-individual for which the covariates are at their mean values. In the presentcase, that would correspond to a tumor thickness of 1.86 mm and a gen-der of 1.39 (!). Notice that we have been sloppy in not defining sex as afactor variable, but that would not actually give a different result (coxphsubtracts the means of the regressors before fitting, so a 1/2 coding is thesame as 0/1, which is what a factor with treatment contrasts gives you).However, you can use the newdata argument of survfit to specify adata frame for which you want to calculate survival curves.

258 14. Survival analysis

0 500 1000 1500 2000 2500 3000

0.0

0.2

0.4

0.6

0.8

1.0

Figure 14.3. Baseline survival curves (ulcerated and nonulcerated tumors) instratified Cox regression.

14.6 Exercises

14.1 In the graft.vs.host data set, estimate the survival function forpatients with or without GVHD. Test the hypothesis that the survivalis the same in both groups. Extend the analysis by including the otherexplanatory variables.

14.2 With the Cox model in the last section of the text, generate a plotwith estimated survival curves for men with nonulcerated tumors ofthicknesses 0.1, 0.2, and 0.5 mm (three curves in one plot). Hint: survfitobjects can be indexed with [] to extract individual strata.

14.3 Fit Cox models to the stroke data with age and sex as predictorsand with sex alone. Explain the difference.

14.4 With the split data from Exercise 10.4, you can fit a Cox model withdelayed entry to the stroke data; help(Surv) shows how to set up theSurv object in that case. Refit the model(s) from the previous exercise.

15Rates and Poisson regression

Epidemiological studies often involve the calculation of rates, typicallyrates of death or incidence rates of a chronic or acute disease. This isbased upon counts of events occurring within a certain amount of time.The Poisson regression method is often employed for the statistical analy-sis of such data. However, data that are not actually counts of events butrather measurements of time until an event (or nonevent) can be analyzedby a technique which is formally equivalent.

15.1 Basic ideas

The data that we wish to analyze can be in one of two forms. They canbe in aggregate form as an observed count x based on a number of person-years T. Often the latter is an approximation based on tables of populationsize. There may of course be more than one group, and we may wish toformulate various models describing the rates in different groups.

We may also have individual-level data, in which for each subject we havea time under observation Ti and a 0/1 indicator xi of whether the subjecthas had an event. The aggregate data can be thought of as being x = ∑ xiand T = ∑ Ti, where the sums are over all individuals in the group.

P. Dalgaard, Introductory Statistics with R,DOI: 10.1007/978-0-387-79054-1_15, © Springer Science+Business Media, LLC 2008

260 15. Rates and Poisson regression

15.1.1 The Poisson distribution

The Poisson distribution can be described as the limiting case of the bino-mial distributions when the size parameter N increases while the expectednumber of successes λ = N p is fixed. This is useful to describe rare eventin large populations. The resulting distribution has point probabilities

f (x) =λx

x!e−λ x = 0, 1, . . .

The distribution is theoretically unbounded, although the probabilities forlarge x will be very small. In R, the Poisson distribution is available via thefunctions dpois, ppois, etc.

In the context of epidemiological data, the parameter of interest is usuallythe expected counts per unit of observed time; i.e., the rate at which eventsoccur. This enables the comparison of populations that may be of differ-ent size or observed for different lengths of time. Accordingly, we mayparameterize the Poisson distribution using

ρ = λ/T

Notice that parts of the literature use λ to denote the rate. The notationused here is chosen so as to stay compatible with the argument name indpois.

The Poisson likelihood

Models for Poisson data can be fitted by the method of maximumlikelihood. If we parameterize in terms of ρ, the log-likelihood becomes

l(ρ) = constant + x log ρ − ρT

which is maximized when ρ = x/T. The log-likelihood can be generalizedto models involving several counts by summing terms of the same form.

15.1.2 Survival analysis with constant hazard

In this section, for convenience, we use terminology appropriate for mor-tality studies, although the event may be many things other than the deathof the subject.

Individual-level data are essentially survival data as described in Chap-ter 14, except for changes in notation. One difference, though, is that inthe analysis of rates it is often reasonable to assume that the hazard doesnot change over time, or at least not abruptly so. Rates tend to be obtainedover rather short individual time periods, and the origin of the timescale

15.1 Basic ideas 261

is not usually keyed to a life-changing event such as disease onset or majorsurgery.

If the hazard is constant, then the distribution of the lifetime is theexponential distribution with density ρe−ρt and survival function e−ρt.

Likelihood analysis

Likelihoods for censored data can be constructed using terms that areeither the probability density at the time of death or the survival prob-ability in the case of censoring. In the constant-hazard case, the two kindsof terms differ only in the presence of the factor ρ, which we may conve-niently encode using the event indicator xi so that the log-likelihood termsare

l(ρ) = xi log ρ − ρTi

Except for the constant, which does not depend on ρ, these terms are for-mally identical to a Poisson likelihood, where the count is 1 (death) orzero (censoring). This is the crucial “trick” that allows survival data withconstant hazard to be analyzed by Poisson regression methods.

The trick can be extended to hazards that are only piecewise constant.Suppose the lifetime of an individual is subdivided as Ti = T

(1)i + · · · +

T(k)i , where the hazard is assumed constant during each section of time.The corresponding log-likelihood term is

l(ρ1, . . . , ρk) =k

∑j=1

(x(j)i log ρj − ρj T(j)i )

in which the first k − 1 of the x(j)i will be 0, and only the last one, x(k)i ,

can be either 0 or one. The point of writing it in this elaborate form isthat it then becomes obvious that the likelihood contribution might as wellhave come from k different individuals where the first k − 1 had censoredobservations.

This is the rationale behind time-splitting techniques where the obser-vation time of one subject is divided into observations for multiplepseudo-individuals.

It should be noted that although the models with (piecewise) constant haz-ard can be fitted and analyzed by likelihood techniques, pretending thatthe data have come from a Poisson distribution, this does not extend toall aspects of the model. For instance following a cohort to extinction willlead to a fixed total number of events by definition, whereas the corre-sponding Poisson model implies that the total event count has a Poisson

262 15. Rates and Poisson regression

distribution. Both types of models deal in rates, counts per time, but thedifference is to what extent the random variation lies in the counts or in theamount of time. When data are frequently censored (i.e., the event is rare),the survival model becomes well approximated by the Poisson model.

15.2 Fitting Poisson models

The class of generalized linear models (see Section 13.1) also includes thePoisson distribution, which by default uses a log link function. This is themathematically convenient option and also a quite natural choice since itallows the linear predictor to span the entire real line. We can use this toformulate models for the log rates of the form

log ρ = β0 + β1 x1 + β2 x2 + . . . βk xkor, since glm needs a model for the expected counts rather than rates,

log λ = β0 + β1 x1 + β2 x2 + . . . βk xk + log T

A feature of many Poisson models is that the model contains an offset inthe linear predictor, log T in this case. Notice that this is not the same asincluding the term as a regression variable since the regression coefficientis fixed at 1.

The following example was used by Erling B. Andersen in 1977. It in-volves the rates of lung cancer by age in four Danish cities and may befound as eba1977 in the ISwR package.

> names(eba1977)[1] "city" "age" "pop" "cases"> attach(eba1977)

To fit a model that has multiplicative effects of age and city on therate of lung cancer cases, we use the glm function in much the sameway as in logistic regression. Of course, we need to change the familyargument to accommodate Poisson-distributed data. We also need to in-corporate an offset to account for the different sizes and age structures ofthe populations in the four cities.

> fit <- glm(cases~city+age+offset(log(pop)), family=poisson)> summary(fit)Call:glm(formula = cases ~ city + age + offset(log(pop)), family=poisson)

Deviance Residuals:Min 1Q Median 3Q Max

-2.63573 -0.67296 -0.03436 0.37258 1.85267

15.2 Fitting Poisson models 263

Coefficients:Estimate Std. Error z value Pr(>|z|)

(Intercept) -5.6321 0.2003 -28.125 < 2e-16 ***cityHorsens -0.3301 0.1815 -1.818 0.0690 .cityKolding -0.3715 0.1878 -1.978 0.0479 *cityVejle -0.2723 0.1879 -1.450 0.1472age55-59 1.1010 0.2483 4.434 9.23e-06 ***age60-64 1.5186 0.2316 6.556 5.53e-11 ***age65-69 1.7677 0.2294 7.704 1.31e-14 ***age70-74 1.8569 0.2353 7.891 3.00e-15 ***age75+ 1.4197 0.2503 5.672 1.41e-08 ***—Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for poisson family taken to be 1)

Null deviance: 129.908 on 23 degrees of freedomResidual deviance: 23.447 on 15 degrees of freedomAIC: 137.84

Number of Fisher Scoring iterations: 5

The offset was included in the model formula in this case. Alternatively, itcould have been given as a separate argument as in

glm(cases~city+age, offset = log(pop), family=poisson)

The table labelled “Coefficients:” contains regression coefficients for thelinear predictor along with standard errors and z tests. These can be in-terpreted in the same way as in ordinary multiple regression or logisticregression. Since both variables are factors and we are using treatmentcontrasts (see Section 12.3), the coefficients indicate differences in the lograte (i.e., the log of the rate ratio) compared with the city of Fredericia andwith the 50–54-year-olds, respectively.

The intercept term refers to the log rate for the group of 50–54-year-oldsin Fredericia. Notice that because we used the population size rather thanthe number of person-years in the offset and the data cover the years 1968–1971, this rate will effectively be per 4 person-years.

A goodness-of-fit statistic is provided by comparing the residual devianceto a χ2 distribution on the stated degrees of freedom. This statistic is gen-erally considered valid if the expected count in all cells is larger than 5.Accordingly,

> min(fitted(fit))[1] 6.731286> pchisq(deviance(fit), df.residual(fit), lower=F)[1] 0.07509017

264 15. Rates and Poisson regression

and we see that the model fits the data acceptably. Of course, we couldalso just have read off the residual deviance and degrees of freedom fromthe summary output:

> pchisq(23.45, 15, lower=F)[1] 0.07504166

From the coefficient table, it is obvious that there is an age effect, but it isless clear whether there is a city effect. We can perform χ2 tests for eachterm by using drop1 and looking at the changes in the deviance.

> drop1(fit, test="Chisq")Single term deletions

Model:cases ~ city + age + offset(log(pop))

Df Deviance AIC LRT Pr(Chi)<none> 23.447 137.836city 3 28.307 136.695 4.859 0.1824age 5 126.515 230.903 103.068 <2e-16 ***…

We see that the age term is significant, hardly surprisingly, but the cityterm apparently is not. However, if you can argue a priori that Fredericiacould be expected to have a higher cancer rate than the three other cities,then it could be warranted to combine the three other cities into one andperform an analysis as below.

> fit2 <- glm(cases~(city=="Fredericia")+age+offset(log(pop)),+ family=poisson)> anova(fit, fit2, test="Chisq")Analysis of Deviance Table

Model 1: cases ~ city + age + offset(log(pop))Model 2: cases ~ (city == "Fredericia") + age + offset(log(pop))Resid. Df Resid. Dev Df Deviance P(>|Chi|)

1 15 23.44752 17 23.7001 -2 -0.2526 0.8814> drop1(fit2, test="Chisq")Single term deletions

Model:cases ~ (city == "Fredericia") + age + offset(log(pop))

Df Deviance AIC LRT Pr(Chi)<none> 23.700 134.088city == "Fredericia" 1 28.307 136.695 4.606 0.03185 *age 5 127.117 227.505 103.417 < 2e-16 ***…

According to this, you may combine the three cities other than Fredericia,and, once this is done, Fredericia does indeed appear to be significantly

15.2 Fitting Poisson models 265

different from the others. Alternatively, you can look at the coefficients infit2 directly

> summary(fit2)…Coefficients:

Estimate Std. Error z value Pr(>|z|)(Intercept) -5.9589 0.1809 -32.947 < 2e-16 ***city == "Fredericia"TRUE 0.3257 0.1481 2.200 0.0278 *age55-59 1.1013 0.2483 4.436 9.17e-06 ***age60-64 1.5203 0.2316 6.564 5.23e-11 ***age65-69 1.7687 0.2294 7.712 1.24e-14 ***age70-74 1.8592 0.2352 7.904 2.71e-15 ***age75+ 1.4212 0.2502 5.680 1.34e-08 ***…

and see the p-value of 0.0278. This agrees with the 0.03185 from drop1;you cannot expect the two p-values to be perfectly equal since they relyon different asymptotic approximations. If you really push it, you can ar-gue that a one-sided test with half the p-value is appropriate since youwould only expect Fredericia to be more harmful than the others, not less.However, the argumentation becomes tenuous, and in his paper Ander-sen outlines the possibility of testing Fredericia against the other cities butstops short of providing any p-value, stating that in his opinion “there isno reason to believe a priori that Fredericia is the more dangerous city”.

It is sometimes preferred to state the results of Poisson regression analy-sis in terms of rate ratios by taking exp() of the estimates (this parallelsthe presentation of logistic regression analysis in terms of odds ratios inSection 13.4). The intercept term is not really a ratio but a rate, and fornonfactor covariates it should be understood that the coefficient is the rel-ative change per unit change in the covariate. Because of the nonlineartransformation, standard errors are not useful; instead one can calculateconfidence intervals for the coefficients as follows:

> cf <- coefficients(summary(fit2))> est <- cf[,1]> s.e. <- cf[,2]> rr <- exp(cbind(est, est – s.e.*qnorm(.975), est+ + s.e.*qnorm(.975) ))> colnames(rr) <- c("RateRatio", "CI.lo","CI.hi")> rr

RateRatio CI.lo CI.hi(Intercept) 0.002582626 0.001811788 0.003681423city == "Fredericia"TRUE 1.384992752 1.036131057 1.851314957age55-59 3.008134852 1.849135187 4.893571521age60-64 4.573665854 2.904833526 7.201245496age65-69 5.863391064 3.740395488 9.191368903age70-74 6.418715646 4.047748963 10.178474731age75+ 4.142034525 2.536571645 6.763637070

266 15. Rates and Poisson regression

Actually, we can do better by using the confint function. This calcu-lates confidence intervals by profiling the likelihood function instead ofusing the approximation with the normal distribution inherent in the useof asymptotic standard errors. This is done like this:

> exp(cbind(coef(fit2), confint(fit2)))Waiting for profiling to be done…

2.5 % 97.5 %(Intercept) 0.002582626 0.001776461 0.003617228city == "Fredericia"TRUE 1.384992752 1.029362341 1.841224091age55-59 3.008134852 1.843578634 4.902339637age60-64 4.573665854 2.912314045 7.248143959age65-69 5.863391064 3.752718226 9.256907108age70-74 6.418715646 4.053262281 10.234338998age75+ 4.142034525 2.527117848 6.771833979

In the present case, we are well within the regime where the asymptoticnormal approximation works well, so there is little difference between thetwo displays. However, in some cases where some expected cell counts arelow and one or several coefficients are poorly determined, the differencecan be substantial.

15.3 Computing rates

We return to the Welsh nickel worker data discussed in Chapter 10. In thatsection, we discussed how to split the individual lifetime data into smallerpieces that could reasonably be merged with the standard mortality tablein the ewrates data.

The result of this initial data restructuring is in the nickel.expand dataset. It contains data from a lot of short time intervals like this:

> head(nickel.expand)agr ygr id icd exposure dob age1st agein ageout lung

1 0 31162 0 31163 0 31164 0 31165 0 31166 0 3116

15.3 Computing rates 267

The same individuals reappear later in the data at older ages. For example,all data for the individual with id number 325 are

> subset(nickel.expand, id==325)agr ygr id icd exposure dob age1st agein ageout lung

1 20 1931 325 0 0 1910.5 14.0737 23.7465 25.0000 613 25 1931 325 0 0 1910.5 14.0737 25.0000 30.0000 14172 30 1936 325 0 0 1910.5 14.0737 30.0000 35.0000 30391 35 1941 325 0 0 1910.5 14.0737 35.0000 40.0000 81728 40 1946 325 434 0 1910.5 14.0737 40.0000 43.0343 236

nasal other1 0 311613 0 3024172 1 3188391 1 3549728 3 3643

Accordingly, this subject enters the study at age 23.7 and we follow himthrough five age groups until his death at age 43.

The variable ygr reflects the year of entry into the interval, so even thoughthe subject dies in 1953, the last record is coded as belonging to the years1946–1950.

Subject no. 325 has the icd code 434 in his last record. This refers to theInternational Classification of Diseases (version 7) and indicates “Otherand unspecified diseases of the heart” as the cause of death. For the pur-poses of this chapter, we are primarily interested in lung cancer, which hascodes 162 and 163, so we define a variable to indicate whether this is thecause of death. (Expect a warning about masking the lung data set uponattaching.)

> nickel.expand <- within(nickel.expand,+ lung.cancer <- as.numeric(icd %in% c(162,163)))> attach(nickel.expand)

The %in% operator returns a logical vector that is TRUE when the corre-sponding element of the operand on the left is contained in the vector thatis the operand on the right and FALSE in all other cases. Use of this op-erator is slightly dangerous in the case of an NA element in icd, but inthese particular data, there are none. We convert the result to zero or onesince we are going to pretend that it is a Poisson count later on (this is notstrictly necessary). Notice that by using lung.cancer as the endpoint,we treat death from all other causes, including “unknown”, as censoring.

Each record provides ageout – agein person-years of risk time, so totabulate the risk times, we can just do as follows:

> pyr <- tapply(ageout-agein,list(ygr,agr), sum)> print(round(pyr), na.print="-")

268 15. Rates and Poisson regression

20 25 30 35 40 45 50 55 60 65 70 75 801931 3 86 268 446 446 431 455 323 159 23 4 – -1936 – – 100 327 504 512 503 472 314 130 20 5 -1941 – – 0 105 336 481 482 445 368 235 80 14 31946 – – – – 102 335 461 404 369 263 157 43 101951 – – – – – 95 299 415 334 277 181 92 311956 – – – – – – 89 252 364 257 181 101 521961 – – – – – – – 71 221 284 150 104 441966 – – – – – – – – 66 168 208 93 511971 – – – – – – – – – 57 133 131 541976 – – – – – – – – – – 31 68 53

Notice that there are many NA entries in cells that no subject ever entered.The subjects in the study were born between 1864 and 1910, so there is alarge block missing in the lower left and a smaller block in the upper right.The na.print option to print allows you to represent these missingvalues by a string that is less visually imposing than the default "NA".

