The Basics of S and S- PLUS

Statistics and Computing Series Editors: J. Chambers W. Eddy W. Hardle S. Sheather L. Tiemey Springer New York Berlin H...

Author: Andreas Krause | Melvin Olson

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Statistics and Computing Series Editors: J. Chambers W. Eddy W. Hardle S. Sheather L. Tiemey

Springer New York Berlin Heidelberg Barcelona Hong Kong London Milan Paris Singapore Tokyo

Statistics and Computing Gentle: Numerical Linear Algebra for Applications in Statistics. Gentle: Random Number Generation and Monte Carlo Methods. Hardle/Klinke/Turlach: XploRe: An Interactive Statistical Computing Environment. Krause/Olson: The Basics of S and S-PLUS, 2nd Edition. Lange: Numerical Analysis for Statisticians. Loader: Local Regression and Likelihood. d Ruanaidh/Fitzgerald: Numerical Bayesian Methods Applied to Signal Processing. Pannatier: VARIOWIN: Software for Spatial Data Analysis in 2D. Pinheiro/Bates: Mixed=Effects Models in S and S-PLUS. Venables/Ripley: Modem Applied Statistics with S-PLUS, 3rd Edition. Venables/Ripley: S Programming. Wilkinson: The Grammar of Graphics.

Andreas Krause Melvin Olson

The Basics of S and S-PLUS Second Edition

With 94 Illustrations

Springer

Andreas Krause Novartis Pharma AG 4002 Basel Switzerland [email protected]

Melvin Olson Allergan, Inc. (eff. 5/2000) 2525 Dupont Drive P.O. Box 19534 Irvine, CA 92623-9534 USA olson_mel vin @ allergan.com

Series Editors: J. Chambers Bell Labs, Lucent Technologies 600 Mountain Ave. Murray Hill, NJ 07974 USA

W, Eddy Department of Statistics Carnegie Mellon University Pittsburgh, PA 15213 USA

S. Sheather Australian Graduate School of Management University of New South Wales Sydney, NSW 2052 Australia

L. Tiemey School of Statistics University of Minnesota Vincent Hall Minneapolis, MN 55455 USA

W. Hiirdle Institut ftir Statistik und Okonometrie Humboldt-Universitat zu Berlin Spandauer Str. 1 D-10178 Berlin Germany

Library of Congress Cataloging-in-Publication Data Krause, Andreas. The basics of S and S-PLUS / Andreas Krause, Melvin Olson.—2nd ed. p. cm. — (Statistics and Computing) Includes bibliographical references and index, ISBN 0-387-98961-7 (.softcover) 1. S-PLUS. 2. Mathematical statistics—Data processing. I. Olson, Melvin. II. Title. III. Series. QA276.4.K73 2000 99-056074 519.5'0285'53—dc21

© 2000, 1997 Springer-Verlag New York, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as under.stood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.

ISBN 0-387-98961-7 Springer-Verlag New York Beriin Heidelberg

SPIN 10750178

Preface to the Second Edition

Since the release of the first edition 2 years ago, S and S-Plus have undergone major changes. The S Language, upon which S-Plus is based, has been completely redesigned and rewritten, new functionality has been added, sometimes replacing the old. This is reflected by changes to the book in the appropriate places and a new chapter on object-oriented programming. S-Plus for Windows has experienced the most visible changes by means of a new and extended graphical user interface (GUI), and a split into the standard version (GUI-only use without command line) and the professional version. The extended possibilities of the GUI are described in a greatly expanded chapter on the interface. The Trellis approach to graphics, which involves conditioning on factors of interest, has become the de facto standard in S-Plus. This can be seen by a smaller chapter on standard graphics and a new chapter on Trellis graphics. This edition of the book is based on S-Plus 2000 for Windows and S-Plus 5.1 for UNIX. Users of previous versions might encounter differences in a few places due to differences between releases. Where necessary, differences between S-Plus for Windows and for UNIX are indicated. Finally, valuable comments and suggestions were provided by many people. We would like to thank all of them, especially J¨ urgen Bock (Basel), Floril`ene Dupont (Geneva), Christian Keller (Basel), Brian Ripley (Oxford), Anthony Rossini (Seattle), and Rodney Wong (Davis). We would also like to thank MathSoft DAPD Seattle for their support and provision of early versions of S-Plus. Basel and Freiburg September 1999

Andreas Krause and Melvin Olson

Preface to the First Edition

S-Plus is a powerful tool for interactive data examination, creation of graphs, and implementation of customized routines based on the S language of Lucent Technologies (formerly AT&T Bell Labs). The modern concept together with its flexibility makes it appealing to data analysts from all scientific fields. Among its many strengths are its powerful but flexible graphics routines, and its facility for writing functions whereby you may either modify an existing function or create one of your own. Most standard statistical techniques and many nonstandard ones besides are already programmed in S-Plus, making exploratory and formal statistical analysis very easy. Our book was written to give a solid but quick introduction to the SPlus environment. If you have never used S-Plus before, this book will get you up to speed quickly, as it is full of examples and insightful remarks on how S-Plus works. Nevertheless, we start from first principles and make few assumptions about prior knowledge of computing and statistics. Although our book is primarily intended for the S-Plus novice, the material covered extends to more advanced topics and contains many hints beneficial to those people who already have some knowledge of S-Plus or are not aware of the many enhancements that have been made to the system over the past several years. The structure of our book, with its detailed exercises and solutions at the end of each chapter, reflects its origin from lecture notes and homework exercises. This is an obvious way to use the book in the future. However, the exercises were kept as an integral part of the book to encourage active thinking on the part of the reader, as opposed to the more passive learn-

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Preface to the First Edition

ing obtained by merely reading it. Comparison of your solutions to ours is strongly encouraged, as there is no single or correct approach to most problems. The best way to use the book is interactively; that is, you should read the book while seated at your computer and repeat all of the examples as you read them. For getting an overview, reading all the text material should take no more than a day or two, while working through the examples will take some additional days. If used as a basis for lectures, typically one chapter forms the material for a 90-minute lecture, and the exercises repeat and reinforce the information contained in that chapter. Using this book to lay the foundation, the reader is thus enabled to use S-Plus with all its basic capabilities. With the groundwork in place, it is then possible to tackle the really sophisticated functions and to write your own. If a specific problem is not covered in the book, you should be able to find the answer in one of the voluminous manuals on S-Plus. This book has undergone many revisions and reflects not only our own experiences, but also those of the many people who influenced it by giving comments and proposing alternative ways to approach the subject. Students of the Statistics Department at the University of Dortmund, as well as students of the Economics Department of the University of Basel, were most helpful. Many more people influenced the various stages of this book and contributed directly or indirectly. They are far too numerous to mention. We would like to thank all of them. Special thanks go to Axel Benner (Heidelberg), Tom Cook (Wisconsin), Peter Hartmann (St. Gallen), Thomas T. K¨ otter (Berlin), and Tony Rossini (South Carolina). We also owe a big thanks to the S-Plus developers and distributors. Doug Martin, Brian Clark, and Charlie Roosen at the Data Analysis Products Division of MathSoft, Inc. (formerly Statistical Sciences) in Seattle, and Reinhard Sy at GraS (Berlin) provided generous support and an early beta version of the latest S-Plus release. The editors at Springer-Verlag (New York), John Kimmel and Martin Gilchrist, provided very helpful comments during a continuous good cooperation. And, last but not least, it is important to mention that this book would not have been possible without the tolerance and support of our wives, Michaela Jahn and Sabine Olson. We dedicate this book to them.

