The R Book

The R Book

The R Book Michael J. Crawley Imperial College London at Silwood Park, UK

Copyright © 2007

John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England Telephone +44 1243 779777

Email (for orders and customer service enquiries): [email protected] Visit our Home Page on www.wiley.com All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except under the terms of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London W1T 4LP, UK, without the permission in writing of the Publisher. Requests to the Publisher should be addressed to the Permissions Department, John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England, or emailed to [email protected], or faxed to (+44) 1243 770620. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the Publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought. Other Wiley Editorial Offices John Wiley & Sons Inc., 111 River Street, Hoboken, NJ 07030, USA Jossey-Bass, 989 Market Street, San Francisco, CA 94103-1741, USA Wiley-VCH Verlag GmbH, Boschstr. 12, D-69469 Weinheim, Germany John Wiley & Sons Australia Ltd, 42 McDougall Street, Milton, Queensland 4064, Australia John Wiley & Sons (Asia) Pte Ltd, 2 Clementi Loop #02-01, Jin Xing Distripark, Singapore 129809 John Wiley & Sons Canada Ltd, 6045 Freemont Blvd, Mississauga, ONT, L5R 4J3 Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. Anniversary Logo Design: Richard J. Pacifico British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN-13: 978-0-470-51024-7 Typeset in 10/12pt Times by Integra Software Services Pvt. Ltd, Pondicherry, India Printed and bound in Great Britain by Antony Rowe Ltd, Chippenham, Wiltshire This book is printed on acid-free paper responsibly manufactured from sustainable forestry in which at least two trees are planted for each one used for paper production.

Contents Preface 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

Getting Started Essentials of the R Language Data Input Dataframes Graphics Tables Mathematics Classical Tests Statistical Modelling Regression Analysis of Variance Analysis of Covariance Generalized Linear Models Count Data Count Data in Tables Proportion Data Binary Response Variables Generalized Additive Models Mixed-Effects Models Non-linear Regression Tree Models Time Series Analysis Multivariate Statistics Spatial Statistics Survival Analysis Simulation Models Changing the Look of Graphics

vii 1 9 97 107 135 183 195 279 323 387 449 489 511 527 549 569 593 611 627 661 685 701 731 749 787 811 827

References and Further Reading

873

Index

877

Preface R is a high-level language and an environment for data analysis and graphics. The design of R was heavily influenced by two existing languages: Becker, Chambers and Wilks’ S and Sussman’s Scheme. The resulting language is very similar in appearance to S, but the underlying implementation and semantics are derived from Scheme. This book is intended as an introduction to the riches of the R environment, aimed at beginners and intermediate users in disciplines ranging from science to economics and from medicine to engineering. I hope that the book can be read as a text as well as dipped into as a reference manual. The early chapters assume absolutely no background in statistics or computing, but the later chapters assume that the material in the earlier chapters has been studied. The book covers data handling, graphics, mathematical functions, and a wide range of statistical techniques all the way from elementary classical tests, through regression and analysis of variance and generalized linear modelling, up to more specialized topics such as spatial statistics, multivariate methods, tree models, mixed-effects models and time series analysis. The idea is to introduce users to the assumptions that lie behind the tests, fostering a critical approach to statistical modelling, but involving little or no statistical theory and assuming no background in mathematics or statistics. Why should you switch to using R when you have mastered a perfectly adequate statistical package already? At one level, there is no point in switching. If you only carry out a very limited range of statistical tests, and you don’t intend to do more (or different) in the future, then fine. The main reason for switching to R is to take advantage of its unrivalled coverage and the availability of new, cutting edge applications in fields such as generalized mixedeffects modelling and generalized additive models. The next reason for learning R is that you want to be able to understand the literature. More and more people are reporting their results in the context of R, and it is important to know what they are talking about. Third, look around your discipline to see who else is using R: many of the top people will have switched to R already. A large proportion of the world’s leading statisticians use R, and this should tell you something (many, indeed, contribute to R, as you can see below). Another reason for changing to R is the quality of back-up and support available. There is a superb network of dedicated R wizards out there on the web, eager to answer your questions. If you intend to invest sufficient effort to become good at statistical computing, then the structure of R and the ease with which you can write your own functions are major attractions. Last, and certainly not least, the product is free. This is some of the finest integrated software in the world, and yet it is yours for absolutely nothing. Although much of the text will equally apply to S-PLUS, there are some substantial differences, so in order not to confuse things I concentrate on describing R. I have made no attempt to show where S-PLUS is different from R, but if you have to work in S-PLUS, then try it and see if it works.