The corresponding counts of lung cancer cases are obtained as

> count <- tapply(lung.cancer, list(ygr, agr), sum)> print(count, na.print="-")

20 25 30 35 40 45 50 55 60 65 70 75 801931 0 0 0 0 0 4 2 2 2 0 0 – -1936 – – 0 0 2 3 4 6 5 1 0 0 -1941 – – 0 0 0 3 7 5 6 3 2 0 01946 – – – – 0 0 8 7 6 2 2 0 01951 – – – – – 0 3 3 9 6 1 0 01956 – – – – – – 0 4 3 6 1 2 01961 – – – – – – – 0 1 1 3 2 11966 – – – – – – – – 2 0 0 1 01971 – – – – – – – – – 0 0 2 21976 – – – – – – – – – – 0 1 1

and the cancer rates can be obtained as the ratio of the counts to the risktime. These are small, so we multiply by 1000 to get rates per 1000 person-years.

> print(round(count/pyr*1000, 1), na.print="-")20 25 30 35 40 45 50 55 60 65 70 75 80

1931 0 0 0 0 0 9.3 4.4 6.2 12.6 0.0 0.0 – -1936 – – 0 0 4 5.9 7.9 12.7 15.9 7.7 0.0 0.0 -1941 – – 0 0 0 6.2 14.5 11.2 16.3 12.8 25.0 0.0 0.01946 – – – – 0 0.0 17.4 17.3 16.3 7.6 12.8 0.0 0.01951 – – – – – 0.0 10.0 7.2 27.0 21.7 5.5 0.0 0.01956 – – – – – – 0.0 15.9 8.2 23.4 5.5 19.8 0.01961 – – – – – – – 0.0 4.5 3.5 19.9 19.3 22.81966 – – – – – – – – 30.1 0.0 0.0 10.7 0.01971 – – – – – – – – – 0.0 0.0 15.2 36.81976 – – – – – – – – – – 0.0 14.6 19.0

15.3 Computing rates 269

Comparison of these rates with those in ewrates suggests that they arevery high. However, this kind of display has the disadvantage that it hidesthe actual counts on which the rates are based. For instance, the lowerpart of the column for 80–84-year-olds jumps by roughly 20 units for eachadditional case since there are only about 50 person-years per cell.

It may be better to compute the expected counts in each cell based onthe standard mortality table and then compare that to the actual counts.Since we have already merged in the ewrates data, this is just a matterof multiplying each piece of risk time by the rate. We need to divide by1e6 (i.e., 106 = 1000000) since the standard rates are given per millionperson-years.

> expect.count <- tapply(lung/1e6*(ageout-agein),+ list(ygr,agr), sum)> print(round(expect.count, 1), na.print="-")

20 25 30 35 40 45 50 55 60 65 70 75 801931 0 0 0 0 0.1 0.1 0.2 0.2 0.1 0.0 0.0 – -1936 – – 0 0 0.1 0.1 0.2 0.3 0.2 0.1 0.0 0.0 -1941 – – 0 0 0.1 0.2 0.3 0.4 0.4 0.2 0.1 0.0 0.01946 – – – – 0.0 0.2 0.4 0.5 0.6 0.5 0.2 0.0 0.01951 – – – – – 0.1 0.4 0.8 0.9 0.8 0.5 0.2 0.01956 – – – – – – 0.1 0.6 1.2 1.0 0.7 0.3 0.11961 – – – – – – – 0.2 0.8 1.4 0.7 0.5 0.11966 – – – – – – – – 0.2 0.9 1.3 0.6 0.21971 – – – – – – – – – 0.3 0.9 1.0 0.31976 – – – – – – – – – – 0.2 0.6 0.4

The observed counts are clearly much larger than expected. We can sum-marize them by calculating the overall SMR (standardized mortality rate),which is simply the ratio of the total number of cases to the total expectednumber of cases.

> expect.tot <- sum(lung/1e6*(ageout-agein))> expect.tot[1] 24.19893> count.tot <- sum(lung.cancer)> count.tot[1] 137> count.tot/expect.tot[1] 5.661408

That is, this data set has almost six times as many cancer deaths as youwould expect from the mortality of the general population.

270 15. Rates and Poisson regression

15.4 Models with piecewise constant intensities

We can formulate the SMR analysis as a “Poisson” regression model in thesense of Section 15.1.2. The assumption behind the SMR is that there is aconstant rate ratio to the standard mortality, so we can fit a model withonly an intercept while having an offset, which is the log of the expectedcount. This is not really different from modelling rates — the populationmortality ρi is just absorbed into the offset, log ρi + log Ti = log ρi Ti.

> fit <- glm(lung.cancer ~ 1, poisson,+ offset = log((ageout-agein)*lung/1e6))> summary(fit)…Coefficients:

Estimate Std. Error z value Pr(>|z|)(Intercept) 1.73367 0.08544 20.29 <2e-16 ***—Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for poisson family taken to be 1)

Null deviance: 1175.6 on 3723 degrees of freedomResidual deviance: 1175.6 on 3723 degrees of freedomAIC: 1451.6

Number of Fisher Scoring iterations: 7

Notice that this is based on individual data; the dependent variablelung.cancer is zero or one. We could have aggregated the data accord-ing to the cross-classification of agr and ygr and analyzed the number ofcases in each cell. This would have allowed glm to run much faster, but onthe other hand it would then not be possible to add individual covariatessuch as age at first exposure.

In this case, we cannot use the deviances for model checking both becausethe expected counts per cell are very small and because we do not actu-ally have Poisson-distributed data. However, the standard error and thep-value should be reliable if the assumptions hold.

The connection between this analysis and the SMR can be seen immedi-ately from

> exp(coef(fit))(Intercept)

5.661408

This value is exactly the SMR value from the previous section.

15.4 Models with piecewise constant intensities 271

We can analyze the data more thoroughly using regression methods. As afirst approach, we investigate whether the SMR is constant over year andage groups using a multiplicative Poisson model.

We need to simplify the groupings because some of the groups containvery few cases. By calculating the marginal tables of counts, we get someidea of what to do.

> tapply(lung.cancer, agr, sum)20 25 30 35 40 45 50 55 60 65 70 75 800 0 0 0 2 10 24 27 34 19 9 8 4> tapply(lung.cancer, ygr, sum)1931 1936 1941 1946 1951 1956 1961 1966 1971 197610 21 26 25 22 16 8 3 4 2

To get at least 10 cases per level, we combine all values of agr up to 45 (i.e.,ages less than 50) and also those from 70 and up. Similarly, we combineall values of ygr for the periods from 1961 onwards.

> detach()> nickel.expand <- within(nickel.expand,{+ A <- factor(agr)+ Y <- factor(ygr)+ lv <- levels(A)+ lv[1:6] <- "< 50"+ lv[11:13] <- "70+"+ levels(A) <- lv+ lv <- levels(Y)+ lv[7:10] <- "1961ff"+ levels(Y) <- lv+ rm(lv)+ })> attach(nickel.expand)

Notice that this is a case where the within function (see Section 2.1.8)works better than transform because it allows more flexibility, includingthe creation of temporary variables such as lv.

We can analyze the effect of A and Y on the mortality ratio by buildinga log-additive model in the usual way. Notice that we still use the orig-inal grouping in the calculation of the offset; it is only the SMR that isassumed to be the same for everyone below 50, etc. We use drop1 to testthe significance of the two factors.

> fit <- glm(lung.cancer ~ A + Y, poisson,+ offset=log((ageout-agein)*lung/1e6))> drop1(fit, test="Chisq")Single term deletions

Model:lung.cancer ~ A + Y

272 15. Rates and Poisson regression

Df Deviance AIC LRT Pr(Chi)<none> 1069.73 1367.73A 5 1073.81 1361.81 4.08 0.5376Y 6 1118.50 1404.50 48.77 8.29e-09 ***—Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

So it seems that we do not need the age grouping in the model, butthe year grouping is needed. Accordingly, we fit a model with Y alone,and by dropping the intercept, we get a parameterization with a separateintercept for each level of Y.

> fit <- glm(lung.cancer ~ Y – 1, poisson,+ offset=log((ageout-agein)*lung/1e6))> summary(fit)…Coefficients:

Estimate Std. Error z value Pr(>|z|)Y1931 2.6178 0.3162 8.279 < 2e-16 ***Y1936 3.0126 0.2182 13.805 < 2e-16 ***Y1941 2.7814 0.1961 14.182 < 2e-16 ***Y1946 2.2787 0.2000 11.394 < 2e-16 ***Y1951 1.8038 0.2132 8.461 < 2e-16 ***Y1956 1.3698 0.2500 5.479 4.27e-08 ***Y1961ff 0.4746 0.2425 1.957 0.0504 …..

The regression coefficients may again be recognized as log-SMR values, asthe following demonstrates:

> round(exp(coef(fit)), 1)Y1931 Y1936 Y1941 Y1946 Y1951 Y1956 Y1961ff13.7 20.3 16.1 9.8 6.1 3.9 1.6

> expect.count <- tapply(lung/1e6*(ageout-agein), Y, sum)> count <- tapply(lung.cancer, Y, sum)> cbind(count=count, expect=round(expect.count,1),+ SMR= round(count/expect.count, 1))

count expect SMR1931 10 0.7 13.71936 21 1.0 20.31941 26 1.6 16.11946 25 2.6 9.81951 22 3.6 6.11956 16 4.1 3.91961ff 17 10.6 1.6

The advantage of using the regression approach is that it provides a frame-work in which you can formulate statistical tests and investigate the effectof multiple regression variables simultaneously.

15.4 Models with piecewise constant intensities 273

Breslow and Day analyzed the nickel data in their seminal book (Bres-low and Day, 1987) on the analysis of cohort studies. In their analysis,they split the individual risk times according to three criteria, two of thembeing age and period, to match the standard mortality table, but theyalso treat time from employment as a time-dependent covariate with apiecewise constant effect, which requires that the person-year be split fur-ther according to the interval boundaries. They then represent time effectsusing three variables: time since, age at, and year of first employment,TFE, AFE, and YFE, respectively. In addition, they include a measure ofexposure level.

The following analysis roughly reproduces the Breslow and Day analysis.It is not completely similar because we settle for splitting time accord-ing to agr only and use the age at entry into each interval to define theTFE variable as well as for choosing the relevant standard mortality rates.However, to enable some comparison of results, we define cut groups ina manner that is similar to that of Breslow and Day.

> detach()> nickel.expand <- within(nickel.expand,{+ TFE <- cut(agein-age1st, c(0,20,30,40,50,100), right=F)+ AFE <- cut(age1st, c(0, 20, 27.5, 35, 100), right=F)+ YFE <- cut(dob + age1st, c(0, 1910, 1915, 1920, 1925),right=F)+ EXP <- cut(exposure, c(0, 0.5, 4.5, 8.5, 12.5, 25), right=F)+ })> attach(nickel.expand)

Some relabelling of group levels might be called for — e.g., the levels forEXP are really 0, 0.5–4, 4.5–8, 8.5–12, 12.5+ — but let us not make more ofit than necessary.

We fit a multiplicative model and test the significance of the individualterms as follows:

> fit <- glm(lung.cancer ~ TFE + AFE + YFE + EXP, poisson,+ offset=log((ageout-agein)*lung/1e6))> drop1(fit, test="Chisq")Single term deletions

Model:lung.cancer ~ TFE + AFE + YFE + EXP

Df Deviance AIC LRT Pr(Chi)<none> 1052.91 1356.91TFE 4 1107.33 1403.33 54.43 4.287e-11 ***AFE 3 1054.99 1352.99 2.08 0.5560839YFE 3 1058.06 1356.06 5.15 0.1608219EXP 4 1071.98 1367.98 19.07 0.0007606 ***

This suggests that the two major terms are TFE and EXP, whereas AFEand YFE could be taken out of the model. Notice, though, that it cannot be

274 15. Rates and Poisson regression

concluded from the above that both can be removed. In principle, one ofthem could become significant when the other is removed. This does nothappen in this case, though.

The table of coefficients looks like this:

> summary(fit)…Coefficients:

Estimate Std. Error z value Pr(>|z|)(Intercept) 2.36836 0.55716 4.251 2.13e-05 ***TFE[20,30) -0.21788 0.36022 -0.605 0.545284TFE[30,40) -0.77184 0.36529 -2.113 0.034605 *TFE[40,50) -1.87583 0.41707 -4.498 6.87e-06 ***TFE[50,100) -2.22142 0.55068 -4.034 5.48e-05 ***AFE[20,27.5) 0.28506 0.31524 0.904 0.365868AFE[27.5,35) 0.21961 0.34011 0.646 0.518462AFE[35,100) -0.10818 0.44412 -0.244 0.807556YFE[1910,1915) 0.04826 0.27193 0.177 0.859137YFE[1915,1920) -0.56397 0.37585 -1.501 0.133483YFE[1920,1925) -0.42520 0.30017 -1.417 0.156614EXP[0.5,4.5) 0.58373 0.21200 2.753 0.005897 **EXP[4.5,8.5) 1.03175 0.28364 3.638 0.000275 ***EXP[8.5,12.5) 1.18345 0.37406 3.164 0.001557 **EXP[12.5,25) 1.28601 0.48236 2.666 0.007674 **…

A dose-response pattern and a declining effect of time since first employ-ment seem to be present.

The results may be more readily interpreted if they are given in terms ofratios and confidence intervals. These can be obtained in exactly the sameway as in the analysis of the eba1977 data.

15.5 Exercises

15.1 In the bcmort data set, we defined the period and area factorsin Exercise 10.2. Fit a Poisson regression model to the data with age,period, and area as descriptors, as well as the three two-factor interac-tion terms. The interaction between period and area can be interpretedas the effect of screening.

15.2 With the split stroke data from Exercise 10.4, fit a Poisson regres-sion model corresponding to a constant hazard in each interval and withmultiplicative effects of age and sex.

16Nonlinear curve fitting

Curve fitting problems occur in many scientific areas. The typical caseis that you wish to fit the relation between some response y and aone-dimensional predictor x, by adjusting a (possibly multidimensional)parameter β. That is,

y = f (x; β) + error

in which the “error” term is usually assumed to contain independent nor-mally distributed terms with a constant standard deviation σ. The class ofmodels can be easily extended to multivariate x and somewhat less easilyto models with nonconstant error variation, but we settle for the simplecase.

Chapter 6 described the special case of a linear relation

y = β0 + β1 x + error

and we discussed the fitting of polynomials by including quadratic andhigher-order terms in Section 12.1. There are other techniques, notablytrigonometric regression and spline regression, that can also be formu-lated in linear form and handled by software for multiple regressionanalysis like lm.

However, sometimes linear methods are inadequate. The common case isthat you have a priori knowledge of the form of the function. This maycome from theoretical analysis of an underlying physical and chemical

P. Dalgaard, Introductory Statistics with R,DOI: 10.1007/978-0-387-79054-1_16, © Springer Science+Business Media, LLC 2008

276 16. Nonlinear curve fitting

system, and the parameters of the relation have a specific meaning in thattheory.

The method of least squares makes good sense even when the relationbetween data and parameters is not linear. That is, we can estimate β byminimizing

SSD(β) = ∑(y − f (x; β))2

There is no explicit formula for the location of the minimum, but the min-imization can be performed numerically by algorithms that we describeonly superficially here. This general technique is also known as nonlin-ear regression analysis. For an in-depth treatment of the topic, a standardreference is Bates and Watts (1988).

If the model is “well-behaved” (to use a deliberately vague term), then themodel can be approximated by a linear model in the vicinity of the opti-mum, and it then makes sense to calculate approximate standard errorsfor the parameter estimates.

Most of the available optimization algorithms build on the same idea oflinearization; i.e.,

y − f (x; β + δ) ≈ y − f (x; β) + D f δ

in which D f denotes the gradient matrix of derivatives of f with respect toβ. This effectively becomes a design matrix of a linear model, and you canproceed from a starting guess at β to find an approximate least squaresfit of δ. Then you replace β by β + δ and repeat until convergence. Varia-tions on this basic algorithm include numerical computation of D f andtechniques to avoid instability if the starting guess is too far from theoptimum.

To perform the optimization in R, you can use the nls function, which isbroadly similar to lm and glm.

16.1 Basic usage

In this section, we use a simulated data set just so that we know what weare doing. The model is a simple exponential decay.

> t <- 0:10> y <- rnorm(11, mean=5*exp(-t/5), sd=.2)> plot(y ~ t)

The simulated data can be seen in Figure 16.1.

16.1 Basic usage 277

0 2 4 6 8 10

Figure 16.1. Simulated exponential decay.

We now fit the model to data using nls. Unlike lm and glm, the modelformula for nls does not use the special codings for linear terms, groupingfactors, interactions, etc. Instead, the right-hand side is an explicit expres-sion to calculate the expected value of the left-hand side. This can dependon external variables as well as the parameters, so we need to specifywhich is which. The simplest way to do this is to specify a named vector(or a named list) of starting values.

> nlsout <- nls(y ~ A*exp(-alpha*t), start=c(A=2, alpha=0.05))> summary(nlsout)

Formula: y ~ A * exp(-alpha * t)

Parameters:Estimate Std. Error t value Pr(>|t|)

A 4.97204 0.21766 22.84 2.80e-09 ***alpha 0.20793 0.01572 13.23 3.35e-07 ***—Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.2805 on 9 degrees of freedom

Number of iterations to convergence: 5

278 16. Nonlinear curve fitting

Achieved convergence tolerance: 2.223e-06

Notice that nls treats t as a variable and not a parameter because it is notmentioned in the start argument. Whenever the fitting algorithm needsto evaluate A * exp(-alpha * t), t is taken from the variable in theglobal environment, whereas A and alpha are varied by the algorithm.