Basel and Geneva May 1997

Andreas Krause and Melvin Olson

Contents

Preface to the Preface to the Figures . . . Tables . . . .

Second Edition First Edition . . . . . . . . . . . . . . . . . . .

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1 Introduction 1.1 The History of S and S-Plus . . . . . . . . . . . . . . . 1.2 S-Plus on Different Operating Systems . . . . . . . . . 1.3 Notational Conventions . . . . . . . . . . . . . . . . . . .

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2 Windows User Interface 2.1 Introduction . . . . . . . . . . . . 2.2 System Overview . . . . . . . . . 2.2.1 Using a Mouse . . . . . . . 2.2.2 Object Explorer . . . . . . 2.2.3 Commands Window . . . 2.2.4 Toolbars . . . . . . . . . . 2.2.5 Graph Sheets . . . . . . . 2.2.6 Script Window . . . . . . 2.3 Getting Started with the Interface 2.3.1 Importing Data . . . . . . 2.3.2 Graphs . . . . . . . . . . . 2.3.3 Data and Statistics . . . . 2.3.4 Customizing the Toolbars 2.3.5 Workspaces . . . . . . . .

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2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19

Detailed Use of the Windows Interface Object Explorer . . . . . . . . . . . . . Help . . . . . . . . . . . . . . . . . . . Data Export . . . . . . . . . . . . . . . Working Directory . . . . . . . . . . . Data Import . . . . . . . . . . . . . . . Data Summaries . . . . . . . . . . . . . Graphs . . . . . . . . . . . . . . . . . . Trellis Graphs . . . . . . . . . . . . . . Linear Regression . . . . . . . . . . . . Powerpoint . . . . . . . . . . . . . . . . Excel . . . . . . . . . . . . . . . . . . . Script Window . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . Solutions . . . . . . . . . . . . . . . . .

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3 A First Session 3.1 General Information . . . . . . . . . . . 3.1.1 Starting and Quitting . . . . . . 3.1.2 The Help System . . . . . . . . 3.1.3 Before Beginning . . . . . . . . 3.2 Simple Structures . . . . . . . . . . . . 3.2.1 Arithmetic Operators . . . . . . 3.2.2 Assignments . . . . . . . . . . . 3.2.3 The Concatenate Command: c . 3.2.4 The Sequence Command: seq . 3.2.5 The Replicate Command: rep . 3.3 Mathematical Operations . . . . . . . 3.4 Use of Brackets . . . . . . . . . . . . . 3.5 Logical Values . . . . . . . . . . . . . . 3.6 Review . . . . . . . . . . . . . . . . . . 3.7 Exercises . . . . . . . . . . . . . . . . . 3.8 Solutions . . . . . . . . . . . . . . . . .

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4 A Second Session 4.1 Constructing and Manipulating Data 4.1.1 Matrices . . . . . . . . . . . . 4.1.2 Arrays . . . . . . . . . . . . . 4.1.3 Data Frames . . . . . . . . . . 4.1.4 Lists . . . . . . . . . . . . . . 4.2 Introduction to Functions . . . . . . 4.3 Introduction to Missing Values . . . . 4.4 Merging Data . . . . . . . . . . . . . 4.5 Putting It All Together . . . . . . . .

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5 Graphics 5.1 Basic Graphics Commands . . . . . . . . . . . . 5.2 Graphics Devices . . . . . . . . . . . . . . . . . 5.2.1 Working with Multiple Graphics Devices 5.3 Plotting Data . . . . . . . . . . . . . . . . . . . 5.3.1 The plot Command . . . . . . . . . . . 5.3.2 Modifying the Data Display . . . . . . . 5.3.3 Modifying Figure Elements . . . . . . . . 5.4 Adding Elements to Existing Plots . . . . . . . 5.4.1 Functions to Add Elements to Graphs . 5.4.2 More About abline . . . . . . . . . . . 5.4.3 More on Adding Axes . . . . . . . . . . 5.4.4 Adding Text to Graphs . . . . . . . . . . 5.5 Setting Options . . . . . . . . . . . . . . . . . . 5.6 Figure Layouts . . . . . . . . . . . . . . . . . . 5.6.1 Layouts Using Trellis Graphs . . . . . . 5.6.2 Matrices of Graphs . . . . . . . . . . . . 5.6.3 Multiple Screens Graphs . . . . . . . . . 5.6.4 Figures of Specified Size . . . . . . . . . 5.7 Exercises . . . . . . . . . . . . . . . . . . . . . . 5.8 Solutions . . . . . . . . . . . . . . . . . . . . . .

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6 Trellis Graphics 6.1 An Example . . . . . . . . . . . . . . . . . . . . . . 6.2 Trellis Basics . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Trellis Syntax . . . . . . . . . . . . . . . . . 6.2.2 Trellis Functions . . . . . . . . . . . . . . . 6.2.3 Displaying and Storing Graphs . . . . . . . 6.3 Output Devices . . . . . . . . . . . . . . . . . . . . 6.4 Customizing Trellis Graphs . . . . . . . . . . . . . . 6.4.1 Setting Options . . . . . . . . . . . . . . . . 6.4.2 Arranging the Layout of a Trellis Graph . . 6.4.3 Layout . . . . . . . . . . . . . . . . . . . . . 6.4.4 Ordering of Graphs . . . . . . . . . . . . . . 6.4.5 Changing Graph Elements . . . . . . . . . . 6.4.6 Modifying Panel Strips . . . . . . . . . . . . 6.4.7 Arranging Several Graphs on a Single Page 6.4.8 Updating Existing Trellis Graphs . . . . . . 6.4.9 Writing Panel Functions . . . . . . . . . . . 6.5 Further Hints . . . . . . . . . . . . . . . . . . . . . 6.5.1 Preparing Data to Use for Trellis . . . . . . 6.5.2 The subset Option . . . . . . . . . . . . . .

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7 Exploring Data 7.1 Descriptive Data Exploration . . . . . . . . . . . . . . . 7.2 Graphical Exploration . . . . . . . . . . . . . . . . . . . 7.3 Distributions and Related Functions . . . . . . . . . . . 7.4 Confirmatory Statistics and Hypothesis Testing . . . . . 7.5 Missing and Infinite Values . . . . . . . . . . . . . . . . . 7.5.1 Testing for Missing Values . . . . . . . . . . . . . 7.5.2 Supplying Data with Missing Values to Functions 7.5.3 Missing Values in Graphs . . . . . . . . . . . . . 7.5.4 Infinite Values . . . . . . . . . . . . . . . . . . . . 7.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . .

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8 Statistical Modeling 8.1 Introductory Examples . . . . . . . . . 8.1.1 Regression . . . . . . . . . . . . 8.1.2 Regression Diagnostics . . . . . 8.2 Statistical Models . . . . . . . . . . . . 8.3 Model Syntax . . . . . . . . . . . . . . 8.4 Regression . . . . . . . . . . . . . . . . 8.4.1 Linear Regression and Modeling 8.4.2 ANOVA . . . . . . . . . . . . . 8.4.3 Logistic Regression . . . . . . . 8.4.4 Survival Data Analysis . . . . . 8.4.5 Endnote . . . . . . . . . . . . . 8.5 Exercises . . . . . . . . . . . . . . . . . 8.6 Solutions . . . . . . . . . . . . . . . . .