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PREFACE

Acknowledgements S is an elegant, widely accepted, and enduring software system with outstanding conceptual integrity, thanks to the insight, taste, and effort of John Chambers. In 1998, the Association for Computing Machinery (ACM) presented him with its Software System Award, for ‘the S system, which has forever altered the way people analyze, visualize, and manipulate data’. R was inspired by the S environment that was developed by John Chambers, and which had substantial input from Douglas Bates, Rick Becker, Bill Cleveland, Trevor Hastie, Daryl Pregibon and Allan Wilks. R was initially written by Ross Ihaka and Robert Gentleman at the Department of Statistics of the University of Auckland in New Zealand. Subsequently, a large group of individuals contributed to R by sending code and bug reports. John Chambers graciously contributed advice and encouragement in the early days of R, and later became a member of the core team. The current R is the result of a collaborative effort with contributions from all over the world. Since mid-1997 there has been a core group known as the ‘R Core Team’ who can modify the R source code archive. The group currently consists of Doug Bates, John Chambers, Peter Dalgaard, Robert Gentleman, Kurt Hornik, Stefano Iacus, Ross Ihaka, Friedrich Leisch, Thomas Lumley, Martin Maechler, Duncan Murdoch, Paul Murrell, Martyn Plummer, Brian Ripley, Duncan Temple Lang, Luke Tierney, and Simon Urbanek. R would not be what it is today without the invaluable help of the following people, who contributed by donating code, bug fixes and documentation: Valerio Aimale, Thomas Baier, Roger Bivand, Ben Bolker, David Brahm, Göran Broström, Patrick Burns, Vince Carey, Saikat DebRoy, Brian D’Urso, Lyndon Drake, Dirk Eddelbuettel, Claus Ekström, John Fox, Paul Gilbert, Frank E. Harrell Jr, Torsten Hothorn, Robert King, Kjetil Kjernsmo, Roger Koenker, Philippe Lambert, Jan de Leeuw, Uwe Ligges, Jim Lindsey, Patrick Lindsey, Catherine Loader, Gordon Maclean, John Maindonald, David Meyer, Eiji Nakama, Jens Oehlschaegel, Steve Oncley, Richard O’Keefe, Hubert Palme, Roger D. Peng, Jose C. Pinheiro, Tony Plate, Anthony Rossini, Jonathan Rougier, Deepayan Sarkar, Guenther Sawitzki, Marc Schwartz, Detlef Steuer, Bill Simpson, Gordon Smyth, Adrian Trapletti, Terry Therneau, Rolf Turner, Bill Venables, Gregory R. Warnes, Andreas Weingessel, Morten Welinder, James Wettenhall, Simon Wood and Achim Zeileis. I have drawn heavily on the R help pages in writing this book, and I am extremely grateful to all the R contributors who wrote the help files. Special thanks are due to the generations of graduate students on the annual GLIM course at Silwood. It was their feedback that enabled me to understand those aspects of R that are most difficult for beginners, and highlighted the concepts that require the most detailed explanation. Please tell me about the errors and omissions you find, and send suggestions for changes and additions to [email protected]. M.J. Crawley Ascot September 2006

1 Getting Started

Installing R I assume that you have a PC or an Apple Mac, and that you want to install R on the hard disc. If you have access to the internet then this could hardly be simpler. First go to the site called CRAN (this stands for Comprehensive R Archive Network). You can type its full address, http://cran.r-project.org/ or simply type CRAN into Google and be transported effortlessly to the site. Once there, you need to ‘Download and Install R’ by running the appropriate precompiled binary distributions. Click to choose between Linux, Mac OS and Windows, then follow the (rather different) instructions. You want the ‘base’ package and you want to run the setup program which will have a name like R*.exe (on a PC) or R*.dmg (on a Mac). When asked, say you want to ‘Run’ the file (rather than ‘Save’ it). Then just sit back and watch. If you do not have access to the internet, then get a friend to download R and copy it onto a CD or a memory stick for you.

Running R To run R, just click on the R icon. If there is no icon, go to Programs, then to R, then click on the R icon. The first thing you see is the version number of R and the date of your version. It is a good idea to visit the CRAN site regularly to make sure that you have got the most up-to-date version of R. If you have an old version, it is best to uninstall your current version before downloading the new one. The header explains that there is no warranty for this free software, and allows you to see the list of current contributors. Perhaps the most important information in the header is found under citation() which shows how to cite the R software in your written work. The R Development Core Team has done a huge amount of work and we, the R user community, should pay them due credit whenever we publish work that has used R.