The general form of the output is quite similar to that of glm, so we shallnot dwell too long upon it. One thing that might be noted is that the t testand p-value stated for each parameter are tests for a hypothesis that theparameter is zero, which is often quite meaningless for nonlinear models.

16.2 Finding starting values

In the previous section, we had quite fast convergence, even though theinitial guess of parameters was (deliberately) rather badly off. Unfortu-nately, things are not always that simple; convergence of nonlinear modelscan depend critically on having good starting values. Even when the al-gorithm is fairly robust, we at least need to get the order of magnituderight.

Methods for obtaining starting values will most often rely on an analy-sis of the functional form; common techniques involve transformation tolinearity and the estimation of “landmarks” such asymptotes, maximumpoints, and initial slopes.

To illustrate this, we again consider the Juul data. This time we focus onthe relation between age and height. To obtain a reasonably homogeneousdata set, we look at males only and subset the data to the ages between 5and 20.

> attach(subset(juul2, age<20 & age>5 & sex==1))> plot(height ~ age)

A plot of the data is shown in Figure 16.2. The plot looks linear over a largeportion of its domain, but there is some levelling off at the right end andof course it is basic human biology that we stop growing at some point inthe later teens.

The Gompertz curve is often used to describe growth. It can be expressedin the following form:

y = αe−βe−γx

16.2 Finding starting values 279

5 10 15 20

120

140

160

180

200

age

heig

Figure 16.2. Relationship between age and height in juul2 data set.

The curve has a sigmoid shape, approaching a constant level α as x in-creases and (in principle) zero for large negative x. The β and γ parametersdetermine the location and steepness of the transition.

To obtain starting values for a nonlinear fit, one approach is to notice thatthe relation between y and x is something like log-log linear. Specifically,we can rewrite the relation as

log y = log α − βe−γx

which we may rearrange and take logarithms on both sides again, yielding

log(log α − log y) = log β − γx

That means that if we can come up with a guess for α, then we can guessthe two other parameters by a linear fit to transformed data. Since α isthe asymptotic maximum, a guess of α = 200 could be reasonable. Withthis guess, we can make a plot that should show an approximately linearrelationship (log 200 ≈ 5.3):

280 16. Nonlinear curve fitting

5 10 15 20

−4.0

−3.5

−3.0

−2.5

−2.0

−1.5

−1.0

−0.5

age

log(5

.3 −

log(h

eig

ht)

)

Figure 16.3. Linearized plot of the Gompertz relation when assuming α ≈ 200.

> plot(log(5.3-log(height))~age)Warning message:In log(5.3 – log(height)) : NaNs produced

Notice that we got a warning that an NaN (Not a Number) value was pro-duced. This is because one individual was taller than 200 cm, and wetherefore tried to take the logarithm of a negative value. The linearizedplot shows a clearly nonconstant variance and probably also some asym-metry of the residual distribution, so the assumptions for linear regressionanalysis are clearly violated. However, it is good enough for our purpose,and a linear fit gives

> lm(log(5.3-log(height))~age)

Call:lm(formula = log(5.3 – log(height)) ~ age)

Coefficients:(Intercept) age

0.4200 -0.1538

Warning message:In log(5.3 – log(height)) : NaNs produced

16.2 Finding starting values 281

Accordingly, an initial guess of the parameters is

log α = 5.3log β = 0.42

γ = 0.15

Supplying these guesses to nls and fitting the Gompertz curve yields

> fit <- nls(height~alpha*exp(-beta*exp(-gamma*age)),+ start=c(alpha=exp(5.3),beta=exp(0.42),gamma=0.15))> summary(fit)

Formula: height ~ alpha * exp(-beta * exp(-gamma * age))

Parameters:Estimate Std. Error t value Pr(>|t|)

alpha 2.428e+02 1.157e+01 20.978 <2e-16 ***beta 1.176e+00 1.892e-02 62.149 <2e-16 ***gamma 7.903e-02 8.569e-03 9.222 <2e-16 ***—Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 6.811 on 499 degrees of freedom

Number of iterations to convergence: 8Achieved convergence tolerance: 5.283e-06(3 observations deleted due to missingness)

The final estimates are quite a bit different from the starting values. Thisreflects the crudeness of the estimation methods used for the startingvalues. In particular, we used transformations that were based on themathematical form of the function but did not take the structure of the er-ror variation into account. Also, the important parameter α was obtainedby eye.

Looking at the fitted model, however, it is not reassuring that the finalestimate for α suggests that boys would continue growing until they are243 cm tall (for readers in nonmetric countries, that is almost eight feet!).Possibly, the Gompertz curve is just not a good fit for these data.

We can overlay the original data with the fitted curve as follows(Figure 16.4)

> plot(age, height)> newage <- seq(5,20,length=500)> lines(newage, predict(fit,newdata=data.frame(age=newage)),lwd=2)

282 16. Nonlinear curve fitting

5 10 15 20

120

140

160

180

200

age

heig

Figure 16.4. The fitted Gompertz curve.

The plot suggests that there is a tendency for the dispersion to increasewith increasing fitted values, so we attempt a log-scale fit. This can bedone expediently by transforming both sides of the model formula.

>> fit <- nls(log(height)~log(alpha*exp(-beta*exp(-gamma*age))),+ start=c(alpha=exp(5.3),beta=exp(.12),gamma=.12))> summary(fit)

Formula: log(height) ~ log(alpha * exp(-beta * exp(-gamma * age)))

Parameters:Estimate Std. Error t value Pr(>|t|)

alpha 255.97694 15.03920 17.021 <2e-16 ***beta 1.18949 0.02971 40.033 <2e-16 ***gamma 0.07033 0.00811 8.673 <2e-16 ***—Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.04307 on 499 degrees of freedom

Number of iterations to convergence: 8Achieved convergence tolerance: 2.855e-06(3 observations deleted due to missingness)

16.2 Finding starting values 283

5 10 15 20

4.7

4.8

4.9

5.0

5.1

5.2

5.3

age

log(h

eig

ht)

Figure 16.5. Fitted Gompertz curve on log scale.

> plot(age, log(height))> lines(newage, predict(fit,newdata=data.frame(age=newage)),lwd=2)

On the log-scale plot (Figure 16.5), the distribution around the curve ap-pears to be more stable. The parameter estimates did not change much,although the maximum height is now increased by a further 13 cm(5 inches) and the γ parameter is reduced to compensate.

Closer inspection of the plots (whether on log scale or not), however, re-veals that the Gompertz curve tends to overshoot the data points at theright end, where a much flatter curve would fit the data in the range from15 years upwards. Although visually there is a nice overall fit, this is nothard to obtain for a three-parameter family of curves, and the Gompertzcurves seem unable to fit the characteristic patterns of human growth.

284 16. Nonlinear curve fitting

16.3 Self-starting models

Finding starting values is an art rather than a craft, but once a stablemethod has been found, it may be reasonable to assume that will applyto most data sets from a given model. nls allows the nice feature thatthe procedure for calculating starting values can be embodied in the ex-pressions that are used on the right-hand side of the model formula. Suchfunctions are by convention named starting with “SS”, and R 2.6.2 comeswith 10 of these built-in. In particular, there is in fact an SSgompertzfunction, so we could have saved ourselves much of the trouble of theprevious section by just writing

> summary(nls(height~SSgompertz(age, Asym, b2, b3)))

Formula: height ~ SSgompertz(age, Asym, b2, b3)

Parameters:Estimate Std. Error t value Pr(>|t|)

Asym 2.428e+02 1.157e+01 20.98 <2e-16 ***b2 1.176e+00 1.892e-02 62.15 <2e-16 ***b3 9.240e-01 7.918e-03 116.69 <2e-16 ***…

Notice, though, that the parameterization is different: The parameter b3is actually eγ, whereas the two other parameters are recognized as α andβ.

One minor drawback of self-starting models is that you cannot just trans-form them if you want to see if the model fits better on, for example, a logscale. In other words, this fails:

> nls(log(height) ~ log(SSgompertz(age, Asym, b2, b3)))Error in nlsModel(formula, mf, start, wts) :singular gradient matrix at initial parameter estimates

Calls: nls -> switch -> nlsModelIn addition: Warning message:In nls(log(height) ~ log(SSgompertz(age, Asym, b2, b3))) :No starting values specified for some parameters.

Intializing ‘Asym’, ‘b2’, ‘b3’ to ’1.’.Consider specifying ’start’ or using a selfStart model

The error message means, in essence, that the self-start machinery isturned off, so nls tries a wild guess, setting all parameters to 1, and thenfails to converge from that starting point.

Using expression log(SSgompertz(age, Asym, b2, b3)) to com-pute the expected value of log(height) is not a problem (in itself). Wecan take the starting values from the untransformed fit but this is still notenough to make things work.

16.4 Profiling 285

There is a hitch: SSgompertz returns a gradient attribute along with thefitted values. This is the derivative of the fitted value with respect to eachof the model parameters. This speeds up the convergence process for theoriginal model but is plainly wrong for the transformed model, where itcauses convergence failure. We could patch this up by calculating the cor-rect gradient, but it is expedient simply to discard the attribute by takingas.vector.

> cf <- coef(nls(height ~ SSgompertz(age, Asym, b2, b3)))> summary(nls(log(height) ~+ log(as.vector(SSgompertz(age,Asym, b2, b3))),+ start=as.list(cf)))

Formula: log(height) ~ log(as.vector(SSgompertz(age, Asym, b2, b3)))

Parameters:Estimate Std. Error t value Pr(>|t|)

Asym 2.560e+02 1.504e+01 17.02 <2e-16 ***b2 1.189e+00 2.971e-02 40.03 <2e-16 ***b3 9.321e-01 7.559e-03 123.31 <2e-16 ***…

It is possible to write your own self-starting models. It is not hard onceyou have some experience with R programming, but we shall not gointo details here. The essence is that you need two basic items: themodel expression and a function that calculates the starting values. Youmust ensure that these adhere to some formal requirements, and thena constructor function selfStart can be called to create the actualself-starting function.

16.4 Profiling

We discussed profiling before in connection with glm and logistic re-gression in Section 13.3. For nonlinear regression, there are some slightdifferences: The function that is being profiled is not the likelihood func-tion but the sum of squared deviations, and the approximate confidenceintervals are based on the t distribution rather than the normal distribu-tion. Also, the plotting method does not by default use the signed versionof the profile, just the square root of the difference in the sum of squareddeviations.

Profiling is designed to eliminate parameter curvature. The same model canbe formulated using different parameterizations (such as when Gompertzcurves could be defined using γ or b3 = eγ). The choice of parameteri-zation can have a substantial influence on whether the distribution of the

286 16. Nonlinear curve fitting

240 260 280 300 320

0.0

1.0

2.0

alpha

1.15 1.20 1.25 1.30

0.0

1.0

2.0

beta

0.05 0.06 0.07 0.08 0.09

0.0

1.0

2.0

gamma

Figure 16.6. Parameter profiles of the Gompertz fit on log scale.

estimate is approximately normal or not, and this in turn means that theuse of symmetric confidence intervals based on the standard errors fromthe model summary can be misleading. Profile-based confidence intervalsdo not depend on parameterization — if you transform a parameter, theends of the confidence interval are just transformed in the same way.

There is, however, also intrinsic curvature of the models. This describeshow far the model is from an approximating linear model. This kind ofcurvature is independent of parameterization and is harder to adjust forthan parameter curvature. The effect of intrinsic curvature is that the tdistribution used for the calculation of profile-based confidence intervalsis not exactly the right distribution to use. Experience suggests that thiseffect is usually much smaller than the distortions caused by parametercurvature.

For the Gompertz fit (after log transformation), we get the plots shown inFigure 16.6.

> par(mfrow=c(3,1))> plot(profile(fit))

16.5 Finer control of the fitting algorithm 287

The plots show that there is a marked curvature for the α and β pa-rameters, reflected in the curved and asymmetric profiles, whereas the γprofile is more linear and symmetric. This is also seen when comparingthe profile-based confidence intervals with those of confint.default,which uses the normal approximation and the approximate standarderrors.

> confint(fit)Waiting for profiling to be done…

2.5% 97.5%alpha 233.49688706 294.76696435beta 1.14429894 1.27416518gamma 0.05505754 0.08575007> confint.default(fit)

2.5 % 97.5 %alpha 226.50064512 285.45322721beta 1.13125578 1.24772846gamma 0.05443819 0.08622691

16.5 Finer control of the fitting algorithm

The Juul example that has been used in this chapter has been quite benignbecause there are a large number of observations and an objective func-tion that is relatively smooth as a function of the parameters. However,convergence problems easily come up in less nice examples. Nonlinearoptimization is simply a tricky topic, to which we have no chance of do-ing justice in this short chapter. The algorithms have several parametersthat can be adjusted in order to help convergence, but since we are notdescribing the algorithms, it is hardly possible to give more than a feelingfor what can be done.

The possibility of supplying a gradient of the fitted curve with respectto parameters was mentioned earlier. If the curve is given by a simplemathematical expression, then the deriv function can even be used togenerate the gradient automatically. If a gradient is not available, then thealgorithm will estimate it numerically; in practice, this often turns out tobe equally fast.

The nls function features a trace argument that, if set to TRUE, allowsyou to follow the parameters and the SSD iteration by iteration. Thisis sometimes useful to get a handle on what is happening, for instancewhether the algorithm is making unreasonably large jumps. To actuallymodify the behaviour, there is a single control argument, which can beset to the return value of nls.control, which in turn has arguments toset iteration limits and tolerances (and more).

288 16. Nonlinear curve fitting

You can switch out the entire fitting method by using the algorithmargument. Apart from the default algorithm, this allows the settings"plinear" and "port". The former allows models of the form

y = ∑i

αi fi(x; βi)

that are partially linear since the αi can be determined by multiple lin-ear regression if the βi are considered fixed. To specify models with morethan one term, you let the expression on the right-hand side of the modelformula return a matrix instead of a vector. The latter algorithm uses aroutine from the PORT library from Lucent Technologies; this in partic-ular allows you to set contraints on parameters by using the upper andlower arguments to nls.

It should be noted that all the available algorithms operate under the im-plicit assumption that the SSD(β) is fairly smooth and well behaved, witha well-defined global minimum and no other local minima nearby. Thereare cases where this assumption is not warranted. In such cases, you mightattack the minimization problem directly using the optim function.

16.6 Exercises

16.1 Try fitting the Gompertz model for girls in the Juul data. Howwould you go about testing whether the same model fits both genders?

16.2 The philion data contain four small-sample EC50 experimentsthat are somewhat tricky to handle. We suggest the model y = ymax/(1 +(x/β)α). It may be useful to transform y by the square root since the dataare counts, and this stabilizes the variance of the Poisson distribution.Consider how to obtain starting values for the model, and fit it with nls.The "port" algorithm seems more stable for these data. For profiling andconfidence intervals, it seems to help if you set the alphamax argumentto 0.2.

16.3 (Theoretical) Continuing with the philion data, consider whathappens if you modify the model to be y = ymax/(1 + x/β)α.

AObtaining and installing R and theISwR package

The way to obtain R is to download it from one of the CRAN (Compre-hensive R Archive Network) sites. The main site is

http://cran.r-project.org/

It has a number of mirror sites worldwide that may be closer to you andgive faster download times.

Installation details tend to vary over time, so you should read theaccompanying documents and any other information offered on CRAN.

Binary distributions

As of this writing, the version for recent variants of Microsoft Win-dows comes as a single R-2.6.2-win32.exe file, on which you simplydouble-click with the mouse and then follow the on-screen instructions.When the process is completed, you will have an entry under Programson the Start menu for invoking R, as well as a desktop icon.

For Linux distributions that use the RPM package format (primarily Red-Hat, Fedora, and SUSE), .rpm files of R and the recommended add-onpackages can be installed using the rpm command and the respectivesystem software management tools. Fedora now has R in its standardrepositories, and it is also in the repository of openSUSE.org. Debian

290 Appendix A. Obtaining and installing R and the ISwR package

packages can be accessed through APT, the Debian package maintenancetool, as can packages for Ubuntu (in both cases, make sure that you getthe r-recommended package). Further details are in the FAQ.

For the Macintosh platforms, only OS X 10.2 and above are supported.Installation is by downloading the disk image R-2.6.2.dmg and double-clicking the “R.mpkg” icon found inside it.

Installation from source

Installation of Rfrom source code is possible on all supported platforms,although not quite trivial on Windows, mainly because the build envi-ronment is not part of the system. On Unix-like systems (Macintosh OSX included), the process can be as simple as unpacking the sources andwriting

./configuremakemake install

The above works on widely used platforms, provided that the relevantcompilers and support libraries are installed. If your system is more eso-teric or you want to use special compilers or libraries, then you may needto dig deeper.

For Windows, the directory src/gnuwin32 has an INSTALL file withdetailed information about the procedure to follow.

Package installation

To work through the examples and exercises in this book, you shouldinstall the ISwR package, which contains the data sets.

Assuming that you are connected to the Internet, you can start R andfrom the Windows and Macintosh versions use their convenient menuinterfaces.

On other platforms, you can type

install.packages("ISwR")

This will give off a harmless warning and install the package in the defaultlocation.

Appendix A. Obtaining and installing R and the ISwR package 291

On Unix and Linux systems you will need superuser permissions to installin the system location. Similarly, you may require administrator access onsome Windows versions.

Otherwise you can set up a private library directory and install intothat. Set the R_LIBS environment variable to use your private librarysubsequently. Further details can be found on the help page for library.

If your R machine is not connected to the Internet, you can also downloadthe package as a file via a different computer. For Windows and Macin-tosh, you should get the binary package (.zip or .tgz extension) andthen installation from a local file is possible via a menu entry. For Unixand Linux, you can issue the following at the shell prompt (the -l optionallows you to give a private library if needed):

R CMD INSTALL ISwR

More information

Information and further Internet resources for R can be obtained fromCRAN and the R homepage at

http://www.r-project.org

Notice in particular the mailing lists, the user-contributed documents, andthe FAQs.

BData sets in the ISwR package1

IgM Immunoglobulin G

DescriptionSerum IgM in 298 children aged 6 months to 6 years.