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9 Programming 9.1 Lists . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Adding and Deleting List Elements . . . . . . 9.1.2 Naming List Elements . . . . . . . . . . . . . 9.1.3 Applying the Same Function to List Elements 9.1.4 Unlisting a List . . . . . . . . . . . . . . . . . 9.1.5 Generating a List by Using split . . . . . . . 9.2 Writing Functions . . . . . . . . . . . . . . . . . . . . 9.2.1 Documenting Functions . . . . . . . . . . . . 9.2.2 Scope of Variables . . . . . . . . . . . . . . . . 9.2.3 Parameters and Defaults . . . . . . . . . . .

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Passing an Unspecified Number of Parameters to a Function . . . . . . 9.2.5 Testing for Existence of an Argument . 9.2.6 Returning Warnings and Errors . . . . 9.2.7 Using Function Arguments in Graphics Iteration . . . . . . . . . . . . . . . . . . . . . 9.3.1 The for Loop . . . . . . . . . . . . . . 9.3.2 The while Loop . . . . . . . . . . . . . 9.3.3 The repeat Loop . . . . . . . . . . . . 9.3.4 Vectorizing a Loop . . . . . . . . . . . 9.3.5 Large Loops . . . . . . . . . . . . . . . Debugging: Searching for Errors . . . . . . . . 9.4.1 Syntax Errors . . . . . . . . . . . . . . 9.4.2 Invalid Arguments . . . . . . . . . . . 9.4.3 Execution or Run-Time Errors . . . . . 9.4.4 Logical Errors . . . . . . . . . . . . . . Output Using the cat Function . . . . . . . . The paste Function . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . Solutions . . . . . . . . . . . . . . . . . . . . .

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10 Object-Oriented Programming 10.1 Creating Classes and Objects 10.2 Creating Methods . . . . . . . 10.3 Debugging . . . . . . . . . . . 10.4 Help . . . . . . . . . . . . . . 10.5 Summary and Overview . . . 10.6 Exercises . . . . . . . . . . . . 10.7 Solutions . . . . . . . . . . . .

. . . . . . . . . . . . Labels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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11 Input and Output 11.1 Reading S-Plus Commands from File . 11.2 Data Import/Export: Easiest Method . . 11.2.1 UNIX Implementation . . . . . . 11.2.2 Windows Implementation . . . . 11.3 Data Import/Export: General Method . 11.4 Data Import/Export: Never-Fail Method 11.5 Reading Data from the Terminal . . . . 11.6 Editing Data . . . . . . . . . . . . . . . . 11.7 Transferring Data . . . . . . . . . . . . . 11.8 Recording a Session . . . . . . . . . . . . 11.9 Exercises . . . . . . . . . . . . . . . . . . 11.10 Solutions . . . . . . . . . . . . . . . . . .

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Contents

12 S-Plus Internals 12.1 How S-Plus Works Under UNIX . . . . . . 12.1.1 The Working Chapter . . . . . . . . 12.1.2 Customization on Start-Up and Exit 12.2 How S-Plus Works Under Windows . . . . 12.2.1 Command Line Options . . . . . . . 12.2.2 Start-up and Exit Functions . . . . . 12.3 Storing Mechanism . . . . . . . . . . . . . . 12.4 Levels of Calls . . . . . . . . . . . . . . . . . 12.5 Exercises . . . . . . . . . . . . . . . . . . . . 12.6 Solutions . . . . . . . . . . . . . . . . . . . .

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333 333 334 334 335 336 336 337 338 341 342

13 Tips and Tricks 13.1 The Process of Developing a Function . . . . . . . . . . 13.1.1 Setting up an Editor and Running the Code in Plus . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Useful Techniques . . . . . . . . . . . . . . . . . . . . . 13.2.1 Housekeeping: Cleaning Up Directories . . . . . 13.2.2 Storing and Restoring Graphical Parameters . . 13.2.3 Naming of Variables and Functions . . . . . . . 13.2.4 Repeating Commands . . . . . . . . . . . . . . 13.3 Greek Letters in Graphs . . . . . . . . . . . . . . . . . 13.4 Batch Jobs . . . . . . . . . . . . . . . . . . . . . . . . . 13.5 Incorporating and Accessing C and Fortran Programs . 13.5.1 Creating Shared Object Files Under UNIX . . . 13.5.2 Creating DLLs Under Windows . . . . . . . . . 13.5.3 Calling C Routines . . . . . . . . . . . . . . . . 13.6 Libraries . . . . . . . . . . . . . . . . . . . . . . . . . . 13.7 Including Graphs in Text Processors . . . . . . . . . . 13.7.1 Generating Graphs on the Windows Clipboard 13.7.2 Generating PostScript Graphs . . . . . . . . . . 13.7.3 PostScript Graphs in Windows Applications . . 13.7.4 PostScript Graphs in TEX . . . . . . . . . . . . 13.7.5 If You Don’t Have a PostScript Printer . . . . . 13.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 13.9 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . .

. S. . . . . . . . . . . . . . . . . . . . .

14 Information Sources on and Around S-Plus 14.1 S-News: Exchanging Information with Other 14.2 The StatLib Server . . . . . . . . . . . . . . 14.3 R: A Public Domain Software . . . . . . . . 14.4 What Next? . . . . . . . . . . . . . . . . . .

. . . .

365 365 366 367 367

15 Bibliography 15.1 Print Bibliography . . . . . . . . . . . . . . . . . . . . .

369 369

Users . . . . . . . . . . . .

. . . .

. . . .

345 345 346 347 347 347 348 348 349 350 351 352 352 353 355 356 356 357 358 358 361 362 363

Contents

15.2

Index

On-Line Bibliography . . . . . . 15.2.1 S-Plus Related Sources 15.2.2 TEX-Related Sources . . 15.2.3 Other Sources . . . . . .

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xv

371 371 372 372 373

Figures

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19 2.20 2.21 2.22 2.23 2.24

The S-Plus screen and its components The S-Plus main toolbar . . . . . . . . The 2D graph palette . . . . . . . . . . The Object Explorer toolbar . . . . . . Filtering data sets for Object Explorer . List of data sets in Object Explorer . . . Details of the Puromycin data set . . . . Data export . . . . . . . . . . . . . . . . Defining a working directory . . . . . . . Data window from Importing Data . . . Deleting a data column . . . . . . . . . Summary Statistics data dialog . . . . . Summary Statistics dialog . . . . . . . . Summary Statistics Report window . . . 2D Plot dialog (symbol) . . . . . . . . . Scatterplot of Puromycin data . . . . . Axis dialog box . . . . . . . . . . . . . . Axis font dialog box . . . . . . . . . . . Annotated scatterplot . . . . . . . . . . Saving a Graph Sheet . . . . . . . . . . Trellis Graphs – Drag and Drop . . . . . A Trellis graph conditioning on State . . Linear Regression dialog . . . . . . . . . Regression formula dialog . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

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. . . . . . . . . . . . . . . . . . . . . . . .