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Below the header you will see a blank line with a > symbol in the left hand margin. This is called the prompt and is R’s way of saying ‘What now?’. This is where you type in your commands, as introduced on p. 9. When working, you will sometimes see + at the left-hand side of the screen instead of >. This means that the last command you typed is incomplete. The commonest cause of this is that you have forgotten one or more brackets. If you can see what is missing (e.g. a final right-hand bracket) then just type the missing character and press enter, at which point the command will execute. If you have made a mistake, then press the Esc key and the command line prompt > will reappear. Then use the Up arrow key to retrieve your last command, at which point you can correct the mistake, using the Left and Right arrow keys.

Getting Help in R The simplest way to get help in R is to click on the Help button on the toolbar of the RGui window. Alternatively, if you are connected to the internet, you can type CRAN in Google and search for the help you need at CRAN. However, if you know the name of the function you want help with, you just type a question mark ? at the command line prompt followed by the name of the function. So to get help on read.table, just type ?read.table Sometimes you cannot remember the precise name of the function, but you know the subject on which you want help (e.g. data input in this case). Use the help.search function (without a question mark) with your query in double quotes like this: help.search("data input") and (with any luck) you will see the names of the R functions associated with this query. Then you can use ?read.table to get detailed help. Other useful functions are find and apropos. The find function tells you what package something is in: find(lowess) [1] "package:stats"

while apropos returns a character vector giving the names of all objects in the search list that match your (potentially partial) enquiry: apropos(lm) [1] [4] [7] [10] [13] [16] [19] [22] [25] [28] [31] [34]

". __C__anova.glm" ". __C__glm.null" "anova.glm" "anova.lmlist" "contr.helmert" "glm.fit" "KalmanForecast" "KalmanSmooth" "lm.fit.null" "lm.wfit.null" "model.matrix.lm" "plot.lm"

". __C__anova.glm.null" ". __C__lm" "anova.glmlist" "anova.mlm" "glm" "glm.fit.null" "KalmanLike" "lm" "lm.influence" "model.frame.glm" "nlm" "plot.mlm"

". __C__glm" ". __C__mlm" "anova.lm" "anovalist.lm" "glm.control" "hatvalues.lm" "KalmanRun" "lm.fit" "lm.wfit" "model.frame.lm" "nlminb" "predict.glm"

GETTING STARTED [37] [40] [43] [46] [49]

"predict.lm" "print.lm" "rstandard.glm" "rstudent.lm" "summary.mlm"

3 "predict.mlm" "residuals.glm" "rstandard.lm" "summary.glm" "kappa.lm"

"print.glm" "residuals.lm" "rstudent.glm" "summary.lm"

Online Help The is a tremendous amount of information about R on the web, but your first port of call is likely to be CRAN at http://cran.r-project.org/ Here you will find a variety of R manuals: • An Introduction to R gives an introduction to the language and how to use R for doing statistical analysis and graphics. • A draft of the R Language Definition documents the language per se – that is, the objects that it works on, and the details of the expression evaluation process, which are useful to know when programming R functions. • Writing R Extensions covers how to create your own packages, write R help files, and use the foreign language (C, C + +, Fortran,    ) interfaces. • R Data Import/Export describes the import and export facilities available either in R itself or via packages which are available from CRAN. • R Installation and Administration, which is self-explanatory. • R: A Language and Environment for Statistical Computing (referred to on the website as ‘The R Reference Index’) contains all the help files of the R standard and recommended packages in printable form. (These manuals are also available in R itself by choosing Help/Manuals (in PDF) from the menu bar.) There are also answers to Frequently Asked Questions (FAQs) and R News, a newsletter which contains interesting articles, book reviews and news of forthcoming releases. The most useful part of the site, however, is the Search facility which allows you to investigate the contents of most of the R documents, functions, and searchable mail archives.

Worked Examples of Functions To see a worked example just type the function name (linear models, lm, in this case) example(lm) and you will see the printed and graphical output produced by the lm function.