UsageIgM

FormatA single numeric vector (g/l).

SourceD.G. Altman (1991), Practical Statistics for Medical Research, Table 3.2,Chapman & Hall.

1Reproduced with permission from the documentation files in the ISwR package.

294 Appendix B. Data sets in the ISwR package

Examplesstripchart(IgM,method="stack")

alkfos Alkaline phosphatase data

DescriptionRepeated measurements of alkaline phosphatase in a randomizedtrial of Tamoxifen treatment of breast cancer patients.

Usagealkfos

FormatA data frame with 43 observations on the following 8 variables.

grp a numeric vector, group code (1=placebo, 2=Tamoxifen).c0 a numeric vector, concentration at baseline.c3 a numeric vector, concentration after 3 months.c6 a numeric vector, concentration after 6 months.c9 a numeric vector, concentration after 9 months.c12 a numeric vector, concentration after 12 months.c18 a numeric vector, concentration after 18 months.c24 a numeric vector, concentration after 24 months.

SourceOriginal data.

ReferencesB. Kristensen et al. (1994), Tamoxifen and bone metabolism in post-menopausal low-risk breast cancer patients: a randomized study.Journal of Clinical Oncology, 12(2):992–997.

Appendix B. Data sets in the ISwR package 295

ashina Ashina’s crossover trial

DescriptionThe ashina data frame has 16 rows and 3 columns. It contains datafrom a crossover trial for the effect of an NO synthase inhibitor onheadaches. Visual analog scale recordings of pain levels were made atbaseline and at five time points after infusion of the drug or placebo.A score was calculated as the sum of the differences from baseline.Data were recorded during two sessions for each patient. Six patientswere given treatment on the first occasion and the placebo on the sec-ond. Ten patients had placebo first and then treatment. The order oftreatment and the placebo was randomized.

Usageashina

FormatThis data frame contains the following columns:

vas.active a numeric vector, summary score when given activesubstance.

vas.plac a numeric vector, summary score when given placebotreatment.

grp a numeric vector code, 1: placebo first, 2: active first.

SourceOriginal data.

ReferencesM.Ashina et al. (1999), Effect of inhibition of nitric oxide synthase onchronic tension-type headache: a randomised crossover trial. Lancet353, 287–289

Examplesplot(vas.active~vas.plac,pch=grp,data=ashina)abline(0,1)

296 Appendix B. Data sets in the ISwR package

bcmort Breast cancer mortality

DescriptionDanish study on the effect of screening for breast cancer.

Usagebcmort

FormatA data frame with 24 observations on the following 4 variables.

age a factor with levels 50-54, 55-59, 60-64, 65-69, 70-74, and75-79.

cohort a factor with levels Study gr., Nat.ctr., Hist.ctr.,and Hist.nat.ctr..

bc.deaths a numeric vector, number of breast cancer deaths.p.yr a numeric vector, person-years under study.

DetailsFour cohorts were collected. The “study group” consists of the pop-ulation of women in the appropriate age range in Copenhagen andFrederiksberg after the introduction of routine mammography screen-ing. The “national control group” consisted of the population in theparts of Denmark in which routine mammography screening wasnot available. These two groups were both collected in the years1991–2001. The “historical control group” and the “historical nationalcontrol group” are similar cohorts from 10 years earlier (1981–1991),before the introduction of screening in Copenhagen and Frederiks-berg. The study group comprises the entire population, not just thoseaccepting the invitation to be screened.

SourceA.H. Olsen et al. (2005), Breast cancer mortality in Copenhagen afterintroduction of mammography screening. British Medical Journal, 330:220–222.

Appendix B. Data sets in the ISwR package 297

bp.obese Obesity and blood pressure

DescriptionThe bp.obese data frame has 102 rows and 3 columns. It containsdata from a random sample of Mexican-American adults in a smallCalifornia town.

Usagebp.obese

FormatThis data frame contains the following columns:

sex a numeric vector code, 0: male, 1: female.obese a numeric vector, ratio of actual weight to ideal weight from

New York Metropolitan Life Tables.bp a numeric vector,systolic blood pressure (mm Hg).

SourceB.W. Brown and M. Hollander (1977), Statistics: A Biomedical Introduc-tion, Wiley.

Examplesplot(bp~obese,pch = ifelse(sex==1, "F", "M"), data = bp.obese)

caesarean Caesarean section and maternal shoe size

DescriptionThe table caesar.shoe contains the relation between caesareansection and maternal shoe size (UK sizes!).

Usagecaesar.shoe

298 Appendix B. Data sets in the ISwR package

FormatA matrix with two rows and six columns.

SourceD.G. Altman (1991), Practical Statistics for Medical Research, Table 10.1,Chapman & Hall.

Examplesprop.trend.test(caesar.shoe["Yes",],margin.table(caesar.shoe,2))

coking Coking data

DescriptionThe coking data frame has 18 rows and 3 columns. It containsthe time to coking in an experiment where the oven width andtemperature were varied.

Usagecoking

FormatThis data frame contains the following columns:width a factor with levels 4, 8, and 12, giving the oven width in

inches.temp a factor with levels 1600 and 1900, giving the temperature in

Fahrenheit.time a numeric vector, time to coking.

SourceR.A. Johnson (1994), Miller and Freund’s Probability and Statistics forEngineers, 5th ed., Prentice-Hall.

Examplesattach(coking)matplot(tapply(time,list(width,temp),mean))detach(coking)

Appendix B. Data sets in the ISwR package 299

cystfibr Cystic fibrosis lung function data

DescriptionThe cystfibr data frame has 25 rows and 10 columns. It containslung function data for cystic fibrosis patients (7–23 years old).

Usagecystfibr

FormatThis data frame contains the following columns:

age a numeric vector, age in years.sex a numeric vector code, 0: male, 1:female.height a numeric vector, height (cm).weight a numeric vector, weight (kg).bmp a numeric vector, body mass (% of normal).fev1 a numeric vector, forced expiratory volume.rv a numeric vector, residual volume.frc a numeric vector, functional residual capacity.tlc a numeric vector, total lung capacity.pemax a numeric vector, maximum expiratory pressure.

SourceD.G. Altman (1991), Practical Statistics for Medical Research, Table 12.11,Chapman & Hall.

ReferencesO’Neill et al. (1983), The effects of chronic hyperinflation, nutritionalstatus, and posture on respiratory muscle strength in cystic fibrosis,Am. Rev. Respir. Dis., 128:1051–1054.

300 Appendix B. Data sets in the ISwR package

eba1977 Lung cancer incidence in four Danish cities 1968–1971

DescriptionThis data set contains counts of incident lung cancer cases andpopulation size in four neighbouring Danish cities by age group.

Usageeba1977

FormatA data frame with 24 observations on the following 4 variables:

city a factor with levels Fredericia, Horsens, Kolding, andVejle.

age a factor with levels 40-54, 55-59, 60-64, 65-69, 70-74, and75+.

pop a numeric vector, number of inhabitants.cases a numeric vector, number of lung cancer cases.

DetailsThese data were “at the center of public interest in Denmark in 1974”,according to Erling Andersen’s paper. The city of Fredericia has asubstantial petrochemical industry in the harbour area.

SourceE.B. Andersen (1977), Multiplicative Poisson models with unequal cellrates, Scandinavian Journal of Statistics, 4:153–158.

ReferencesJ. Clemmensen et al. (1974), Ugeskrift for Læger, pp. 2260–2268.

Appendix B. Data sets in the ISwR package 301

energy Energy expenditure

DescriptionThe energy data frame has 22 rows and 2 columns. It contains dataon the energy expenditure in groups of lean and obese women.

Usageenergy

FormatThis data frame contains the following columns:

expend a numeric vector, 24 hour energy expenditure (MJ).stature a factor with levels lean and obese.

SourceD.G. Altman (1991), Practical Statistics for Medical Research, Table 9.4,Chapman & Hall.

Examplesplot(expend~stature,data=energy)

ewrates Rates of lung and nasal cancer mortality, and totalmortality.

DescriptionEngland and Wales mortality rates from lung cancer, nasal cancer, andall causes, 1936–1980. The 1936 rates are repeated as 1931 rates in orderto accommodate follow-up for the nickel study.

Usageewrates

302 Appendix B. Data sets in the ISwR package

FormatA data frame with 150 observations on the following 5 variables:

year calendar period, 1931: 1931–35, 1936: 1936–40, . . . .age age class, 10: 10–14, 15:15–19, . . . .lung lung cancer mortality rate per 1 million person-yearsnasal nasal cancer mortality rate per 1 million person-yearsother all cause mortality rate per 1 million person-years

DetailsTaken from the “Epi” package by Bendix Carstensen et al.

SourceN.E. Breslow, and N. Day (1987). Statistical Methods in Cancer Research.Volume II: The Design and Analysis of Cohort Studies, Appendix IX. IARCScientific Publications, Lyon.

fake.trypsin Trypsin by age groups

DescriptionThe trypsin data frame has 271 rows and 3 columns. Serum levelsof immunoreactive trypsin in healthy volunteers (faked!).

Usagefake.trypsin

FormatThis data frame contains the following columns:

trypsin a numeric vector, serum-trypsin in ng/ml.grp a numeric vector, age coding. See below.grpf a factor with levels 1: age 10–19, 2: age 20–29, 3: age 30–39, 4:

age 40–49, 5: age 50–59, and 6: age 60–69.

DetailsData have been simulated to match given group means and SD.

Appendix B. Data sets in the ISwR package 303

SourceD.G. Altman (1991), Practical Statistics for Medical Research, Table 9.12,Chapman & Hall.

Examplesplot(trypsin~grp, data=fake.trypsin)

graft.vs.host Graft versus host disease

DescriptionThe gvhd data frame has 37 rows and 7 columns. It contains datafrom patients receiving a nondepleted allogenic bone marrow trans-plant with the purpose of finding variables associated with thedevelopment of acute graft-versus-host disease.

Usagegraft.vs.host

FormatThis data frame contains the following columns:

pnr a numeric vector patient number.rcpage a numeric vector, age of recipient (years).donage a numeric vector, age of donor (years).type a numeric vector, type of leukaemia coded 1: AML, 2: ALL, 3:

CML for acute myeloid, acute lymphatic, and chronic myeloidleukaemia.

preg a numeric vector code indicating whether donor has beenpregnant. 0: no, 1: yes.

index a numeric vector giving an index of mixed epidermal cell-lymphocyte reactions.

gvhd a numeric vector code, graft-versus-host disease, 0: no, 1: yes.time a numeric vector, follow-up timedead a numeric vector code, 0: no (censored), 1: yes

SourceD.G. Altman (1991), Practical Statistics for Medical Research, Exercise12.3, Chapman & Hall.

304 Appendix B. Data sets in the ISwR package

Examplesplot(jitter(gvhd,0.2)~index,data=graft.vs.host)

heart.rate Heart rates after enalaprilat

DescriptionThe heart.rate data frame has 36 rows and 3 columns. It containsdata for nine patients with congestive heart failure before and shortlyafter administration of enalaprilat, in a balanced two-way layout.

Usageheart.rate

FormatThis data frame contains the following columns:

hr a numeric vector, heart rate in beats per minute.subj a factor with levels 1 to 9.time a factor with levels 0 (before), 30, 60, and 120 (minutes after

administration).

SourceD.G. Altman (1991), Practical Statistics for Medical Research, Table 12.2,Chapman & Hall.

Examplesevalq(interaction.plot(time,subj,hr), heart.rate)

hellung Growth of Tetrahymena cells

DescriptionThe hellung data frame has 51 rows and 3 columns. diameter andconcentration of Tetrahymena cells with and without glucose added togrowth medium.

Appendix B. Data sets in the ISwR package 305

Usagehellung

FormatThis data frame contains the following columns:

glucose a numeric vector code, 1: yes, 2: no.conc a numeric vector, cell concentration (counts/ml).diameter a numeric vector, cell diameter (µm).

SourceD. Kronborg and L.T. Skovgaard (1990), Regressionsanalyse, Table 1.1,FADLs Forlag (in Danish).

Examplesplot(diameter~conc,pch=glucose,log="xy",data=hellung)

intake Energy intake

DescriptionThe intake data frame has 11 rows and 2 columns. It contains pairedvalues of energy intake for 11 women.

Usageintake

FormatThis data frame contains the following columns:

pre a numeric vector, premenstrual intake (kJ).post a numeric vector, postmenstrual intake (kJ).

SourceD.G. Altman (1991), Practical Statistics for Medical Research, Table 9.3,Chapman & Hall.

306 Appendix B. Data sets in the ISwR package

Examplesplot(intake$pre, intake$post)

juul Juul’s IGF data

DescriptionThe juul data frame has 1339 rows and 6 columns. It contains a ref-erence sample of the distribution of insulin-like growth factor (IGF-I),one observation per subject in various ages, with the bulk of the datacollected in connection with school physical examinations.

Usagejuul

FormatThis data frame contains the following columns:

age a numeric vector (years).menarche a numeric vector. Has menarche occurred (code 1: no, 2:

yes)?sex a numeric vector (1: boy, 2: girl).igf1 a numeric vector, insulin-like growth factor (µg/l).tanner a numeric vector, codes 1–5: Stages of puberty ad modum

Tanner.testvol a numeric vector, testicular volume (ml).

SourceOriginal data.

Examplesplot(igf1~age, data=juul)

Appendix B. Data sets in the ISwR package 307

juul2 Juul’s IGF data, extended version

DescriptionThe juul2 data frame has 1339 rows and 8 columns; extended versionof juul.

Usagejuul2

FormatThis data frame contains the following columns:

age a numeric vector (years).height a numeric vector (cm).menarche a numeric vector. Has menarche occurred (code 1: no, 2:

yes)?sex a numeric vector (1: boy, 2: girl).igf1 a numeric vector, insulin-like growth factor (µg/l).tanner a numeric vector, codes 1–5: Stages of puberty ad modum

Tanner.testvol a numeric vector, testicular volume (ml).weight a numeric vector, weight (kg).

SourceOriginal data.

Examplesplot(igf1~age, data=juul2)

kfm Breast-feeding data

DescriptionThe kfm data frame has 50 rows and 7 columns. It was collected byKim Fleischer Michaelsen and contains data for 50 infants of age ap-proximately 2 months. They were weighed immediately before and

308 Appendix B. Data sets in the ISwR package

after each breast feeding. and the measured intake of breast milk wasregistered along with various other data.

Usagekfm

FormatThis data frame contains the following columns:

no a numeric vector, identification number.dl.milk a numeric vector, breast-milk intake (dl/24h).sex a factor with levels boy and girl.weight a numeric vector, weight of child (kg).ml.suppl a numeric vector, supplementary milk substitute (ml/24h).mat.weight a numeric vector, weight of mother (kg).mat.height a numeric vector, height of mother (cm).

NoteThe amount of supplementary milk substitute refers to a period beforethe data collection.

SourceOriginal data.

Examplesplot(dl.milk~mat.height,pch=c(1,2)[sex],data=kfm)

lung Methods for determining lung volume

DescriptionThe lung data frame has 18 rows and 3 columns. It contains data onthree different methods of determining human lung volume.

Usagelung

Appendix B. Data sets in the ISwR package 309

FormatThis data frame contains the following columns:

volume a numeric vector, measured lung volume.method a factor with levels A, B, and C.subject a factor with levels 1–6.

SourceAnon. (1977), Exercises in Applied Statistics, Exercise 4.15, Dept. ofTheoretical Statistics, Aarhus University.

malaria Malaria antibody data

DescriptionThe malaria data frame has 100 rows and 4 columns.

Usagemalaria

FormatThis data frame contains the following columns:

subject subject code.age age in years.ab antibody level.mal a numeric vector code, Malaria: 0: no, 1: yes.

DetailsA random sample of 100 children aged 3–15 years from a village inGhana. The children were followed for a period of 8 months. At thebeginning of the study, values of a particular antibody were assessed.Based on observations during the study period, the children were cat-egorized into two groups: individuals with and without symptoms ofmalaria.

SourceUnpublished data.

310 Appendix B. Data sets in the ISwR package

Examplessummary(malaria)

melanom Survival after malignant melanoma

DescriptionThe melanom data frame has 205 rows and 7 columns. It containsdata relating to the survival of patients after an operation for ma-lignant melanoma, collected at Odense University Hospital by K.T.Drzewiecki.

Usagemelanom

FormatThis data frame contains the following columns:

no a numeric vector, patient code.status a numeric vector code, survival status; 1: dead from melanoma,

2: alive, 3: dead from other cause.days a numeric vector, observation time.ulc a numeric vector code, ulceration; 1: present, 2: absent.thick a numeric vector, tumor thickness (1/100 mm).sex a numeric vector code; 1: female, 2: male.

SourceP.K. Andersen, Ø. Borgan, R.D. Gill, and N. Keiding (1991), StatisticalModels Based on Counting Processes, Appendix 1, Springer-Verlag.

Examplesrequire(survival)plot(survfit(Surv(days,status==1),data=melanom))

Appendix B. Data sets in the ISwR package 311

nickel Nickel smelters in South Wales

DescriptionThe data concern a cohort of nickel smelting workers in South Wales,with information on exposure, follow-up period, and cause of death.

Usagenickel

FormatA data frame containing 679 observations of the following 7 variables:

id subject identifier (numeric).icd ICD cause of death if dead, 0 otherwise (numeric).exposure exposure index for workplace (numeric)dob date of birth (numeric).age1st age at first exposure (numeric).agein age at start of follow-up (numeric).ageout age at end of follow-up (numeric).

DetailsTaken from the “Epi” package by Bendix Carstensen et al. For com-parison purposes, England and Wales mortality rates (per 1,000,000per annum) from lung cancer (ICDs 162 and 163), nasal cancer (ICD160), and all causes, by age group and calendar period, are suppliedin the data set ewrates.

SourceN.E. Breslow and N. Day (1987). Statistical Methods in Cancer Research.Volume II: The Design and Analysis of Cohort Studies, IARC ScientificPublications, Lyon.