8 10 12 17 18 19 20 22 24 25 26 27 28 29 31 32 32 33 34 35 37 38 39 40

xviii

Figures

2.25 2.26 2.27 2.28 2.29 2.30 2.31 2.32 2.33 2.34 2.35 2.36 2.37 2.38 2.39 2.40 2.41 2.42 2.43 2.44 2.45

Regression Report window . . . . . . . . . Combining graphs onto one Graph Sheet . The Script window . . . . . . . . . . . . . Script window toolbar . . . . . . . . . . . Inserting a column of data . . . . . . . . . Loess smoother curves . . . . . . . . . . . Variable transformation for regression . . Regression output using log(CONC) . . . Residual plots using log(CONC) . . . . . Regression formula using STATE . . . . . Regression output using STATE . . . . . Regression formula using interaction term Regression output using interaction term . Available data objects . . . . . . . . . . . Database filter . . . . . . . . . . . . . . . Object Explorer . . . . . . . . . . . . . . . Plot Symbol dialog . . . . . . . . . . . . . Geyser scatterplot . . . . . . . . . . . . . The Annotate palette . . . . . . . . . . . Plot Line dialog . . . . . . . . . . . . . . . Geyser scatterplot with annotation . . . .

. . . . . . . . . . . . . . . . . . . . .

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41 43 46 47 50 51 52 53 54 55 55 56 56 57 58 59 60 61 62 62 63

5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11

Different options for the type parameter of the plot function 118 An example of a graph layout . . . . . . . . . . . . . . . . 120 A customized graph . . . . . . . . . . . . . . . . . . . . . 121 The S-Plus definition of figure and plot regions . . . . . 126 A customized graph layout . . . . . . . . . . . . . . . . . 130 A customized display of the Geyser data set . . . . . . . . 131 Trigonometric functions . . . . . . . . . . . . . . . . . . . 133 Geometric figures (a): a rectangle. . . . . . . . . . . . . . 135 Geometric figures (b): A circle . . . . . . . . . . . . . . . 136 Geometric figures (c): A spiral . . . . . . . . . . . . . . . 137 Lissajous Figures . . . . . . . . . . . . . . . . . . . . . . . 138

6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10

An example of a Trellis graph layout . . . . . . . . . . . . Heights of singers for voice parts . . . . . . . . . . . . . . Customizing a Trellis graph arrangement . . . . . . . . . . Graph ordering in a Trellis arrangement . . . . . . . . . . A multigraph page of Trellis displays . . . . . . . . . . . . A multigraph page of Trellis displays with different size . Trellis display with customized panel function . . . . . . . Trellis display of data spread, mean and standard deviations Trellis histograms with added key of derived statistics. . . Trellis display of data spread, mean, and standard deviations (2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

140 141 148 149 151 152 155 160 163 166

Figures

xix

6.11 Trellis display of data spread, mean, and standard deviations (3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

167

7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11 7.12 7.13 7.14 7.15 7.16 7.17

Histogram of the variable yield in the Barley data set . . Histogram of the barley yield by site . . . . . . . . . . . . The barley yield by site and year . . . . . . . . . . . . . Boxplot display of the barley yield for all sites, by year . . Boxplot display of the barley yield by site and year . . . . The Barley data in boxplot displays, categorized by site . The Old Faithful Geyser data . . . . . . . . . . . . . . . . The Geyser data in subsets . . . . . . . . . . . . . . . . . A perspective plot of the Geyser data . . . . . . . . . . . An image plot of the Geyser data . . . . . . . . . . . . . . A contour plot of the Geyser data . . . . . . . . . . . . . A graphical comparison of three distributions . . . . . . . A graph of the Binomial distribution . . . . . . . . . . . . A graphical display of the t-test . . . . . . . . . . . . . . . Different histogram displays of Normally distributed data The car data set . . . . . . . . . . . . . . . . . . . . . . . Estimating pi using a simulation technique . . . . . . . .

183 184 185 186 187 188 190 191 194 195 196 200 201 203 213 217 222

8.1 8.2 8.3 8.4

Regression diagnostic plots . . . . . . . Typical regression diagnostic plots . . . GAM regression diagnostic plots . . . . Design and interaction plots for ANOVA

. . . .

226 250 251 255

9.1

Two-dimensional functions . . . . . . . . . . . . . . . . . .

290

10.1 Country data graph for a single country. . . . . . . . . . . 10.2 Country data graph for two countries . . . . . . . . . . . . 10.3 Country data graph for four countries . . . . . . . . . . .

301 302 306

11.1 Capital development with a given interest rate . . . . . .

332

13.1 The “play” button to run S-Plus scripts . . . . . . . . .

346

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. . . .

Tables

1.1 1.2

Systems supported by S-Plus . . . . . . . . . . . . . . . . Notational conventions . . . . . . . . . . . . . . . . . . . .

4 5

2.1

File types available with the import facility . . . . . . . .

11

3.1 3.2 3.3 3.4

Arithmetic operations on vectors Different brackets in S-Plus . . Symbols for logical operators . . Truncation functions . . . . . . .

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75 76 78 86

5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10

The most common output devices . . . Commands for graphics devices . . . . . Options for type in the plot command Options for plotting functions . . . . . . Commands that add to existing graphs . Some examples for the use of abline . . Optional parameters for plotting axes . Optional parameters for plotting axes . Optional parameters for adding text . . Layout parameters . . . . . . . . . . . .

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115 117 118 120 122 123 124 124 125 126

6.1 6.2 6.3

Trellis functions . . . . . . . . . . . . . . . . . . . . . . . . Trellis graphics devices . . . . . . . . . . . . . . . . . . . . Trellis plot options . . . . . . . . . . . . . . . . . . . . . .

143 145 147

. . . .

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xxii

Tables

6.4

Trellis panel functions . . . . . . . . . . . . . . . . . . . .

7.1 7.2 7.3 7.4 7.5 7.6 7.7

153

Descriptive statistics functions . . . . . . . . . . . . . . . Tabulation and split functions . . . . . . . . . . . . . . . . Graphical data exploration functions . . . . . . . . . . . . Multivariate graphical data exploration functions . . . . . Multivariate dynamic graphics functions . . . . . . . . . . Categorization of distribution-related functions in S-Plus Distribution-related functions in S-Plus(1): Discrete distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8 Distribution-related functions in S-Plus(2): Continuous distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 7.9 Statistical tests in S-Plus . . . . . . . . . . . . . . . . . . 7.10 Estimating pi using a simulation technique . . . . . . . .

176 177 182 192 195 197

8.1 8.2 8.3 8.4 8.5

Swiss fertility data set . . Selected statistical models Model syntax . . . . . . . Kyphosis data frame . . . Leukemia data frame . . .

. . . . .

224 228 229 235 238

9.1 9.2

Some commands in browser, debugger, and recover . . Control sequences in S-Plus . . . . . . . . . . . . . . . .

285 286

11.1 File types and UNIX . . . . 11.2 File types and Windows . .

. . . . . . in S-Plus . . . . . . . . . . . . . . . . . .

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extensions with the import/export . . . . . . . . . . . . . . . . . . . . extensions with the import/export . . . . . . . . . . . . . . . . . . . .

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facility . . . . facility . . . .

in . in .