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Demonstrations of R Functions These can be useful for seeing the range of things that R can do. Here are some for you to try: demo(persp) demo(graphics) demo(Hershey) demo(plotmath)

Libraries in R To use one of the libraries (listed in Table 1.1), simply type the library function with the name of the library in brackets. Thus, to load the spatial library type library(spatial)

Table 1.1. Libraries used in this book that come supplied as part of the base package of R. lattice MASS mgcv nlme nnet spatial survival

lattice graphics for panel plots or trellis graphs package associated with Venables and Ripley’s book entitled Modern Applied Statistics using S-PLUS generalized additive models mixed-effects models (both linear and non-linear) feed-forward neural networks and multinomial log-linear models functions for kriging and point pattern analysis survival analysis, including penalised likelihood

Contents of Libraries It is easy to use the help function to discover the contents of library packages. Here is how you find out about the contents of the spatial library: library(help=spatial) Information on package "spatial" Package: spatial Description: Functions for kriging and point pattern analysis.

followed by a list of all the functions and data sets. You can view the full list of the contents of a library using objects with search() like this. Here are the contents of the spatial library: objects(grep("spatial",search())) [1] [5] [9] [13]

"anova.trls" "gaucov" "plot.trls" "ppregion"

"anovalist.trls" "Kaver" "ppgetregion" "predict.trls"

"correlogram" "Kenvl" "ppinit" "prmat"

"expcov" "Kfn" "pplik" "Psim"

GETTING STARTED [17] "semat" [21] "surf.gls" [25] "variogram"

5 "sphercov" "surf.ls"

"SSI" "Strauss" "trls.influence" "trmat"

Then, to find out how to use, say, Ripley’s K (Kfn), just type ?Kfn

Installing Packages and Libraries The base package does not contain some of the libraries referred to in this book, but downloading these is very simple. Run the R program, then from the command line use the install.packages function to download the libraries you want. You will be asked to highlight the mirror nearest to you for fast downloading (e.g. London), then everything else is automatic. The packages used in this book are install.packages("akima") install.packages("chron") install.packages("Ime4") install.packages("mcmc") install.packages("odesolve") install.packages("spdep") install.packages("spatstat") install.packages("tree") If you want other libraries, then go to CRAN and browse the list called ‘Packages’ to select the ones you want to investigate.

Command Line versus Scripts When writing functions and other multi-line sections of input you will find it useful to use a text editor rather than execute everything directly at the command line. I always use Word for this, because it is so easy to keep a copy of all the output and graphics produced by R using Copy and Paste. Other people prefer to use R’s own built-in editor. It is accessible from the RGui menu bar. Click on File then click on New script. At this point R will open a window entitled Untitled - R Editor. You can type and edit in this, then when you want to execute a line or group of lines, just highlight them and press Ctrl + R (the Control key and R together). The lines are automatically transferred to the command window and executed. By pressing Ctrl + S you can save the contents of the R Editor window in a file that you will have to name. It will be given a .R file extension automatically. In a subsequent session you can click on File/Open script    when you will see all your saved .R files and can select the one you want to open.

Data Editor There is a data editor within R that can be accessed from the menu bar by selecting Edit/Data editor   . You provide the name of the matrix or dataframe containing the material you

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want to edit (this has to be a dataframe that is active in the current R session, rather than one which is stored on file), and a Data Editor window appears. Alternatively, you can do this from the command line using the fix function (e.g. fix(data.frame.name)). Suppose you want to edit the bacteria dataframe which is part of the MASS library: library(MASS) attach(bacteria) fix(bacteria) The window has the look of an Excel spreadsheet, and you can change the contents of the cells, navigating with the cursor or with the arrow keys. My preference is to do all of my data preparation and data editing in Excel itself (because that is what it is good at). Once checked and edited, I save the data from Excel to a tab-delimited text file (∗ .txt) that can be imported to R very simply using the function called read.table (p. 98). One of the most persistent frustrations for beginners is that they cannot get their data imported into R. Things that typically go wrong at the data input stage and the necessary remedial actions are described on p. 98.

Changing the Look of the R Screen The default settings of the command window are inoffensive to most people, but you can change them if you don’t like them. The Rgui Configuration Editor under Edit/GUI preferences    is used to change the look of the screen. You can change the colour of the input line (default is red), the output line (default navy) or the background (default white). The default numbers of rows (25) and columns (80) can be changed, and you have control over the font (default Courier New) and font size (default 10).

Significance Stars If these worry you, then turn them off. Significance stars are shown by default next to the p values in the output of statistical models. gg log(42/7.3) [1] 1.749795

Each line can have at most 128 characters, so if you want to give a lengthy instruction or evaluate a complicated expression, you can continue it on one or more further lines simply by ending the line at a place where the line is obviously incomplete (e.g. with a trailing comma, operator, or with more left parentheses than right parentheses, implying that more right parentheses will follow). When continuation is expected, the prompt changes from > to + > 5+6+3+6+4+2+4+8+ + 3+2+7 [1] 50

Note that the + continuation prompt does not carry out arithmetic plus. If you have made a mistake, and you want to get rid of the + prompt and return to the > prompt, then either press the Esc key or use the Up arrow to edit the last (incomplete) line.