312 Appendix B. Data sets in the ISwR package

nickel.expand Nickel smelters in South Wales, expanded

DescriptionThe data concern a cohort of nickel smelting workers in South Wales,with information on exposure, follow-up period, and cause of death,as in the nickel data. This version has follow-up times split ac-cording to age groups and is merged with the mortality rates inewrates.

Usagenickel.expand

FormatA data frame with 3724 observations on the following 12 variables:

agr age class: 10: 10–14, 15: 15–19, . . . .ygr calendar period, 1931: 1931–35, 1936: 1936–40, . . . .id subject identifier (numeric).icd ICD cause of death if dead, 0 otherwise (numeric).exposure exposure index for workplace (numeric).dob date of birth (numeric).age1st age at first exposure (numeric).agein age at start of follow-up (numeric).ageout age at end of follow-up (numeric).lung lung cancer mortality rate per 1 million person-years.nasal nasal cancer mortality rate per 1 million person-years.other all cause mortality rate per 1 million person-years.

SourceComputed from nickel and ewrates data sets.

philion Dose response data

DescriptionFour small experiments with the purpose of estimating the EC50 of abiological dose-response relation.

Appendix B. Data sets in the ISwR package 313

Usagephilion

FormatA data frame with 30 observations on the following 3 variables:

experiment a numeric vector; codes 1 through 4 denote the experi-ment number.

dose a numeric vector, the dose.response a numeric vector, the response (counts).

DetailsThese data were discussed on the R mailing lists, initially suggest-ing a log-linear Poisson regression, but actually a relation like y =ymax/(1 + (x/β)α) is more suitable.

SourceOriginal data from Vincent Philion, IRDA, Québec.

Referenceshttp://tolstoy.newcastle.edu.au/R/help/03b/1121.html

react Tuberculin reactions

DescriptionThe numeric vector react contains differences between two nurses’determinations of 334 tuberculin reaction sizes.

Usagereact

FormatA single vector, differences between reaction sizes in mm.

314 Appendix B. Data sets in the ISwR package

SourceAnon. (1977), Exercises in Applied Statistics, Exercise 2.9, Dept. ofTheoretical Statistics, Aarhus University.

Exampleshist(react) # not good because of discretization effects…plot(density(react))

red.cell.folate Red cell folate data

DescriptionThe folate data frame has 22 rows and 2 columns. It contains dataon red cell folate levels in patients receiving three different methodsof ventilation during anesthesia.

Usagered.cell.folate

FormatThis data frame contains the following columns:

folate a numeric vector, folate concentration (µg/l).ventilation a factor with levels N2O+O2,24h: 50% nitrous oxide

and 50% oxygen, continuously for 24 hours; N2O+O2,op: 50%nitrous oxide and 50% oxygen, only during operation; O2,24h:no nitrous oxide but 35%–50% oxygen for 24 hours.

SourceD.G. Altman (1991), Practical Statistics for Medical Research, Table 9.10,Chapman & Hall.

Examplesplot(folate~ventilation,data=red.cell.folate)

Appendix B. Data sets in the ISwR package 315

rmr Resting metabolic rate

DescriptionThe rmr data frame has 44 rows and 2 columns. It contains the restingmetabolic rate and body weight data for 44 women.

Usagermr

FormatThis data frame contains the following columns:

body.weight a numeric vector, body weight (kg).metabolic.rate a numeric vector, metabolic rate (kcal/24hr).

SourceD.G. Altman (1991), Practical Statistics for Medical Research, Exercise11.2, Chapman & Hall.

Examplesplot(metabolic.rate~body.weight,data=rmr)

secher Birth weight and ultrasonography

DescriptionThe secher data frame has 107 rows and 4 columns. It contains ultra-sonographic measurements of fetuses immediately before birth andtheir subsequent birth weight.

Usagesecher

316 Appendix B. Data sets in the ISwR package

FormatThis data frame contains the following columns:

bwt a numeric vector, birth weight (g).bpd a numeric vector, biparietal diameter (mm).ad a numeric vector, abdominal diameter (mm).no a numeric vector, observation number.

SourceD. Kronborg and L.T. Skovgaard (1990), Regressionsanalyse, Table 3.1,FADLs Forlag (in Danish).Secher et al. (1987), European Journal of Obstetrics, Gynecology, andReproductive Biology, 24: 1–11.

Examplesplot(bwt~ad, data=secher, log="xy")

secretin Secretin-induced blood glucose changes

DescriptionThe secretin data frame has 50 rows and 6 columns. It contains datafrom a glucose response experiment.

Usagesecretin

FormatThis data frame contains the following columns:

gluc a numeric vector, blood glucose level.person a factor with levels A–E.time a factor with levels 20, 30, 60, 90 (minutes since injection), and

pre (before injection).repl a factor with levels a: 1st sample; b: 2nd sample.time20plus a factor with levels 20+: 20 minutes or longer since

injection; pre: before injection.time.comb a factor with levels 20: 20 minutes since injection; 30+:

30 minutes or longer since injection; pre: before injection.

Appendix B. Data sets in the ISwR package 317

DetailsSecretin is a hormone of the duodenal mucous membrane. An extractwas administered to five patients with arterial hypertension. Primaryregistrations (double determination) of blood glucose were on graphpaper and later quantified with the smallest of the two measurementsrecorded first.

SourceAnon. (1977), Exercises in Applied Statistics, Exercise 5.8, Dept. ofTheoretical Statistics, Aarhus University.

stroke Estonian stroke data

DescriptionAll cases of stroke in Tartu, Estonia, during the period 1991–1993, withfollow-up until January 1, 1996.

Usagestroke

FormatA data frame with 829 observations on the following 10 variables.

sex a factor with levels Female and Male.died a Date, date of death.dstr a Date, date of stroke.age a numeric vector, age at stroke.dgn a factor, diagnosis, with levels ICH (intracranial haemorrhage),

ID (unidentified). INF (infarction, ischaemic), SAH (subarchnoidhaemorrhage).

coma a factor with levels No and Yes, indicating whether patient wasin coma after the stroke.

diab a factor with levels No and Yes, history of diabetes.minf a factor with levels No and Yes, history of myocardial infarc-

tion.han a factor with levels No and Yes, history of hypertension.obsmonths a numeric vector, observation times in months (set to 0.1

for patients dying on the same day as the stroke).dead a logical vector, whether patient died during the study.

318 Appendix B. Data sets in the ISwR package

SourceOriginal data.

ReferencesJ. Korv, M. Roose, and A.E. Kaasik (1997). Stroke Registry of Tartu,Estonia, from 1991 through 1993. Cerebrovascular Disorders 7:154–162.

tb.dilute Tuberculin dilution assay

DescriptionThe tb.dilute data frame has 18 rows and 3 columns. It containsdata from a drug test involving dilutions of tuberculin.

Usagetb.dilute

FormatThis data frame contains the following columns:

reaction a numeric vector, reaction sizes (average of diameters) fortuberculin skin pricks.

animal a factor with levels 1–6.logdose a factor with levels 0.5, 0, and -0.5.

DetailsThe actual dilutions were 1:100, 1:100

√10, 1:1000. Setting the middle

one to 1 and using base-10 logarithms gives the logdose values.

SourceAnon. (1977), Exercises in Applied Statistics, part of Exercise 4.15, Dept.of Theoretical Statistics, Aarhus University.

Appendix B. Data sets in the ISwR package 319

thuesen Ventricular shortening velocity

DescriptionThe thuesen data frame has 24 rows and 2 columns. It containsventricular shortening velocity and blood glucose for type 1 diabeticpatients.

Usagethuesen

FormatThis data frame contains the following columns:

blood.glucose a numeric vector, fasting blood glucose (mmol/l).short.velocity a numeric vector, mean circumferential shorten-

ing velocity (%/s).

SourceD.G. Altman (1991), Practical Statistics for Medical Research, Table 11.6,Chapman & Hall.

Examplesplot(short.velocity~blood.glucose, data=thuesen)

tlc Total lung capacity

DescriptionThe tlc data frame has 32 rows and 4 columns. It contains data onpretransplant total lung capacity (TLC) for recipients of heart-lungtransplants by whole-body plethysmography.

Usagetlc

320 Appendix B. Data sets in the ISwR package

FormatThis data frame contains the following columns:

age a numeric vector, age of recipient (years).sex a numeric vector code, female: 1, male: 2.height a numeric vector, height of recipient (cm).tlc a numeric vector, total lung capacity (l).

SourceD.G. Altman (1991), Practical Statistics for Medical Research, Exercise12.5, 10.1, Chapman & Hall.

Examplesplot(tlc~height,data=tlc)

vitcap Vital capacity

DescriptionThe vitcap data frame has 24 rows and 3 columns. It contains dataon vital capacity for workers in the cadmium industry. It is a subset ofthe vitcap2 data set.

Usagevitcap

FormatThis data frame contains the following columns:

group a numeric vector; group codes are 1: exposed > 10 years, 3: notexposed.

age a numeric vector, age in years.vital.capacity a numeric vector, vital capacity (a measure of

lung volume) in liters.

SourceP. Armitage and G. Berry (1987), Statistical Methods in Medical Research,2nd ed., Blackwell, p.286.

Appendix B. Data sets in the ISwR package 321

Examplesplot(vital.capacity~age, pch=group, data=vitcap)

vitcap2 Vital capacity, full data set

DescriptionThe vitcap2 data frame has 84 rows and 3 columns. Age and vitalcapacity for workers in the cadmium industry.

Usagevitcap2

FormatThis data frame contains the following columns:

group a numeric vector; group codes are 1: exposed > 10 years, 2:exposed < 10 years, 3: not exposed.

age a numeric vector, age in years.vital.capacity a numeric vector, vital capacity (a measure of

lung volume) (l).

SourceP. Armitage and G. Berry (1987), Statistical Methods in Medical Research,2nd ed., Blackwell, p.286.

Examplesplot(vital.capacity~age, pch=group, data=vitcap2)

wright Comparison of Wright peak-flow meters

DescriptionThe wright data frame has 17 rows and 2 columns. It contains dataon peak expiratory flow rate with two different flow meters on eachof 17 subjects.

322 Appendix B. Data sets in the ISwR package

Usagewright

FormatThis data frame contains the following columns:

std.wright a numeric vector, data from large flow meter (l/min).mini.wright a numeric vector, data from mini flow meter (l/min).

SourceJ.M. Bland and D.G. Altman (1986), Statistical methods for assess-ing agreement between two methods of clinical measurement, Lancet,1:307–310.

Examplesplot(wright)abline(0,1)

zelazo Age at walking

DescriptionThe zelazo object is a list with four components.

Usagezelazo

FormatThis is a list containing data on age at walking (in months) for fourgroups of infants:

active test group receiving active training; these children had theirwalking and placing reflexes trained during four three-minutesessions that took place every day from their second to theireighth week of life.

passive passive training group; these children received the sametypes of social and gross motor stimulation, but did not havetheir specific walking and placing reflexes trained.

Appendix B. Data sets in the ISwR package 323

none no training; these children had no special training, but weretested along with the children who underwent active or passivetraining.

ctr.8w eighth-week controls; these children had no training andwere only tested at the age of 8 weeks.

NoteWhen asked to enter these data from a text source, many students willuse one vector per group and will need to reformat data into a dataframe for some uses. The rather unusual format of this data set mimicsthat situation.

SourceP.R. Zelazo, N.A. Zelazo, and S. Kolb (1972), “Walking” in thenewborn, Science, 176: 314–315.

CCompendium

Elementary

Commands

ls() or objects() List objects in workspacerm(object) Delete objectsearch() Search path

Variable names

Combinations of letters, digits, and period. Must not start with adigit. Avoid starting with period.

Assignments

<- Assign value to variable-> Assignment “to the right”<<- Global assignment (in functions)

326 Appendix C. Compendium

Operators

Arithmetic

+ Addition- Subtraction, sign* Multiplication/ Division^ Raise to power%/% Integer division%% Remainder from integer division

Logical and relational

== Equal to!= Not equal to< Less than> Greater than<= Less than or equal to>= Greater than or equal to

is.na(x) Missing?& Logical AND| Logical OR! Logical NOT

& and | are elementwise. See “Programming” (p. 336) for && and ||.

Appendix C. Compendium 327

Vectors and data types

Generating

numeric(25) 25 zeroscharacter(25) 25 × ""logical(25) 25 × FALSE

seq(-4,4,0.1) Sequence: −4.0, −3.9, 3.8, . . . , 3.9, 4.01:10 Same as seq(1,10,1)

c(5,7,9,13,1:5) Concatenation: 5 7 9 13 1 2 3 4 5rep(1,10) 1 1 1 1 1 1 1 1 1 1gl(3,2,12) Factor with 3 levels, repeat each level in blocks

of 2, up to length 12 (i.e., 1 1 2 2 3 3 1 1 2 2 3 3)

Coercion

as.numeric(x) Convert to numericas.character(x) Convert to text stringas.logical(x) Convert to logicalfactor(x) Create factor from vector x

For factors, see also “Tabulation, grouping, and recoding” (p. 331).

Data frames

data.frame(height =c(165,185), weight =c(90,65))

Data frame with two named vectors

data.frame(height,weight)

Collect vectors into data frame

dfr$var Select vector var in data frame dfrattach(dfr) Put data frame in search pathdetach() — and remove it from pathAttached data frames always come after .GlobalEnv in the search path.Attached data frames are copies; subsequent changes to dfr have no effect.

328 Appendix C. Compendium

Numerical functions

Mathematical

log(x) Logarithm of x, natural (base-e)logarithm

log10(x) Base-10 logarithmexp(x) Exponential function ex

sin(x) Sinecos(x) Cosinetan(x) Tangentasin(x) Arcsin (inverse sine)acos(x)atan(x)min(x) Smallest value in vector

min(x1,x2,…) Minimum over several vectors (onenumber)

max(x) Largest value in vectorrange(x) Like c(min(x),max(x))

pmin(x1,x2,…) Parallel (elementwise) minimumover multiple equally long vectors

pmax(x1,x2,…) Parallel maximumlength(x) Number of elements in vector

sum(complete.cases(x)) Number of nonmissing elements invector

Statistical

mean(x) Averagesd(x) Standard deviationvar(x) Variance

median(x) Medianquantile(x,p) Quantiles

cor(x,y) Correlation

Appendix C. Compendium 329

Indexing/selection

x[1] First elementx[1:5] Subvector containing first five

elementsx[c(2,3,5,7,11)] Element nos. 2, 3, 5, 7, and 11

x[y<=30] Selection by logical expressionx[sex=="male"] Selection by factor variable

i <- c(2,3,5,7,11); x[i] Selection by numeric variablel <- (y<=30); x[l] Selection by logical variable

Matrices and data frames

m[4,] Fourth rowm[,3] Third column

dfr[dfr$var<=30,] Partial data framesubset(dfr,var<=30) Same, often simpler

Input of data

data(name) Built-in data setread.table("filename") Read from external file

Common arguments to read.table

header=TRUE First line has variable namessep="," Data are separated by commasdec="," Decimal point is comma

na.strings="." Missing value is dot

Variants of read.table

read.csv("filename") Comma-separatedread.delim("filename") Tab-delimitedread.csv2("filename") Semicolon-separated, comma

decimal pointread.delim2("filename") Tab-delimited, comma decimal

pointThese all set header=TRUE.

330 Appendix C. Compendium

Missing values

Functions

is.na(x) Logical vector. TRUE where xhas NA.

complete.cases(x1,x2,…) Missing neither in x1, nor x2,nor. . . .

Arguments to other functions

na.rm= In statistical functions, removemissing if TRUE, return NA ifFALSE.

na.last= In sort; TRUE, FALSE and NAmean, respectively, “last”,“first”, and “throw away”.

na.action= In lm, etc., values na.fail,na.omit, na.exclude; also inoptions("na.action").

na.print= In summary andprint.default; how torepresent NA in output.

na.strings= In read.table(); code(s) forNA in input.

Appendix C. Compendium 331

Tabulation, grouping, and recoding

table(f1,…) (Cross)-tabulationxtabs(~ f1 + …) ditto, formula interface

ftable(f1 ~ f2 + …) “Flat” tablestapply(x,f,mean) Table of means

aggregate(df,list(f),mean) Means for several variablesby(df, list(f), summary) Summarize data frame by group

factor(x) Convert vector to factorcut(x,breaks) Groups from cutpoints for

continuous variable

Arguments to factor

levels Values of x to code. Use if somevalues are not present in data orif the order would be wrong.

labels Values associated with factorlevels.

exclude Values to exclude. Default NA.Set to NULL to have missingvalues included as a level.

Arguments to cut

breaks Cutpoints. Note that values of xoutside breaks give NA.

labels Names for groups. Default is(0,30], etc.

right Right endpoint included?(FALSE: left)

Recoding factors

levels(f) <- names New level nameslevels(f) <- list( Combining levels

new1=c("old1","old2")new2="old3")

332 Appendix C. Compendium

Statistical distributions

Normal distribution

dnorm(x) Densitypnorm(x) Cumulative distribution

function, P(X ≤ x)qnorm(p) p-quantile, x : P(X ≤ x) = prnorm(n) n (pseudo-)random normally

distributed numbers

Distributions

pnorm(x,mean,sd) Normalplnorm(x,mean,sd) Lognormal

pt(x,df) Student’s tpf(x,n1,n2) F distributionpchisq(x,df) χ2

pbinom(x,n,p) Binomialppois(x,lambda) Poissonpunif(x,min,max) Uniformpexp(x,rate) Exponential

pgamma(x,shape,scale) Gammapbeta(x,a,b) Beta

Same convention (d-q-r) for density, quantiles, and random numbers as for normaldistribution.