13.1 S-Plus types and C/Fortran equivalents . . . . . . . . . .

197 198 204 222

321 322 353

1 Introduction

Over the years, the S language and S-Plus have undergone many changes. Since its development in the mid-seventies, the three main authors of S, Rick Becker, John Chambers, and Allan Wilks, have enhanced the entire language considerably. All their work was done at Bell Labs with the original goal of defining a language to make it easier to do repetitive tasks in data analysis, like calculating a linear model. In the following years, many people contributed to the S project in one form or another. People outside Bell Labs also became aware of the interesting development and took part in it, and this is to a great extent the way S and S-Plus are still developed today. A very lively user community works with and on S/S-Plus, and they especially appreciate the S style of working in an integrated environment. Special strengths are the extremely modern and flexible language, which has many elements of an interactive C, Lisp, and APL, and good graphics capabilities. It is noteworthy that the authors do not consider S as a primarily statistical system. The system was developed to be flexible and interactive, especially designed for doing data analysis of an exploratory nature, which began to boom in the late seventies after the release of Tukey’s book (1977) on the subject. Most of the statistical functionality was added later, and many statistics routines like estimation, regression, and testing were incorporated by the S-Plus team. This chapter describes the development of S and S-Plus over the years, clarifies the differences between the two, and points to some further references.

2

1. Introduction

1.1 The History of S and S-Plus The S Language, and some years later the New S, were developed at AT&T Bell Labs in the late seventies and early eighties, mainly by Rick Becker and John Chambers. Some years later, Allan Wilks joined the core team. Since then, several other people have been involved in the project. Becker (1994) describes in great detail the foundation and development of S and points out some future directions. The year 1976 can perhaps be viewed as the founding year of S. First concepts were developed and implemented. At first, the “system” consisted of a library of routines together with an interface, such that the kernel code itself could be kept unmodified. In 1981, the S team decided to rewrite the system in C and port it to the UNIX operating system. Since 1981, the source code has been available for interested people outside Bell Labs. The next years revealed a strong increasing interest among statisticians and data analysts in using the system, which was called S by then. It is remarkable that the important steps in the development of S were all marked by books, such that S users today talk about the days of the Brown Book, the Blue Book, and the White Book. The Green Book marks the most recent milestone. In 1984, as the interest in S began to rise, a real manual was necessary. The first book, today referred to as the Brown Book, was written by Becker and Chambers (1984). This version of S is now referred to as “Old S,” as no version numbers existed at the time. The QPE (Quantitative Programming Environment) developed by John Chambers set a milestone in the development of S. In 1988, it introduced the function concept (replacing the former macros), and new programming concepts were added. This work is described in the Blue Book (Becker, Chambers, and Wilks, 1988). During all these years, the user community added substantial functionality to S, and many sophisticated techniques like tree regression, nonparametric smoothing, survival analysis, object-oriented programming, and new model formulation became a part of S. This step in the development was manifested and accompanied by the White Book (Chambers and Hastie, 1992). Version 4 of S came out in 1998 and is described in full detail in Chambers (1998), the Green Book. The object-oriented paradigm forms the basis of the entire language, documentation is integrated into an object, and the general paradigm “everything is an object” is followed throughout the language. In 1987, Douglas Martin at the University of Washington, Seattle, founded a small company to make S more popular. He realized that the major drawback of S was the need of professional support for end-users. Hence, he started the company Statistical Sciences, Inc. (StatSci), a division of MathSoft, Inc., since 1994. StatSci added more functionality to S, ported

1.2. S-Plus on Different Operating Systems

3

it to other hardware platforms, and provided the necessary support for technical and statistical problems. The enhanced version of S received a new name: S-Plus. S-Plus helped popularize S among nontechnical people. StatSci ported S-Plus to the only non-UNIX platform, releasing S-Plus for DOS in 1989 and S-Plus for Windows in 1993. Up to version 3, S-Plus for DOS/Windows and UNIX provided essentially the same functionality. S-Plus Version 4 was only released for the Windows platform. It was enhanced by a new graphical user interface (GUI) and a graphics system based on the Axum package. It adopts the Windows standard and gives a menu-based interface to S-Plus. Windows-specific functionality like the creation of PowerPoint slides or direct data exchange with other programs via DLLs was added. Much of the S-Plus functionality is available via menus and buttons, and the graphics are shown in an editable graph sheet. A major strength is that all functionality that is accessible via the menus can also be called from the command line. S-Plus comes with a script window where the corresponding command line input is shown. In fall 1998, S-Plus for Windows was released in version 4.5, which is split into a “Professional Edition” that comprises the full functionality and a menu-only version called “S-Plus Standard.” Version 5 of S-Plus was only released on UNIX systems in late 1998 and 1999. It is based on S Version 4 as described in Chambers (1998). For the first time, the Linux system for Intel PC is supported. S-Plus 2000 for Windows systems (analogous to Office 2000 by Microsoft) was released in mid-1999, still based on S Version 3. The next release for Windows will (probably) be based on S Version 4. S is still the heart of the system, and the core S team continues to work on the S system. The whole S functionality is incorporated in S-Plus and enhanced, and today the S system is no longer publically available. In the remainder of the book, we will use S-Plus as the standard reference.

1.2 S-Plus on Different Operating Systems Sometimes it is important to know about differences in software on various hardware systems or under different operating systems. This can be the case if you work on more than one computer system with S-Plus (and therefore need to exchange data files) or if you want to be informed about the differences before deciding in favor of a specific system. In this section, we discuss some details about different systems supported by S-Plus. In addition, the chapter provides some basic information about the general setup of files and structures in S-Plus. More information on the S-Plus internal workings can be found in Sections 12.1 and 12.2.

4

1. Introduction

At present, S-Plus supports two major operating systems: UNIX (with most of its variants) and Windows (with its variants). Table 1.1 summarizes the currently supported hardware and operating systems.

Table 1.1. Systems supported by S-Plus

AIX 4.3.1 Intel-based Linux Windows 95, 98, NT, 2000

DEC UNIX 4.0 IRIX 6.2, 6.5

HPUX 10.2, 11.0 Solaris 2.5, 2.6 (SPARC)

As the S source code is no longer available, machines not binary compatible to the ones supported are not able to run S or S-Plus. In those cases, the R system offers an alternative. We will get to the details in Section 14.3 (page 367). S-Plus has minimum requirement specifications regarding main memory and hard disk size. The disk space needed depends on the version and operating system, but is typically between 50 and 100 Megabytes. As a side note, S-Plus consumes and releases memory dynamically during a session, depending on the needs. Therefore, it does not run out of memory until there is no more main memory and swap space available. If S-Plus runs out of main memory (RAM), the operating system assigns virtual memory (i.e., hard disk space) as a substitute. As this slows down the execution time dramatically, the machine should be equipped with a reasonable amount of memory. For improving performance, main memory is the first speedup factor. If you are not satisfied with the performance, watch for permanent hard disk access while executing commands, or use a monitoring tool (like “top” under UNIX or the “Task Manager” under Windows) to track swapping activity.

Differences Between Versions S-Plus has some differences in its implementation between the UNIX and the Windows versions. The most visible difference is the user interface. SPlus for Windows has a typical Windows-like menu-based user interface. Many graphical functions can be started using the toolbar interface. Graphics are editable using point and click. Data and graphics transfer from and to other Windows applications (like Excel and Powerpoint) is part of the system. S-Plus for UNIX allows easy integration of C and Fortran code. S-Plus for Windows allows one to dynamically link libraries (DLLs). S-Plus for Windows comes in two editions: “Professional” and “Standard.” The standard version does not provide command line access.