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Two or more expressions can be placed on a single line so long as they are separated by semi-colons: 2+3; 5*7; 3-7 [1] 5 [1] 35 [1] -4

From here onwards and throughout the book, the prompt character > will be omitted. The material that you should type on the command line is shown in Arial font. Just press the Return key to see the answer. The output from R is shown in Courier New font, which uses absolute rather than proportional spacing, so that columns of numbers remain neatly aligned on the page or on the screen.

Built-in Functions All the mathematical functions you could ever want are here (see Table 2.1). The log function gives logs to the base e e = 2718282, for which the antilog function is exp log(10) [1] 2.302585

exp(1) [1] 2.718282

If you are old fashioned, and want logs to the base 10, then there is a separate function log10(6) [1] 0.7781513

Logs to other bases are possible by providing the log function with a second argument which is the base of the logs you want to take. Suppose you want log to base 3 of 9: log(9,3) [1] 2

The trigonometric functions in R measure angles in radians. A circle is 2 radians, and this is 360 , so a right angle 90  is /2 radians. R knows the value of  as pi: pi [1] 3.141593

sin(pi/2) [1] 1

cos(pi/2) [1] 6.123032e-017

Notice that the cosine of a right angle does not come out as exactly zero, even though the sine came out as exactly 1. The e-017 means ‘times 10−17 ’. While this is a very small

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Table 2.1. Mathematical functions used in R. Function

Meaning

log(x) exp(x) log(x,n) log10(x) sqrt(x) factorial(x) choose(n,x) gamma(x) lgamma(x) floor(x) ceiling(x) trunc(x)

log to base e of x antilog of x ex  log to base n of x log to base 10 of x square root of x x! binomial coefficients n!/(x! n − x!) x, for real x x − 1!, for integer x natural log of x greatest integer < x smallest integer > x closest integer to x between x and 0 trunc(1.5) = 1, trunc(-1.5) = −1 trunc is like floor for positive values and like ceiling for negative values round the value of x to an integer give x to 6 digits in scientific notation generates n random numbers between 0 and 1 from a uniform distribution cosine of x in radians sine of x in radians tangent of x in radians inverse trigonometric transformations of real or complex numbers inverse hyperbolic trigonometric transformations of real or complex numbers the absolute value of x, ignoring the minus sign if there is one

round(x, digits=0) signif(x, digits=6) runif(n) cos(x) sin(x) tan(x) acos(x), asin(x), atan(x) acosh(x), asinh(x), atanh(x) abs(x)

number it is clearly not exactly zero (so you need to be careful when testing for exact equality of real numbers; see p. 77).

Numbers with Exponents For very big numbers or very small numbers R uses the following scheme: 1.2e3

means 1200 because the e3 means ‘move the decimal point 3 places to the right’

1.2e-2

means 0.012 because the e-2 means ‘move the decimal point 2 places to the left’

3.9+4.5i

is a complex number with real (3.9) and imaginary (4.5) parts, and i is the square root of −1.

Modulo and Integer Quotients Integer quotients and remainders are obtained using the notation %/% (percent, divide, percent) and %% (percent, percent) respectively. Suppose we want to know the integer part of a division: say, how many 13s are there in 119:

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119 %/% 13 [1] 9

Now suppose we wanted to know the remainder (what is left over when 119 is divided by 13): in maths this is known as modulo: 119 %% 13 [1] 2

Modulo is very useful for testing whether numbers are odd or even: odd numbers have modulo 2 value 1 and even numbers have modulo 2 value 0: 9 %% 2 [1] 1

8 %% 2 [1] 0

Likewise, you use modulo to test if one number is an exact multiple of some other number. For instance to find out whether 15 421 is a multiple of 7, ask: 15421 %% 7 == 0 [1] TRUE

Rounding Various sorts of rounding (rounding up, rounding down, rounding to the nearest integer) can be done easily. Take 5.7 as an example. The ‘greatest integer less than’ function is floor floor(5.7) [1] 5

and the ‘next integer’ function is ceiling ceiling(5.7) [1] 6

You can round to the nearest integer by adding 0.5 to the number then using floor. There is a built-in function for this, but we can easily write one of our own to introduce the notion of function writing. Call it rounded, then define it as a function like this: rounded