Appendix C. Compendium 333

Statistical standard methods

Continuous response

t.test One- and two-sample t testspairwise.t.test Pairwise comparisons

cor.test Correlationvar.test Comparison of two variances

(F test)lm(y ~ x) Regression analysislm(y ~ f) One-way analysis of variance

lm(y ~ f1 + f2) Two-way analysis of variancelm(y ~ f + x) Analysis of covariance

lm(y ~ x1 + x2 + x3) Multiple regression analysisbartlett.test Bartlett’s test (k variances)

Nonparametric:wilcox.test One- and two-sample

Wilcoxon testskruskal.test Kruskal–Wallis testfriedman.test Friedman’s two-way analysis

of variancecor.test variants:

method="kendall" Kendall’s τmethod="spearman" Spearman’s ρ

Discrete response

binom.test Binomial test (incl. sign test)prop.test Comparison of proportions

prop.trend.test Test for trend in relativeproportions

fisher.test Exact test in small tableschisq.test χ2 test

glm(y ~ x1+x2+x3, binomial) Logistic regression

334 Appendix C. Compendium

Models

Model formulas

~ Described by+ Additive effects: Interaction* Main effects + interaction

(a*b = a + b + a:b)-1 Remove intercept

Classifications are represented by descriptive variable being a factor.

Linear, nonlinear, and generalized linear models

lm.out <- lm(y ~ x) Fit model and save resultsummary(lm.out) Coefficients, etc.anova(lm.out) Analysis of variance tablefitted(lm.out) Fitted valuesresid(lm.out) Residuals

predict(lm.out, newdata) Predictions for new data frameglm(y ~ x, binomial) Logistic regressionglm(y ~ x, poisson) Poisson regressionnls(y ~ a*exp(-b*x), Nonlinear regressionstart=c(a=5, b=.2))

Diagnostics

rstudent(lm.out) Studentized residualsdfbetas(lm.out) Change in β if obs. removeddffits(lm.out) Change in fit if obs. removed

Survival analysis

S <- Surv(time, ev) Create survival objectsurvfit(S) Kaplan–Meier estimate

plot(survfit(S)) Survival curvesurvdiff(S ~ g) (Log-rank) test for equal

survival curvescoxph(S ~ x1 + x2) Cox’s proportional hazards

model

Appendix C. Compendium 335

Graphics

Standard plots

plot() Scatterplot (and more)hist() Histogram

boxplot() Box-and-whiskers plotstripplot() Stripplotbarplot() Bar diagramdotplot() Dot diagrampiechart() Cakes. . .

interaction.plot() Interaction plot

Plotting elements

lines() Linesabline() Line given by intercept and slope

(and more)points() Points

segments() Line segmentsarrows() Arrows (N.B.: angle=90 for error

bars)axis() Axisbox() Frame around plot

title() Title (above plot)text() Text in plotmtext() Text in marginlegend() List of symbols

These are all added to existing plots.

Graphical parameters

pch Symbol (plotting character)mfrow, mfcol Several plots on one (multif rame)xlim, ylim Plot limitslty,lwd Line type/width

col Colourcex, mex Character size and line spacing in

marginsSee the help page for par for more details.

336 Appendix C. Compendium

Programming

Conditional executionif(p<0.05)

print("Hooray!")

— with alternative

if(p<0.05)print("Hooray!")

elseprint("Bah.")

Loop over listfor(i in 1:10)

print(i)

Loop

i <- 1while(i<10) {

print(i)i <- i + 1

}

User-defined function

f <- function(a,b,doit=FALSE){if (doit)

a + belse

In flow control, one uses a && b and a || b, where b is onlycomputed if necessary; that is, if a then b else FALSE andif a then TRUE else b.

DAnswers to exercises

1.1 One possibility is

x <- y <- c(7, 9, NA, NA, 13)all(is.na(x) == is.na(y)) & all((x == y)[!is.na(x)])

Notice that FALSE & NA is FALSE, so the case of different NA patterns ishandled correctly.

1.2 Factor x gets treated as if it contained the integer codes.

x <- factor(c("Huey", "Dewey", "Louie", "Huey"))y <- c("blue", "red", "green")xy[x]

(This is useful, e.g., when selecting plot symbols.)

1.3

juul.girl <- juul[juul$age >=7 & juul$age < 14 & juul$sex == 2,]summary(juul.girl)

1.4 The levels with the same name are collapsed into one.

1.5 sapply(1:10, function(i) mean(rexp(20)))

2.1 To insert 1.23 between x[7] and x[8]:

338 Appendix D. Answers to exercises

x <- 1:10z <- append(x, 1.23, after=7)z

Otherwise, consider

z <- c(x[1:7],1.23,x[8:10])z

or, more generally, to insert v just after index k (the boundary cases requiresome care),

v <- 1.23; k <- 7i <- seq(along=x)z <- c(x[i <= k], v, x[i > k])z

2.2 (First part only) Use

write.table(thuesen, file="foo.txt")# edit the fileread.table("foo.txt", na.strings=".")

write.table(thuesen, file="foo.txt", na=".")read.table("foo.txt", na.strings=".")

(Notice that if you do not edit the file in the first case, then the secondcolumn gets read as a character vector.)

3.1

1 – pnorm(3)1 – pnorm(42, mean=35, sd=6)dbinom(10, size=10, prob=0.8)punif(0.9) # this one is obvious…1 – pchisq(6.5, df=2)

It might be better to use lower.tail=FALSE instead of subtracting from1 in (a), (b), and (e). Notice that question (c) is about a point probability,whereas the others involve the cumulative distribution function.

3.2 Evaluate each of the following. Notice that the standard normal canbe used for all questions.

pnorm(-2) * 2qnorm(1-.01/2)qnorm(1-.005/2)qnorm(1-.001/2)qnorm(.25)qnorm(.75)

Appendix D. Answers to exercises 339

Again, lower.tail can be used in some cases.

3.3 dbinom(0, size=10, prob=.2)

3.4 Either of the following should work:

rbinom(10, 1, .5)ifelse(rbinom(10, 1, .5) == 1, "H", "T")c("H", "T")[1 + rbinom(10, 1, .5)]

The first one gives a 0/1 result, the two others H/T like the sample exam-ple in the text. One advantage of using rbinom is that its prob argumentcan be a vector, so you can have different probabilities of success for eachelement of the result.

4.1 For example,

x <- 1:5 ; y <- rexp(5,1) ; opar <- par(mfrow=c(2,2))plot(x, y, pch=15) # filled squareplot(x, y, type="b", lty="dotted")plot(x, y, type="b", lwd=3)plot(x, y, type="o", col="blue")par(opar)

4.2 Use a filled symbol, and set the fill colour equal to the plotbackground:

plot(rnorm(10),type="o", pch=21, bg="white")

4.3 You can use qqnorm with plot.it=F and get a return value fromwhich you can extract the range information (you could of course also getthis “by eye”).

x1 <- rnorm(20)x2 <- rnorm(10)+1q1 <- qqnorm(x1, plot.it=F)q2 <- qqnorm(x2, plot.it=F)xr <- range(q1$x, q2$x)yr <- range(q1$y, q2$y)qqnorm(x1, xlim=xr, ylim=yr)points(q2, col="red")

Here, qqnorm is used for the basic plot to get the labels right. Thenpoints is used with q2 for the overlay.Setting type="l" gives a messy plot because the values are not plottedin order. The remedy is to use sort(x1) and sort(x2).

4.4 The breaks occur at integer values, as do the data. Data on the bound-ary are counted in the column to the left of it, effectively shifting the

340 Appendix D. Answers to exercises

histogram half a unit left. The truehist function allows you to specify abetter set of breaks.

hist(react)library(MASS)truehist(react,h=1,x0=.5)

4.5 The thing to notice is the linear interpolation between data points:

z <- runif(5)curve(quantile(z,x), from=0, to=1)

5.1 The distribution appears reasonably normal, with some discretiza-tion effect and two weak outliers, one at each end. There is a significantdifference from zero (t = −7.75, p = 1.1 × 10−13).

qqnorm(react)t.test(react)

5.2 t.test(vital.capacity~group,conf=0.99,data=vitcap).The fact that age also differs by group may cause bias.

5.3 This is quite parallel to t.test usage

wilcox.test(react)wilcox.test(vital.capacity~group, data=vitcap)

5.4 The following builds a post-vs.-pre plot, a difference-vs.-average)(Bland-Altman) plot, and a histogram and a Q-Q plot of the differences.

attach(intake) ; opar <- par(mfrow=c(2,2))plot(post ~ pre) ; abline(0,1)plot((post+pre)/2, post – pre,

ylim=range(0,post-pre)); abline(h=0)hist(post-pre)qqnorm(post-pre)detach(intake)par(opar)

5.5 The outliers are the first and last observations in the (sorted) datavector and can be removed as follows

shapiro.test(react)shapiro.test(react[-c(1,334)])qqnorm(react[-c(1,334)])

The test comes out highly significant even with outliers removed becauseit picks up the discretization effect in the otherwise nearly straight-lineqqnorm plot.

Appendix D. Answers to exercises 341

5.6 A paired t test is appropriate if there is no period effect. However,even with a period effect (assumed additive), you would expect the dif-ference between the two periods to be the same in both groups if therewere no effect of treatment. This can be used to test for a treatment effect.

attach(ashina)t.test(vas.active, vas.plac, paired=TRUE)t.test((vas.active-vas.plac)[grp==1],

(vas.plac-vas.active)[grp==2])

Notice that the subtraction is reversed in one group. Observe that theconfidence interval in the second case is for twice the treatment effect.

5.7 This is the sort of thing replicate is for. The plot at the end showsa P-P plot with logarithmic axes, showing that extreme p-values tend tobe exaggerated.

t.test(rnorm(25))$p.value #repeat 10xt.test(rt(25,df=2))$p.value #repeat 10xt.test(rexp(25), mu=1)$p.value #repeat 10xx <- replicate(5000, t.test(rexp(25), mu=1)$p.value)qqplot(sort(x),ppoints(5000),type="l",log="xy")

6.1 The following gives both elementary and more general answers.Notice the use of confint.

fit <- lm(metabolic.rate ~ body.weight, data=rmr)summary(fit)811.2267 + 7.0595 * 70 # , or:predict(fit, newdata=data.frame(body.weight=70))qt(.975,42)7.0595 + c(-1,1) * 2.018 * 0.9776 # , or:confint(fit)

6.2 summary(lm(sqrt(igf1)~age,data=juul,subset=age>25))

6.3 We can fit a linear model and plot the data as follows:

summary(lm(log(ab)~age, data=malaria))plot(log(ab)~age, data=malaria)

The plot appears to show a cyclic pattern. It is unclear whether it reflectsa significant departure from the model, though. Malaria is a disease withepidemic behaviour, so cycles are plausible.

6.4 (This could be elaborated by wrapping the random number genera-tion in a function, etc.)

rho <- .90 ; n <- 100x <- rnorm(n)

342 Appendix D. Answers to exercises

y <- rnorm(n, rho * x, sqrt(1 – rho^2))plot(x, y)cor.test(x, y)cor.test(x, y, method="spearman")cor.test(x, y, method="kendall")

You will most likely find that the Kendall correlation is somewhat smallerthan the two others.

7.1

walk <- unlist(zelazo) # or c(..,recursive=TRUE)group <- factor(rep(1:4,c(6,6,6,5)), labels=names(zelazo))summary(lm(walk ~ group))t.test(zelazo$active,zelazo$ctr.8w) # first vs. lastt.test(zelazo$active,unlist(zelazo[-1])) # first vs. rest

7.2 A and C differ with B intermediate, not significantly different fromeither. (The B–C comparison is not available from the summary, but dueto the balanced design, the standard error of that difference is 0.16656 likethe two others.)

fit <- lm(volume~method+subject, data=lung)anova(fit)summary(fit)

7.3

kruskal.test(walk ~ group)wilcox.test(zelazo$active,zelazo$ctr.8w) # first vs. lastwilcox.test(zelazo$active,unlist(zelazo[-1])) # first vs. restfriedman.test(volume ~ method | subject, data=lung)wilcox.test(lung$volume[lung$method=="A"],

lung$volume[lung$method=="C"], paired=TRUE) # etc.

7.4 (Only the square-root transform is shown; you can do the same forlog-transformed and untransformed data.)

attach(juul)tapply(sqrt(igf1),tanner, sd, na.rm=TRUE)plot(sqrt(igf1)~jitter(tanner))oneway.test(sqrt(igf1)~tanner)

The square root looks nice, logarithms become skewed in the oppositedirection. The transformations do not make much of a difference for thetest. It is, however, a problem that strong age effects are being ignored,particularly within Tanner stage 1.

8.1 With 10 patients, p = 0.1074. Fourteen or more are needed forsignificance at level 0.05.

Appendix D. Answers to exercises 343

binom.test(0, 10, p=.20, alt="less")binom.test(0, 13, p=.20, alt="less")binom.test(0, 14, p=.20, alt="less")

8.2 Yes, it is highly significant.

prop.test(c(210,122),c(747,661))

8.3 The confidence interval (from prop.test) is (−0.085, 0.507)

M <- matrix(c(23,7,18,13),2,2)chisq.test(M)fisher.test(M)prop.test(M)

8.4 The following is a simplified analysis, which uses fisher.testbecause of the small cell counts:

tbl <- c(42, 157, 47, 62, 4, 15, 4, 1, 8, 28, 9, 7)dim(tbl) <- c(2,2,3)dimnames(tbl) <- list(c("A","B"),

c("not pierced","pierced"),c("ok","broken","cracked"))

ftable(tbl)fisher.test(tbl["B",,]) # slice analysisfisher.test(tbl["A",,])fisher.test(margin.table(tbl,2:3)) # marginal

You may wish to check that there is little or no effect of egg size onbreakage, so that the marginal analysis is defensible. You could also trycollapsing the “broken” and “cracked” categories.

8.5 The curve shows substantial discontinuities where probability massis shifted from one tail to the other and also a number of local minima. Aconfidence region could be defined as those p against which there is nosignificant evidence at level α, but for some α that is not an interval.

p <- seq(0,1,0.001)pval <- sapply(p,function(p)binom.test(3,15,p=p)$p.value)plot(p,pval,type="l")

9.1 The estimated sample size is 6.29 or 8.06 per group depending onwhether you use one- or two-sided testing. The approximate formulagives 6.98 for the two-sided case. The reduction in power due to the un-balanced sampling can be accounted for by reducing delta by the ratioof the two SEDM.

power.t.test(power=.8,delta=.30,sd=.20)power.t.test(power=.8,delta=.30,sd=.20,alt="one.sided")

344 Appendix D. Answers to exercises

(qnorm(.975)+qnorm(.8))^2*2*(.2/.3)^2 # approx. formulapower.t.test(n=8, delta=.30, sd=.20) # power with eq.sized2 <- .30 * sqrt(2/8) / sqrt(1/6+1/10) # corr.f.uneq. sizepower.t.test(n=8, delta=d2, sd=.20)

9.2 This is straightforward:

power.prop.test(power=.9, p1=.6, p2=.75)power.prop.test(power=.8, p1=.6, p2=.75)

9.3 Notice that the noncentral t distribution is asymmetric, with a ratherheavy right tail.

curve(dt(x-3, 25), from=0, to=5)curve(dt(x, 25, 3), add=TRUE)

9.4 This causes the “power” at zero effect size (i.e., under the null hy-pothesis) to be half the significance level, in contradiction to theory. Forany relevant true effect size, the difference is immaterial.

9.5 The power in that case is approximately 0.50; exactly so if thevariance is assumed known.

10.1

attach(thuesen)f <- cut(blood.glucose, c(4, 7, 9, 12, 20))levels(f) <- c("low", "intermediate", "high", "very high")

10.2

bcmort2 <- within(bcmort,{period <- area <- cohortlevels(period) <- rep(c("1991-2001","1981-1991"), each=2)levels(area) <- rep(c("Cph+Frb","Nat"),2)

})summary(bcmort2)

10.3 One way is the following (for later use, we also make sure thatvariables are converted to factors):

ashina.long <- reshape(ashina, direction="long",varying=1:2, timevar="treat")

ashina.long <- within(ashina.long, {m <- matrix(c(2,1,1,2),2)id <- factor(id)treat <- factor(treat)grp <- factor(grp)period <- factor(m[cbind(grp,treat)])

Appendix D. Answers to exercises 345

rm(m)})

Notice the use of array indexing. Alternatively, an ifelse construct can beused; e.g., the following (notice that (3 – grp) is 2 when grp is 1 andvice versa):

within(ashina.long,period2 <- ifelse(treat != "active",

as.numeric(grp), 3 – as.numeric(grp)))

Arithmetic involving grp does not work after it was converted to a factor,hence the conversion with as.numeric.

10.4 This can be done a little more easily than in the nickel example byusing subset and transform. It also helps that all observation periodsstart at time zero in this case.

stroke.trim <- function(t1, t2)subset(transform(stroke,

entry=t1, exit=pmin(t2, obsmonths),dead=dead & obsmonths <= t2),

entry < exit)stroke2 <- do.call(rbind, mapply(stroke.trim,

c(0,0.5,2,12), c(0.5,2,12,Inf), SIMPLIFY=F))table(stroke$dead)table(stroke2$dead)

Notice the use of mapply here. This is like sapply and lapply but al-lows the function to have multiple arguments. Alternatively, one couldarrange for stroke.trim to have a single interval argument and uselapply on a list of such intervals.The tabulation at the end is a “sanity check” to show that we have thesame number of deaths but many more censored cases after time-splitting.

11.1 The model with both diameters has a residual error of 0.107, com-pared with 0.128 using abdominal diameter alone and 0.281 with nopredictors at all. If a fetus is scaled isotropically, a cubic relation withweight is expected, and you could speculate that this is reflected in thesum of coefficients when using log scales.

summary(lm(log(bwt) ~ log(bpd) + log(ad), data=secher))summary(lm(log(bwt) ~ log(ad), data=secher))

11.2 If you use attach(tlc), the tlc variable will mask the data frameof the same name, which makes it awkward to access the data frame ifyou need to. If the data frame is in the global environment rather than in

346 Appendix D. Answers to exercises

a package, you get the opposite problem, masking of the variable by thedata frame. The simplest workaround is to avoid attach.

pairs(tlc)summary(lm(log(tlc) ~ ., data=tlc))opar <- par(mfrow=c(2,2))plot(lm(log(tlc) ~ ., data=tlc), which=1:4)

drop1(lm(log(tlc) ~ ., data=tlc))drop1(lm(log(tlc) ~ . – age, data=tlc))

par(mfrow=c(1,1))plot(log(tlc) ~ height, data=tlc)par(mfrow=c(2,2))plot(lm(tlc ~ ., data=tlc), which=1:4) # slightly worsepar(opar)

Some new variations of model formulas were introduced above. A doton the right-hand side in this context means “everything not used on theleft-hand side” within the scope of the data frame. A minus term is re-moved from the model. In other words, … ~ . – age is the same as… ~ sex + height.