1.3. Notational Conventions

5

1.3 Notational Conventions By now you must be eager to get started, but it might be worth reviewing these few notational conventions first, as they are used throughout the book. To begin with, you must be aware that when running S-Plus, you will be asked for a new command with the greater than sign: >. We use the > in the book to denote the S-Plus prompt. Also, if a single command extends over one line of input, the prompt changes to the plus sign: +. A preview of this is shown below. > This is where an S-Plus command would appear and + notice that the prompt changes on the second line. If a longer listing or function is shown, the prompt is typically omitted, as we do not assume interactive input, and reading the code is easier. There are occasional examples of commands to either the UNIX or DOS shells. For these examples, no prompt is used. All commands in S-Plus are actually calls to functions. To highlight the occurrence of a function in S-Plus, or one of the parameters to it, we have written them in a special font as with the example, print. When presenting commands, we sometimes include descriptive text. The descriptive text is written in S-Plus syntax for comments. Anything after the number sign # until the end of the line is treated as a comment and not interpreted as a command. A summary of these conventions is found in Table 1.2. Table 1.2. Notational conventions

Convention

Explanation

> + Commands No prompt # placeholders

S-Plus prompt Command has continued onto next line Typewriter font For calls to UNIX or DOS shell Comment symbol indicating start of a comment Italic. You need to replace them by an appropriate expression, like filename, which needs to be replaced by a valid file name Menu entries and buttons are referred to in this font Notes point out something important, like a practical example, an application, or an exception. They end with the symbol ⊳

Menu Note

2 Windows User Interface

2.1 Introduction It has been fashionable lately for statistical software packages to become interactive and/or “point-and-click.” S-Plus has always been interactive but has only recently added a Windows-based point-and-click interface. The Windows user now has the best of both worlds available, the ease of the graphical user interfact (GUI) combined with the detailed commands and control offered by the command line environment. The advantage of the GUI for the novice of S-Plus is that you don’t have to know the syntax of S-Plus to get started. All you need is a little familiarity with typical Windows software and a data set in some sort of standard format. The point-and-click approach to using S-Plus is not available on UNIX platforms. With these platforms, the Commands Window is the system which does not rely on any menus or toolbars. As the complete functionality of S-Plus is only available through the Commands Window anyway (or its counterpart, the Script Window), using S-Plus on these platforms represents no loss of performance, although perhaps a small sacrifice of ease of use. Users of UNIX or Linux should skip to the next chapter which is where the more detailed treatment of the Commands Window begins. The approach we will take in this chapter is to quickly introduce the S-Plus system design under Windows, show the briefest of explanations of how it functions, and finish with describing in detail the various tasks that will be needed to complete a data analysis, from data input to printing

8

2. Windows User Interface

and saving the results. It is by no means intended to be an extensive or exhaustive exploration of the GUI, merely a way of familiarizing you with its structure, where to find things, and, most importantly, where and what to try for more detailed options.

2.2 System Overview When you open S-Plus under Windows by double-clicking on the S-Plus icon, you are greeted by a screen layout as shown in Figure 2.1. This may vary slightly according to the version of S-Plus you are using. The main elements that are visible include the Object Explorer, the Commands Window, the menus, and the toolbar. Optionally, a graphics window can be opened. A short description of each of these components is given in the next several subsections.1

Figure 2.1. The S-Plus screen and its components, the Object Explorer, the Commands Window, and the toolbars.

1 According

present.

to the version of S-Plus being used, some components might not be

2.2. System Overview

9

The general layout of the S-Plus system is similar to many popular windows systems in that it has pull-down menus at the top and toolbars just below the menus. To use such a system it is useful to be a little familiar with basic point-and-click operations and how to use a mouse. For those who are not that comfortable with window- and icon-based software, the following subsections provide a crash introduction to the essentials. The pull-down menus across the top are used to group categories of commands or options to be set. Under the File menu, we find actions relating to files (Open, Close, Import, Save) as well as to exiting the system. The toolbars below the menus contain buttons that are convenient shortcuts to commands found through the layers in the menus. Some of the toolbar buttons (e.g., plots2d) open a palette containing a myriad of options to complete your task. An online help facility is available through the main menu. The typical approach to handling the Help system (i.e., search by index or topic) has been used. The big advantage of the Help system in S-Plus is that it includes online manuals. We encourage liberal use of the help system.

2.2.1 Using a Mouse Using a mouse efficiently is important to get the most out of the system. Clicking once on the left mouse button is usually used to highlight an item in a list (e.g., a file out of a list of files) or to select a menu heading or button from a toolbar. You will not always be able to guess at the function of a toolbar button merely by looking at the icon, but if you are at a loss, simply position the mouse over the button in question and a short text description will appear below it. Double-clicking on the left mouse button is used to select and execute. Examples include double-clicking on a file name to select it and start the function, or on a part of a graph to select and edit it. After having selected an item by clicking once with the left mouse button, the right mouse button opens a context menu, which changes depending on the item selected.

2.2.2 Object Explorer The Object Explorer is used to get an overview of what is available on the system, including data, functions, and graphs. It operates in much the same way as the Windows Explorer in that there is a tree-like structure in the left pane and the details are displayed in the right pane. The Object Explorer can be opened either from the menu or from a toolbar button. Not only can it show what objects exist, but with data, for example, the Object Explorer is used to view (browse) it and even edit it. Double-click on the data you want to view or edit and a spreadsheet will open containing the selected data. Once the spreadsheet is open, the data can be edited.

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2.2.3 Commands Window The Commands Window is actually the heart of the S-Plus system. Every command that is performed via menus and buttons can be issued as a command directly from the prompt in the Commands Window. In addition, there are many functions that can only be run in the Commands Window. As examples, the functions of programming, personalized functions, data subsets, and logical operations are only available, or more extensively available, through the use of the Commands Window.

2.2.4 Toolbars The main toolbar contains many familiar commands, including Open, Save, Print, Copy, Paste, and Undo and is shown in Figure 2.2. In S-Plus, however, one may also open the Commands Window and Object Explorer, open a new Graph Sheet, open the History Log, create 2D and 3D graphs, and so forth.

Figure 2.2. The S-Plus main toolbar.

An additional feature of the toolbars in S-Plus is that they are contextsensitive (smart). Open a Graph Sheet, for example, and extra buttons will appear in the toolbar area that are specific to Graph Sheets.

2.2.5 Graph Sheets Graphs are drawn in windows referred to as Graph Sheets. Starting with Version 4.0, components of graphs can be edited and redefined by doubleclicking on the component of interest. Labels can be changed, colors modified, axes redefined, and more, all through menus and dialog boxes available simply by clicking in the Graph Sheet. The Insert menu is useful for adding features and components to graphs which already exist in a Graph Sheet, and the Format menu is useful for changing the design of many components of an existing graph.

2.2.6 Script Window The Script Window also consists of two panes, one on top and one on bottom. The top pane can be used as a development space in which to create or fine-tune a section of code. When the commands in the top pane are run, the output from them appears in the bottom pane. In function, the Script Window is similar to the Commands Window, the difference

2.3. Getting Started with the Interface

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being that the former runs commands in segments, whereas the latter is completely interactive and executes commands one at a time as they are input. More details of the layout of the Script Window and how it operates are provided in Section 2.16.