11.3 The regression coefficient describes a value to be added for females.

11.4 age is highly significant in the first analysis but only borderlinesignificant (p = 0.06) in the second analysis after removing height andweight. You would expect similar results, but the number of observationsdiffers in the two cases, due to missing observations.

summary(lm(sqrt(igf1) ~ age, data=juul2, subset=(age >= 25)))anova(lm(sqrt(igf1) ~ age + weight + height,

data=juul2, subset=(age >= 25)))

11.5 sex is treated as a binary indicator for girls. Notice that there areeffects both of the mother’s and the child’s size. The reason why heightrather than weight of the mother enters into the equation is somewhatobscure, but one could speculate that weight is an unreliable indicatorshortly after pregnancy.

summary(lm(dl.milk ~ . – no, data=kfm))summary(lm(dl.milk ~ . – no – mat.weight, data=kfm))summary(lm(dl.milk ~ . – no – mat.weight – sex, data=kfm))summary(lm(dl.milk ~ weight + mat.height, data=kfm))

The variations on model formulas used here were described in thesolution to Exercise 11.2.

12.1 Using ashina.long from Exercise 10.3,

Appendix D. Answers to exercises 347

fit.ashina <- lm(vas ~ id + period + treat, data=ashina.long)drop1(fit.ashina, test="F")anova(fit.ashina)

attach(ashina)dd <- vas.active – vas.plact.test(dd[grp==1], -dd[grp==2], var.eq=T)t.test(dd[grp==1], dd[grp==2], var.eq=T)

Notice that the imbalance in group sizes makes the tests for period andtreatment effects order-dependent. The t tests are equivalent to the F testsfrom drop1 but not those from anova.

12.2

attach(tb.dilute)anova(lm(reaction ~ animal + logdose))ld <- c(0.5, 0, -0.5)[logdose]anova(lm(reaction ~ animal + ld))summary(lm(reaction ~ animal + ld))4.7917 + 0.6039 * qt(c(.025,.975), 11)# or:confint(lm(reaction ~ animal + ld))["ld",]

slopes <- reaction[logdose==0.5] – reaction[logdose==-0.5]t.test(slopes)

anova(lm(reaction ~ animal*ld))

Notice that the formula for the fitted slope is β̂ = ∑ xy/ ∑ x2 since x̄ = 0,which reduces to taking differences. (The calculation does rely on databeing in the right order.)The confidence interval is wider in the t test, reflecting that slopesmay vary between rats and that there are fewer degrees of freedom forestimating the variation.The final ANOVA contains a test for parallel slopes, and the F statistic isless than one, so in these data the slopes vary less than expected and theDF must be the important issue for the confidence interval.

12.3 This can be varied indefinitely, but consider these examples:

model.matrix(~ a:b) ; lm(z ~ a:b)model.matrix(~ a * b) ; lm(z ~ a * b)model.matrix(~ a:x) ; lm(z ~ a:x)model.matrix(~ a * x) ; lm(z ~ a * x)model.matrix(~ b * (x + y)) ; lm(z ~ b * (x + y))

The first model is singular because indicator variables are created for allfour groups, but the intercept is not removed. R will only reduce the setof design variables for an interaction term between categorical variables

348 Appendix D. Answers to exercises

when one of the main effects is present. There are no singularities in eitherof the two cases involving a categorical and a continuous variable, but thefirst one has one parameter less (common-intercept model).The last example has a “coincidental” singularity (x and y are propor-tional within each level of b) that R has no chance of detecting.

12.4 The models can be illustrated by plotting the fitted values againsttime with separate symbols for each person; e.g.,

tt <- c(20,30,60,90,0)[time]plot(fitted(model4)~tt,pch=as.character(person))

With model1 there is no imposed structure, model2 is completely addi-tive so that the individual traces are parallel to each other, model3 allowsthe jump from the “pre” value to the value at 20 minutes to vary betweenindividuals, and finally model4 is like model3 except that there is nochange after 30 minutes (traces become horizontal). So model3 is nestedin model1 and both model2 and model4 are nested in model3, but thereis no nesting relation between model2 and model4.

12.5

bp.obese <- transform(bp.obese,sex=factor(sex, labels=c("M","F")))plot(log(bp) ~ log(obese), pch=c(20,21)[sex], data=bp.obese)summary(lm(log(bp) ~ sex, data=bp.obese))summary(lm(log(bp) ~ sex + log(obese), data=bp.obese))summary(lm(log(bp) ~ sex*log(obese), data=bp.obese))

12.6

vitcap2 <- transform(vitcap2,group=factor(group,labels=c("exp>10","exp<10", "unexp")))

attach(vitcap2)plot(vital.capacity~age, pch=(20:22)[group])vit.fit <- lm(vital.capacity ~ age*group)summary(vit.fit)drop1(vit.fit, test="F")for (i in 1:3) abline(lm(vital.capacity ~ age,

subset=as.numeric(group)==i), lty=i)legend(20, 3.5 ,legend=levels(group), pch=20:22, lty=1:3)

Notice that there is a significant interaction; i.e., the lines are not parallel.

12.7

juul.prepub <- subset(juul, tanner==1)

summary(lm(sqrt(igf1)~age, data=juul.prepub, subset= sex==1))summary(lm(sqrt(igf1)~age, data=juul.prepub, subset= sex==2))

Appendix D. Answers to exercises 349

summary(lm(sqrt(igf1)~age*factor(sex), data=juul.prepub))summary(lm(sqrt(igf1)~age+factor(sex), data=juul.prepub))

12.8

summary(fit.aicopt <- step(lm(dl.milk ~ . – no, data=kfm)))opar <- par(mfrow=c(2,2))plot(fit.aicopt, which=1:4)kfm[32,]summary(kfm)summary(update(fit.aicopt, ~ . – sex))plot(update(fit.aicopt, ~ . – sex – ml.suppl), which=1:4)par(opar)

Observation 32 contains an extremely large value of ml.suppl and there-fore has a large influence on its regression coefficient. Without ml.supplin the model, the Cook’s distances are much smaller.

12.9

juulyoung <- subset(juul, age < 25)juulyoung <- transform(juulyoung,

sex=factor(sex), tanner=factor(tanner))fit.untf <- lm(igf1 ~ age * sex * tanner, data=juulyoung,

na.action=na.exclude)plot(fitted(fit.untf) ~ age, data=juulyoung,

col=c("red","green")[sex])fit.log <- update(fit.untf, log(igf1) ~ .)fit.sqrt <- update(fit.untf, sqrt(igf1) ~ .)opar <- par(mfrow=c(2,2))plot(fit.untf, which=1:4)plot(fit.log, which=1:4)plot(fit.sqrt, which=1:4)par(opar)

13.1

summary(glm(mal~age+log(ab), binomial, data=malaria))

(You may also want to check for interaction.)

13.2

attach(graft.vs.host)type <- factor(type,labels=c("AML", "ALL", "CML"))m1 <- glm(gvhd~rcpage+donage+type+preg+log(index), binomial)m1a <- glm(gvhd~rcpage+donage+type+preg+index, binomial)summary(m1)summary(m1a)

350 Appendix D. Answers to exercises

The coefficient to log(index) is more significant, but the model withindex has a slightly better deviance. There is little hard evidence for ei-ther. The log-transform has the advantage that it reduces the influence oftwo very large values of index.

drop1(m1, test="Chisq")drop1(update(m1, ~ . – rcpage), test="Chisq")drop1(update(m1, ~ . – rcpage – type), test="Chisq")drop1(update(m1, ~ . – rcpage – type – preg), test="Chisq")summary(m2 <- glm(gvhd~donage + log(index), binomial))

Notice that except for log(index) it is essentially an arbitrary decisionwhich variables to put in the final model. Altman (1991) treats the typeclassification as separate binary variables and gets a final model whereALL and AML are combined into one group and includes preg but notdonage.

13.3 For example,

confint(m2)## normal approximation:est <- coefficients(summary(m2))[,1]se <- coefficients(summary(m2))[,2]est + cbind(qnorm(.025)*se, qnorm(.975)*se)confint.default(m2)

Notice that the confint-generated intervals lie asymmetrically aroundthe estimate. In this case, both ends of the interval are shifted away fromzero, in accordance with the fact that the deviance-based tests from drop1have lower p-values than the approximate t tests in summary.

13.4 The model can be fitted as follows

counts <- c(13,40,157,40,21,61)total <- c(108,264,375,310,181,162)age <- gl(3,1,6)type <- gl(2,3,6)anova(glm(counts/total~age+type,weights=total, binomial),

test="Chisq")

The effect of type vanished once age was included, suggesting that itreally is the same disease, which has affected mostly younger (and fitter)subjects in the Eastern region.

13.5

juul.girl <- transform(subset(juul,age>8 & age<20 &complete.cases(menarche)),

menarche=factor(menarche))logit.menarche <- glm(menarche~age+I(age^2)+I(age^3),

binomial, data=juul.girl)

Appendix D. Answers to exercises 351

probit.menarche <- glm(menarche~age+I(age^2)+I(age^3),binomial(probit), data=juul.girl)

summary(logit.menarche)summary(probit.menarche)Age=seq(8,20,.1)newages <- data.frame(age=Age)p.logit <- predict(logit.menarche,newdata=newages,type="resp")p.probit <- predict(probit.menarche,newdata=newages,type="resp")matplot(Age,cbind(p.probit,p.logit),type="l")

14.1

attach(graft.vs.host)plot(survfit(Surv(time,dead)~gvhd))survdiff(Surv(time,dead)~gvhd)summary(coxph(Surv(time,dead) ~ gvhd)) # for comparisonsummary(coxph(Surv(time,dead) ~

gvhd + log(index) + donage + rcpage + preg))

Subsequent elimination suggests that preg might be a better predictorthan gvhd.

14.2

attach(melanom)cox1 <- coxph(Surv(days, status==1) ~

log(thick) + sex + strata(ulc))new <- data.frame(sex=2, thick=c(0.1, 0.2, 0.5))svfit <- survfit(cox1,newdata=new)plot(svfit[2], ylim=c(.985, 1))

14.3

summary(coxph(Surv(obsmonths, dead)~age+sex, data=stroke))summary(coxph(Surv(obsmonths, dead)~sex, data=stroke))with(stroke, tapply(age,sex,mean))

The men were considerably younger than the women when they had theirstroke, which may explain their apparently better survival.

14.4 Using stroke2 from Exercise 10.4,

summary(coxph(Surv(entry, exit, dead)~age+sex, data=stroke2))

Notice that the result is essentially the same as in the unsplit analysis; onlyn and Rsquare are changed.

15.1 Using bcmort2 from Exercise 10.2,

bcfit <- glm(bc.deaths ~ (age + period + area)^2, poisson,offset=log(p.yr), data=bcmort2)

352 Appendix D. Answers to exercises

summary(bcfit)drop1(bcfit, test="Chisq")confint(bcfit, parm="period1981-1991:areaNat")

15.2 Continuing with stroke2 from Exercise 10.4, the only slight com-plication is to convert entry to a factor to specify the relevant timeinterval.

summary(glm(dead~sex+age+factor(entry), poisson,offset=log(exit-entry), data=stroke2))

Notice how similar the results are to the Cox analysis in Exercise 14.3.

16.1 To fit to the data for girls, just copy the procedure for boys. Eventhough the growth curves differ, there is no real reason to redo the startingvalue calculation, so we can fit the model to boys, girls, and both as follows

girls <- subset(juul2, age<20 & age>5 & sex==2)boys <- subset(juul2, age<20 & age>5 & sex==1)young <- subset(juul2, age<20 & age>5)stval <- c(alpha=exp(5.3),beta=exp(0.42),gamma=0.15)fit.boys <- nls(height~alpha*exp(-beta*exp(-gamma*age)),

start=stval, data=boys)fit.girls <- nls(height~alpha*exp(-beta*exp(-gamma*age)),

start=stval, data=girls)fit.young <- nls(height~alpha*exp(-beta*exp(-gamma*age)),

start=stval, data=young)

To test whether we can use the same model for boys and girls, there aretwo approaches. One is to make an F test based on the three fits above:

ms.pooled <- (deviance(fit.boys) + deviance(fit.girls))/(499+625)ms.diff <- (deviance(fit.young) –

deviance(fit.boys) – deviance(fit.girls))/3ms.diff/ms.pooled

This gives F = 90.58 on 3 and 1124 degrees of freedom, which is of coursehighly significant.Alternatively, we can set up the joint model with separate parameters forboys and girls and test whether it fits the data better than the model withthe same parameters, like this:

fit.young2 <- nls(height~(alpha+da*(sex==1))*exp(-(beta+db*(sex==1))*exp(-(gamma+dg*(sex==1))*age)),

start=c(alpha=exp(5.3),beta=exp(0.42),gamma=0.15,da=0, db=0, dg=0), data=young)

summary(fit.young2)anova(fit.young, fit.young2)

Appendix D. Answers to exercises 353

Notice that da, db, and dg represent differences between the parametersfor the two genders. The term sex==1 is 0 for girls and 1 for boys.

16.2 We consider experiment 1 only. Starting values can be eyeballed byusing the observation at zero dose for ymax and the x (dose) value atapproximately ymax/2 as β. The value of α can be guessed as the fixedconstant 1. Below, we use α via its logarithm, called la.

e1 <- subset(philion, experiment==1)fit <- nls(sqrt(response) ~ sqrt(ymax / (1 + (dose/ec50)^exp(la))),

start=list(ymax=28, ec50=.3, la=0), data=e1,lower=c(.1,.0001,-Inf), algorithm="port")

summary(fit)confint(fit)p <- profile(fit, alphamax=.2)par(mfrow=c(3,1))plot(p)confint(p)

16.3 The alternative model has similar tail behaviour but behaves differ-ently when x is close to zero. (In particular, the original model has a termproportional to −xα−1 in its derivative. At zero, this is −∞ when α < 1and 0 when α > 1, so the model describes curves that are either very steepor very flat near zero.) Notice that, in the modified model, β is no longerthe EC50; the latter is now the solution for x of (1 + x/β)α = 2. The twoare connected by a factor of 21/α − 1.

e1 <- subset(philion, experiment==1)fit1 <- nls(sqrt(response) ~ sqrt(ymax / (1 + dose/b)^exp(la)),

start=list(ymax=28, b=.3, la=0), data=e1,lower=c(.1,.0001,-Inf), algorithm="port")

summary(fit1)fit2 <- nls(sqrt(response) ~ sqrt(ymax / (1 +

dose/(d50/(2^(1/exp(la))-1)))^exp(la)),start=list(ymax=28, d50=.3, la=0), data=e1,lower=c(.1,.0001,-Inf), algorithm="port")

summary(fit2)

Here, fit1 and fit2 are equivalent models, except that the latter is pa-rameterized in terms of ec50. We can compare the fitted curve with themodel from the previous exercise as follows:

dd <- seq(0,1,,200)yy <- predict(fit, newdata=data.frame(dose=dd))y1 <- predict(fit2, newdata=data.frame(dose=dd))matplot(dd,cbind(yy,y1)^2, type="l")

(Notice that the fitted values should be squared because of the square-roottransformation in the models.)

Bibliography

Agresti, A. (1990), Categorical Data Analysis, John Wiley & Sons, New York.

Altman, D. G. (1991), Practical Statistics for Medical Research, Chapman &Hall, London.

Andersen, P. K., Borgan, Ø., Gill, R. D., and Keiding, N. (1991), StatisticalModels Based on Counting Processes, Springer-Verlag, New York.

Armitage, P. and Berry, G. (1994), Statistical Methods in Medical Research,3rd ed., Blackwell, Oxford.

Bates, D. M. and Watts, D. G. (1988), Nonlinear regression analysis and itsapplications, John Wiley & Sons, New York.

Becker, R. A., Chambers, J. M., and Wilks, A. R. (1988), The NEW SLanguage, Chapman & Hall, London.

Breslow, N. E. and Day, N. (1987), Statistical Methods in Cancer Research.Volume II: The Design and Analysis of Cohort Studies, IARC ScientificPublications, Lyon.

Campbell, M. J. and Machin, D. (1993), Medical Statistics. A CommonsenseApproach, 2nd ed., John Wiley & Sons, Chichester.

Chambers, J. M. and Hastie, T. J. (1992), Statistical Models in S, Chapman &Hall, London.

Clayton, D. and Hills, M. (1993), Statistical Models in Epidemiology, OxfordUniversity Press, Oxford.

356 Bibliography

Cleveland, W. S. (1994), The Elements of Graphing Data, Hobart Press, NewJersey.

Cochran, W. G. and Cox, G. M. (1957), Experimental Designs, 2nd ed., JohnWiley & Sons, New York.

Cox, D. R. (1970), Analysis of Binary Data, Chapman & Hall, London.

Cox, D. R. and Oakes, D. (1984), Analysis of Survival Data, Chapman &Hall, London.

Everitt, B. S. (1994), A Handbook of Statistical Analyses Using S-PLUS,Chapman & Hall, London.

Hájek, J., Šidák, Z., and Sen, P. K. (1999), Theory of Rank Tests, 2nd ed.,Academic Press, San Diego.

Hald, A. (1952), Statistical Theory with Engineering Applications, John Wiley& Sons, New York.

Hosmer, D. W. and Lemeshow, S. (2000), Applied Logistic Regression, 2nded., John Wiley & Sons, New York.

Johnson, R. A. (1994), Miller & Freund’s Probability & Statistics for Engineers,5th ed., Prentice-Hall, Englewood Cliffs, NJ.

Kalbfleisch, J. D. and Prentice, R. L. (1980), The Statistical Analysis of FailureTime Data, John Wiley & Sons, New York.

Lehmann, E. L. (1975), Nonparametrics, Statistical Methods Based on Ranks,McGraw-Hill, New York.

Matthews, D. E. and Farewell, V. T. (1988), Using and Understanding MedicalStatistics, 2nd ed., Karger, Basel.

McCullagh, P. and Nelder, J. A. (1989), Generalized Linear Models, 2nd ed.,Chapman & Hall, London.

Murrell, P. (2005), R Graphics, Chapman & Hall/CRC, Boca Raton, Florida.

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Venables, W. N. and Ripley, B. D. (2000), S Programming, Springer-Verlag,New York.

Venables, W. N. and Ripley, B. D. (2002), Modern Applied Statistics with S,4th ed., Springer-Verlag, New York.

Weisberg, S. (1985), Applied Linear Regression, 2nd ed., John Wiley & Sons,New York.

Zar, J. H. (1999), Biostatistical Analysis, Prentice Hall, Englewood Cliffs, NJ.

Index

abline, 113and logarithmic scales, 211

acceptance region, 96aggregate, 77alternative, 96, 98

one-sided, 96two-sided, 98

analysis of covariance, 208analysis of deviance, 234analysis of variance, 127

one-way, 127unequal variances, 133

two-way, 137with replications, 207

ANOVA, see analysis of varianceanova, 130, 139, 141, 188, 216ANOVA table

for covariance analysis, 216for multiple regression, 188in regression analysis, 141

apply, 27arrays, 16arrows, 134as.Date, 167as.numeric, 19ASCII files, 47assignment, 3

assignment operator, 4attach, 36ave, 178average, 67

bar plotgrouped, 90stacked, 90

barplot, 89barplotlegend.text, 89, 91

bartlett.test, 136binom.test, 146binomial coefficients, 58Bland–Altman plot, 104Bonferroni correction, 132boxplot, 75

parallel, 80by, 78

c, 14calculator, overgrown, 3cat, 13categorical variables, 18cbind, 18censoring, 249chisq.test, 149, 151

358 Index

expected, 152observed, 152

choose, 57classes, 46combinatorics, 56comment (#), 106comparison operators, 22complete.cases, 115concatenation, 14conditional calculation, 170confidence bands, 117confidence interval, 63, 98confint, 237, 287confint.default, 237Conradsen, Knut, 154console window, 1contingency tables, see tablescontrasts, 132Cook’s distance, 218, 222cor, 122correlation, 120

between parameter estimates, 233Kendall’s τ, 124multiple, 113

adjusted, 113Pearson-, 121Spearman’s ρ, 123

count data, 259Cox model, 256curve, 43, 60curve fitting, 275cut, 163, 245

data, 35data editor, 51data entry, 46data frames, 20

appending, 172components of, 21encoding grouped data, 25merging, 173reshaping, 175splitting, 179

data from other programs, 52data sets, 35data.frame, 20dates, 166

format of, 167decile, 68

degrees of freedom, 102fractional, 102

deleting objects, 32density, 42, 58, 59design, see sample sizedesign matrix, 200design variables, 200detach, 36deviance, 228

null, 233residual, 232, 240

dfbetas, 219dffits, 219difftime object, 168dim, 17dimension, 17distribution, 55

binomial, 58, 61normal approximation, 145

continuous, 58cumulative distribution function,

57, 62empirical, 73

density, 59discrete, 57empirical description of, 71exponential, 261Gaussian, see normalgeometric, 58hypergeometric, 148noncentral t, 156normal, 42, 58, 60point probabilities, 57, 59Poisson, 260quantiles, 63

empirical, 67random numbers, 64uniform, 58

distribution-free methods, 99do.call, 182dotchart, 91drop1, 235, 271dummy variables, 132, 195, 200, 201

edit, 51eggs, 154error bars, 134escape character, 48escape sequence, 13

Index 359

ESS (Emacs Speaks Statistics), 32expand.grid, 229expressions, 9extractor function, 111

F testfor comparison of variances, 103in ANOVA table, 142in regression analysis, 113

factor, 19, 165levels, 19

factors, 18describing regular pattern, 139levels of, 18manipulating levels of, 165ordered, 19reading, 49regular pattern, 229

FALSE, 12figure region, 39filename

of file in package, 50filenames, 48Fisher’s exact test, 148fisher.test, 148fitted, 113fitted line, 113fitted values, 113fix, 52flow control, 44break, 45compound expression, 45for, 45if, 45repeat, 45while, 45

force of mortality, 250frequency, 145Friedman’s test, 141ftable, 86functions, 11

arguments to, 11actual, 11default, 44formal, 11keyword matching, 11named, 11positional matching, 11

generic, 46

generalized linear models, 228gl, 139, 229glm, 229family, 229weights, 230

graphics, 7, 79colour coding, 222for grouped data, 134for repeated measurements, 140pairwise plots, 185

graphics window, 2

hazard function, 250constant, 261piecewise constant, 261

head, 24header, 48help, 34help functions, 34help.search, 34help.start, 34hist, 42, 71, 79breaks, 79freq, 72

histogram, 42, 71, 79Holm correction, 133

I, 196, 213ifelse, 171indexing, 21

negative, 22of data frames, 23with a vector, 21with logical vector, 23with relational expression, 22

inputfrom file, 47

interaction, 140, 206, 216interaction.plot, 140intercept, 109, 112, 198interquartile range (IQR), 68ISwR package, 2

Kaplan–Meier estimator, 251Kruskal–Wallis test, 136kruskal.test, 136

lapply, 26, 170lazy loading, 35

360 Index

least squares, 109, 276legend, 91legend, 91, 209length, 73levels, 19, 165, 166library, 35lifetime data, 249likelihood function, 228

“Poisson” (constant hazardsurvival), 261

Poisson, 260linear regression, 109linearity

grouped data, 202lines, 8, 115link function, 228list, 19lists, 19

components of, 20lm, 110, 129, 139load, 32locator, 91, 209log odds, 227log-rank test, 254logarithmic scale, 210logical expressions, 22

combination of, 22logical operators, 22logistic regression, 227logit, 227loops

implicit, 26ls, 31

Mann–Whitney test, see Wilcoxon test,two-sample

margin coordinates, 39margin.table, 87matlines, 119matrices, 16, 83

binding together, 18row and column names, 17, 84transposition, 17

matrix, 17, 83byrow, 83ncol, 83nrow, 83

maximum likelihood, 228mean, 5

mean, 67mean squares, 128median, 67merge, 173missing values, 14, 23, 68, 115is.na, 23, 115na.exclude, 115na.rm, 69na.rm argument, 122use argument, 122

model formulas, 81, 111, 187, 216arithmetic expressions in, 213interaction terms, 216

model object, 111model search, 190model.matrix, 200multiframe graphics layout, 75, 79multiple comparisons, 131multiple regression, 185

NA, see missing valuesnls, 276nonparametric tests, see distribution-

free methodsnormal distribution, 63, 74null hypothesis, 95number of observations, 73

objects, 9listing of, 31removing, 32

odds ratio, 148, 227, 239one-sided test, 96oneway.test, 133online help, 34

search engine, 34order, 28ordered, 19

p-value, 96packages, 35pairs, 185pairwise comparisons, 131pairwise.t.test, 132par, 42, 79percentile, 68person-years, 259pie, 92pin diagram, 61

Index 361

plot, 7plot layout, 39plot region, 39plot symbol, 7plots

adding toabline, 40axis, 41box, 41lines, 8, 40, 115mtext, 40points, 40text, 40title, 41

combining, 42plotting parameters

axis labels (xlab and ylab), 39axis limits (xlim and ylim), 43, 73,

79colour (col), 79empty plot (type="n"), 40heading (main), 39, 93line plot (type="l"), 60logarithmic axes (log), 210margin expansion (mex), 42margin sizes (mar), 42multiframe layout (mfrow and

mfcol), 75, 79no axes (axes=F), 40pin diagram (type="h"), 61setting with par, 42step function (type="s"), 73subtitle (sub), 39symbol (pch), 7, 209text font (font), 40

pmax, 180pmin, 168, 180polynomial regression, 196power, 155, 156power.prop.test, 155power.t.test, 155predict, 119, 197prediction, 197, 241prediction bands, 117prediction in new data frame, 120print methods, 46probability, 56probability distribution, see

distribution

probability plot, 74product-limit estimator, see

Kaplan–Meier estimatorprofile, 238, 286profile plot, 238, 286profiling, 237glm, 238nls, 285

programming, 44prompt, 2

continuation, 20prop.table, 87prop.test, 146prop.trend.test, 150proportional hazards model, 256proportions, 145

comparison of k groups, 149comparison of two groups, 147

power of, 159test for trend, 149, 235test of simple hypothesis, 145

Q–Q plot, 74, 218qqnorm, 74, 117quantile, 67, 164quartile, 68quote symbols, 12

R2, 113in model without intercept, 199

random numbers, 2, 64random sample, 55

with and without replacement, 56range, 43rate ratio, 265rates, 259, 266rbind, 18, 172read.csv, 51read.csv2, 51read.delim, 51read.delim2, 51read.table, 47colClasses, 51comments, 50conversion, 51field separator, 50header, 48NA strings, 50quotes, 50

362 Index

with unequal field count, 50reading data, 47regression

nonlinear, 276self-starting models, 284

regression analysis, 195diagnostics, 218line through origin, 198linear, 109logistic, 227

for raw data, 239for tabular data, 229

multiple, 185polynomial, 196

regression coefficients, 109, 112, 201interpretation for factors, 132interpretation in covariance

analysis, 214regression lines

comparison of, 212relational expressions, 13rep, 16replicate, 26replication, 16reshape, 175resid, 113residual variation, 113residuals, 109, 112, 113

deviance, 231, 240leave-out-one, 219standardized, 218

rm, 32rnorm, 2rstandard, 219rstudent, 219

sample, 55sample size, 155

comparison of proportions, 161one-sample problems, 161paired tests, 161two-sample problems, 159

sampling, see random samplesapply, 26save, 32save.image, 32saving

command history, 33workspace, 32

scripts, 33windows for editing, 33

sd, 67search, 36search path, 36SEDM, 101segments, 116selection, 22SEM, 96seq, 15, 60sequences of numbers, 15, 60sign test, 62signed-rank test, 99significance level, 96, 156significance stars, 112sink, 32slope, 109SMR (standardized mortality rate),

269sort, 27sorting, 27

by another variable, 28source, 33spaghettigram, 140split, 25, 178staggered entry, 250standard deviation, 5, 67standard error

of differences of means, 101of regression coefficients, 110of the mean, 96

stripchart, 82, 134jitter, 82method, 82

subset, 37select argument, 172

subsetting, 21, 37summarycorr argument, 233of data frame, 69of glm object, 231of lm object, 111, 131of numeric variable, 69of survfit object, 252

summary statistics, 67, 97tables of, 75

Surv objects, 250survdiff, 255survfit, 252

Index 363

plot of, 253survival function, 250, 251system.file, 50

t test, 6, 95approximate power, 158one-sample, 6, 95

power of, 156paired, 104

power of, 156two-sample, 100

power of, 158same variance, 102

t.test, 97alternative, 99mu, 97paired, 105var.equal, 102

tables, 83graphical display, 89marginals, 87r × c, 151relative frequencies, 87statistical analysis, 145

tail, 24tapply, 27, 75ties, 100time splitting, 179transform, 37transposition, 87treatment contrasts, 132, 201TRUE, 12two-sided test, 98type I and type II errors, 156

var, 67var.test, 103

variable names, 4variance, 67

comparison of, 103, 136between linear models, 215

variationbetween rows and columns, 137due to model, 141residual, 141within and between groups, 128

vectors, 4, 12calculations with, 5logical, 12numeric, 4, 12recycling of, 5text- (character), 12

Welch’s test, 101more than two categories, 133

wilcox.test, 99Wilcoxon test, 95

matched-pairs, 106one-sample, 99two-sample, 103

Windows, 2with, 37within, 37working directory, 33workspace, 31

clearing, 32

χ2 test, 149, 151components of, 152

xtabs, 86

Yates correction, 145, 148

springer.com

Software for Data Analysis Programming with R John M. Chambers This book guides the reader through programming with R, beginning with simple interactive use and progressing by gradual stages, starting with simple functions. More advanced programming techniques can be added as needed, allowing users to grow into software contributors, benefiting their careers and the community. R packages provide a powerful mechanism for contributions to be organized and communicated. 2008. Approx. 510 pp. (Statistics and Computing) Hardcover ISBN 978-0-387-75935-7 Time Series Analysis with Applications in R

Jonathan D. Cryer and Kung-Sik Chan Time Series Analysis With Applications in R, Second Ed., presents an accessible approach to understanding time series models and their applications. Although the emphasis is on time domain ARIMA models and their analysis, the new edition devotes two chapters to the frequency domain and three to time series regression models, models for heteroscedasticty, and threshold models. All of the ideas and methods are illustrated with both real and simulated data sets. A unique feature of this edition is its integration with the R computing environment. 2008. 2nd Ed., 494 pp. (Springer Texts in Statistics) Hardcover ISBN 0-387-75958-6

Data Manipulation with R Phil Spector This book presents a wide array of methods applicable for reading data into R, and efficiently manipulating that data. In addition to the built-in functions, a number of readily available packages from CRAN (the Comprehensive R Archive Network) are also covered. All of the methods presented take advantage of the core features of R: vectorization, efficient use of subscripting, and the proper use of the varied functions in R that are provided for common data management tasks. 2008. 164 pp. (Use R) Softcover ISBN 978-0-387-74730-9

Easy Ways to Order► Call: Toll-Free 1-800-SPRINGER ▪ E-mail: [email protected] ▪ Write: Springer, Dept. S8113, PO Box 2485, Secaucus, NJ 07096-2485 ▪ Visit: Your local scientific bookstore or urge your librarian to order.

Cover.jpg
front-matter.pdf

Preface
Contents

fulltext.pdf

Basics

First steps

An overgrown calculator
Assignments
Vectorized arithmetic
Standard procedures
Graphics

R language essentials

Expressions and objects
Functions and arguments
Vectors
Quoting and escape sequences
Missing values
Functions that create vectors
Matrices and arrays
Factors
Lists
Data frames
Indexing
Conditional selection
Indexing of data frames
Grouped data and data frames
Implicit loops
Sorting

Exercises

fulltext_2.pdf

The R environment

Session management

The workspace
Textual output
Scripting
Getting help
Packages
Built-in data
attach and detach
subset, transform, and within

The graphics subsystem

Plot layout
Building a plot from pieces
Using par
Combining plots

R programming

Flow control
Classes and generic functions

Data entry

Reading from a text file
Further details on read.table
The data editor
Interfacing to other programs

Exercises

fulltext_3.pdf

Probability and distributions

Random sampling
Probability calculations and combinatorics
Discrete distributions
Continuous distributions
The built-in distributions in R

Densities
Cumulative distribution functions
Quantiles
Random numbers

Exercises

fulltext_4.pdf

Descriptive statistics and graphics

Summary statistics for a single group
Graphical display of distributions

Histograms
Empirical cumulative distribution
Q–Q plots
Boxplots

Summary statistics by groups
Graphics for grouped data

Histograms
Parallel boxplots
Stripcharts

Tables

Generating tables
Marginal tables and relative frequency

Graphical display of tables

Barplots
Dotcharts
Piecharts

Exercises

fulltext_5.pdf

One- and two-sample tests

One-sample t test
Wilcoxon signed-rank test
Two-sample t test
Comparison of variances
Two-sample Wilcoxon test
The paired t test
The matched-pairs Wilcoxon test
Exercises

fulltext_6.pdf

Regression and correlation

Simple linear regression
Residuals and fitted values
Prediction and confidence bands
Correlation

Pearson correlation
Spearman's
Kendall's

Exercises

fulltext_7.pdf

Analysis of variance and the Kruskal–Wallis test

One-way analysis of variance

Pairwise comparisons and multiple testing
Relaxing the variance assumption
Graphical presentation
Bartlett's test

Kruskal–Wallis test
Two-way analysis of variance

Graphics for repeated measurements

The Friedman test
The ANOVA table in regression analysis
Exercises

fulltext_8.pdf

Tabular data

Single proportions
Two independent proportions
k proportions, test for trend
r c tables
Exercises

fulltext_9.pdf

Power and the computation of sample size

The principles of power calculations

Power of one-sample and paired t tests
Power of two-sample t test
Approximate methods
Power of comparisons of proportions

Two-sample problems
One-sample problems and paired tests
Comparison of proportions
Exercises

fulltext_10.pdf

Advanced data handling

Recoding variables

The cut function
Manipulating factor levels
Working with dates
Recoding multiple variables

Conditional calculations
Combining and restructuring data frames

Appending frames
Merging data frames
Reshaping data frames

Per-group and per-case procedures
Time splitting
Exercises

fulltext_11.pdf

Multiple regression

Plotting multivariate data
Model specification and output
Model search
Exercises

fulltext_12.pdf

Linear models

Polynomial regression
Regression through the origin
Design matrices and dummy variables
Linearity over groups
Interactions
Two-way ANOVA with replication
Analysis of covariance

Graphical description
Comparison of regression lines

Diagnostics
Exercises

fulltext_13.pdf

Logistic regression

Generalized linear models
Logistic regression on tabular data

The analysis of deviance table
Connection to test for trend

Likelihood profiling
Presentation as odds-ratio estimates
Logistic regression using raw data
Prediction
Model checking
Exercises

fulltext_14.pdf

Survival analysis

Essential concepts
Survival objects
Kaplan–Meier estimates
The log-rank test
The Cox proportional hazards model
Exercises

fulltext_15.pdf

Rates and Poisson regression

Basic ideas

The Poisson distribution
Survival analysis with constant hazard

Fitting Poisson models
Computing rates
Models with piecewise constant intensities
Exercises

fulltext_16.pdf

Nonlinear curve fitting

Basic usage
Finding starting values
Self-starting models
Profiling
Finer control of the fitting algorithm
Exercises

back-matter.pdf

Obtaining and installing R and the ISwR package
Data sets in the ISwR package
Compendium
Answers to exercises
Bibliography
Index