2.3 Getting Started with the Interface If you want to use S-Plus having had no introduction, use the menus and toolbar buttons. For the most part, you should find them to be selfexplanatory. However, we have described a few key functions in a bit more detail to get you on your way.

2.3.1 Importing Data You probably have your own data that you want to analyze, so the first thing you have to know is how to import it into S-Plus. There is an Import Data facility located in the File menu. In the Import Data dialog, you will be asked for the name you want for your data (which may be more than eight characters) and have the usual Windows boxes for specifying the name and location of your data file. Pay careful attention to the type of data (Files of Type pull-down menu) that you have and properly define it in the corresponding box. The data file types available are shown in Table 2.1. Table 2.1. File types available with the import facility

File ASCII FoxPro MS Access SAS Transport SPSS Export

Format ASCII formatted dBase Gauss Lotus 1-2-3 Paradox QuattroPro SigmaPlot S-Plus STATA Systat

Excel MatLab SAS SPSS

When the spreadsheet containing newly imported data is closed, the new data automatically appear as a new entry in the Object Explorer.

2.3.2 Graphs Graphs are created by connecting the data shown in the Object Explorer to a graphical function represented by a button in a palette. In the Object Explorer, the data are displayed with its full name, and after a click on the data object’s name, the elements of the selected data set are displayed in the right-hand pane of the window. By clicking on these elements, the

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set of variables to be plotted can be selected. To select a second and third variable, hold down the key and click on the element. Before actually displaying the data graphically, the graph palettes need to be opened, if they are not open yet. Open the 2D or 3D graph palette by using the appropriate icon in the toolbar. The 2D graph palette appears in Figure 2.3 and shows all of the types of 2D graphs available.

Figure 2.3. The 2D graph palette.

Once the palette is open, you might want to move the mouse over the different icons slowly. If the mouse stops for a second, the name of the method represented by the button is displayed. Select the plot type by clicking on it. S-Plus executes the graphical method right away by using the selected data. Note For creating a graph, the order of selection of variables is important. The variable selected first becomes the x-variable, the second variable selected becomes the y-variable, and if a third variable is selected, it becomes the z-variable. ⊳ Once data have been selected, another graphical method can be applied by simply clicking on a different graph button in the palette. Note If you click on a graphics button and a graph is set up but not drawn, this probably means that the data selected and the graph type


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chosen are not compatible. Selecting two variables and clicking on a 3D graphics method is an example of such a situation. ⊳ Once a basic graph has been created, it can be edited. Double-click on the element to modify, for example, the axis. A window opens and offers the possibility of modifying all the components like range, tickmarks, color, width, label, and much more. Using the annotate palette that appears once a Graph Sheet is open, further elements such as text or graphics symbols like rectangles and arrows can be inserted into the graph. A Graph Sheet may be printed using File - Print Graph Sheet... or saved using File - Save. All that is left is to specify the destination file name and directory.

2.3.3 Data and Statistics Data can be summarized by using graphical display techniques, but also by statistics like the mean, median, minimum, and maximum values. There are Data and Statistics menus in the top menu bar that contain a large set of routines for manipulating and transforming a data set (Data) and routines to process the data (Statistics). First, select the statistical method to use, then specify the data set (or vice versa) and the options you want in the menu fields. The method is applied to the data by clicking on the OK button and the results are displayed in a Report Window. In this way, data can be summarized by looking at minimum and maximum values, at quantiles, or at the correlation between variables. Data can be processed by carrying out a t-test, a regression model, or any other method, and tick boxes can be ticked or not to choose options like paired or unpaired or the degree of a polynomial to fit.

2.3.4 Customizing the Toolbars The toolbars and palettes can be customized to some extent. It is possible to select which palettes show up on the screen. Select View from the top menu and click on Toolbars. A window opens which lets you choose which toolbars should be shown. Clicking on any of the selection fields immediately pops up or removes the corresponding palette on the desktop. These settings are stored and reused when S-Plus is started again. Be careful which ones are chosen, remember that the Graph toolbar will open when a graph is created, and so forth. The shapes of the palettes might not be satisfactory, depending on your screen resolution, personal preferences, and more. Clicking on one of the four edges allows you to resize the palettes, which is automatically set such that all icons fit.

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Furthermore, the palette buttons can be displayed in palette form or in toolbar form. Click on the palette and drag it to the top where the primary toolbar is located. Drop it and the palette becomes a horizontal toolbar itself. Dragging the bar away from the top recreates a palette. Adding New Palettes and Buttons The palettes and menu buttons are open for extension. You can add menu buttons to an existing palette and create new palettes. You can generate new toolbars via the View menu by selecting the Toolbars entry and clicking on New. This produces a toolbar dialog box that can be used to create a new toolbar. You can add a new button by selecting the desired toolbar and opening the context menu with the right mouse button. The entry New Button opens up a window where you can specify the name of the button, the text to show when the mouse is over the button, and the S-Plus function to carry out. (You will learn more about functions as you proceed.) You can modify an existing button by selecting the button and opening its context menu (right mouse click). A property dialog displays the current settings.

2.3.5 Workspaces S-Plus gives you the ability to save files from different projects into different areas called workspaces. Using this feature, it is then very easy to switch between the workspaces for your various projects. All elements open during a session, such as Object Explorer and graph sheet, can be saved to a workspace such that on reopening the workspace, everything is as you left it. Since S-Plus keeps all data items ever created, we highly recommend the use of workspaces as an easy way of organizing your work and switching between project files. The actual operation of workspaces is quite straightforward. Under File-Workspaces you have the options of New, Open, and Save. The available workspaces are listed beneath these options and can be clicked directly as opposed to using the Open option. Create a new workspace by choosing the New option, and specifying the directory where it should be located. There is a box to specify Databases to Detach where the default is ticked. Leave it this way to ensure that the previously opened database is detached before the new one is opened. S-Plus will create several subdirectories that it needs, including a Data directory where data are stored. The new workspace is completely empty with not even a blank Object Explorer open. Open a new Object Explorer using File-New-Object Explorer-OK. We want to create some data here so that when we switch between


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workspaces we can verify that they actually perform the way we want them to and keep our projects separate.

Creating Data • Click on Data, then on Random Numbers... • For Data Set:, specify test • For Sample Size:, specify 50 • Choose OK – The first entry should appear in the Object Explorer. • Click on Data, then on Transform • For Data Set:, specify test • For Target Column:, specify Square • For Variable:, specify Sample • For Function:, specify X^2 • Click on Add – Sample^2 appears in Expression: • Choose OK – Can see that the data sheet has been opened and contains the variables Sample (the original Gaussian distributed data) and Square (the square of the variable Sample).

Save the new workspace (File-Workspace-Save...) accepting the default directories and file names as prompted (these have already been specified by you earlier). The name for the workspace is taken from the name of the directory where it was originally created. For example, if the workspace was created in the directory c:\sbook, then its name would be sbook. Switch back to the default workspace (named Workspace) by choosing it from the list under File-Workspace. Does the newly created data set (test) appear in the Object Explorer of this workspace? Switch back to the new workspace. Is test the only data set that appears in the Object Explorer here? Note Warning messages may appear about duplicate databases, detaching, attaching, and position numbers. These messages may be ignored but serve to point out that the database for workspace A is not actually removed when you switch to workspace B, but is simply moved further down the

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search path. This action has two implications. Data sets from a non-active workspace cannot be seen through the Object Explorer but can be accessed. If a data set in the active workspace has the same name as one in a nonactive workspace, the one in the active workspace will be accessed. ⊳ In summary, the workspace feature is a convenient way to help keep yourself organized. We highly recommend that you begin using it right from the start. For practice with workspaces, try creating one for each chapter of the book and keeping chapter-specific data sets in the respective workspaces.

2.4 Detailed Use of the Windows Interface The previous section provides a quick introduction to some of the basic concepts involved with using the graphical user interface (GUI) developed for the Windows version of S-Plus. Brief descriptions of how to perform several tasks are given, but without much in the way of specifics. The details of how to understand and work with the GUI are given throughout this section. We cannot emphasize enough that to really profit from this book, you need to be running S-Plus and following the examples while reading the book. Only by trying something on your own will you really learn it to the extent that you could do it on your own. We have left a lot of tasks to be done by the reader (you!), so as to build on the concepts immediately.

2.5 Object Explorer The Object Explorer plays the same role in S-Plus as the Explorer does in Windows in that the “contents” are displayed in a tree-like diagram on the left-hand side of the screen and the details on the right-hand side. Similar or related objects can be stored in folders, giving structure to the way one works. When an object is put into a folder, a link (shortcut) is created such that the actual location of the object is not changed, just its virtual location. Folders can be dragged and dropped between Object Explorers, between Explorer pages within an Object Explorer, or onto the toolbar as a button. Objects can be classified into three types: computational engine, interface, and document. Computational engine objects include data frames, matrices, lists, functions, and others, and are objects containing data or the functions that are used to process the data. Interface objects are a type that relate to communication with the system and include search paths, menu items, toolbars, dialogs, and the like. Document objects refer to out-

2.5. Object Explorer

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put from the system and include Graph Sheets, Reports, Scripts, and the like. All of the object types mentioned can be saved in folders in the Object Explorer. Folders can be organized according to object type, data set, project, model class, or in any way that seems to be helpful. Open the Object Explorer if it is not already open by clicking the Object Explorer button on the standard toolbar. It should look like the one in Figure 2.4. If you have not yet worked with S-Plus the Object Explorer will be empty and if you have, you will see different object types in the left pane and their details in the right pane.

Figure 2.4. The Object Explorer toolbar.

We want you to follow an example and to get practice exporting a data set so we will take a built-in S-Plus data set, export it, and then import it before we begin analyzing it. First, we have to use the Object Explorer to find the data set. S-Plus uses a system of “positions” that are searched sequentially until the desired object is found. A message that an object has not been found is only issued once all the positions have been searched. Position number 1 is the current working directory. By default, the Object Explorer is used to show the objects in the current working directory. Other positions are used to store general functions, statistical functions, other types of data sets, and the like. The data we want to use are a few positions down in the list because the built-in data sets are not meant to be worked on in a working directory, merely read from a safer location. The search path lists all the positions currently available to S-Plus. By examining the search path, we can find the built-in data sets, locate the one we want, and write it to the hard disk.

S-Plus Data Sets • Click beneath the object types in the left pane of the Object Explorer (or else you will get a subfolder and not a folder) • Create a new folder by clicking on the New Folder button on the Object Explorer toolbar • Type in the name for the folder (use “Examples”) and hit Return • Right-click on the folder

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• Choose Folder. . . • In the Data Objects section, tick Data • Choose tab for Advanced (see Figure 2.5) – File locations in the Databases: section specify the Search Path. Notice that the first database matches the file location that appears in the Commands Window as the current working directory. • In the Databases: section, choose SPLUSpath\STAT\ DATASET where SPLUSpath is the path where S-Plus is installed (typically C:\Program Files\sp2000) – This choice should be in position 6 (i.e., sixth in the list) • Choose OK – A “+” appears next to the Examples folder and the appearance of icon changes – List of all the data sets appears in right-hand pane (see Figure 2.6)

Figure 2.5. Filtering data sets for Object Explorer.

We have just used the Object Explorer to look at the search path and put the contents of one of the positions into a folder so that we can directly access the contents. The Object Explorer can also be used to find more information about the contents of a particular folder. The data set

2.5. Object Explorer

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Figure 2.6. List of data sets displayed in the Object Explorer.

we want to work with is called Puromycin, can you locate it using the Object Explorer? What happens if you now double-click on Puromycin in the Object Explorer? Do you see the similarity of the Object Explorer to the Explorer in Windows? Double-clicking on Puromycin runs the application which, in this case, opens a spreadsheet-like editor that can be used to view and edit the data. By single-clicking on Puromycin in the left-hand pane of the Object Explorer, a list of its variables is shown in the right-hand pane. The Puromycin data set contains the three variables, conc=concentration, vel=velocity, and state, as shown in Figure 2.7.

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Figure 2.7. Details of the Puromycin data set.

2.6 Help At some point you will want to use (or have to use) the Help system in S-Plus and we have reached a point where it is possible to explain this simple procedure. Suppose we feel that the three variable names in the Puromycin data set and their corresponding data types are not terribly informative and we want to know more about it. We can invoke the Help system to find out more.

2.7. Data Export

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Using the Help System • Click on the Help main menu • Choose the S-Plus Help category • Choose Index tab • Type “pur” – Puromycin is highlighted • Choose Display at bottom – Window explaining the Puromycin data set appears

The Help system can be used in the same way to find information on (almost) any other topic.

2.7 Data Export As promised, we’re going to export the Puromycin data, import it again, and then work with it. The process is all menu-driven.

Exporting Data • Click on Puromycin in the left-hand pane of the Object Explorer • Click on the File main menu • Choose Export Data • Choose To File . . . • Specify location for file to go using Save in: • Specify the file name (e.g., puro) using File name: • Specify file type using Save as type: – We chose SAS Files (*.sd2) to make another point later on and suggest that you do as well (you don’t need SAS) – Puromycin is already highlighted in Export From Data Set: (see Figure 2.8) • Choose OK The file should now be created with the name (puro) and type (.sd2) that we specified, residing in the directory of our choice. Use the Windows

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Figure 2.8. Data export.

Explorer to verify that the desired data set has been created. Save the same data in different formats and open them with the appropriate program, for example, using Excel for *.xls files. as an Excel file and open the file in Excel.

2.8 Working Directory When working in a Windows environment, it is often convenient to set up a different “workspace” for each separate study on which you are working. Workspaces can be set up in two different ways: as shown in Section 2.3.5, or by defining multiple S-Plus icons on the desktop. The advantage of the former approach is that you never have to leave S-Plus to switch between project specific directories, whereas the advantage of the latter approach is that the danger of inadvertently working in the wrong directory is minimized. The method of defining and switching between workspaces is easy, flexible, and orderly (see p. 14), but, as people work in different manners, the method of defining a working directory and creating an S-Plus icon will be shown here.

2.8. Working Directory

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Defining a Working Directory Create a copy of the S-Plus shortcut icon by: • Right-click on the Start button in the Windows desktop (lower left) • Choose Open • Double-click on Programs • Double-click on the S-Plus folder • Depress the key, Click on the S-Plus icon, and Drag it to the Windows desktop • Release the mouse and the key and you have a copy of the icon • Right-click on new icon • Choose Properties • Choose tab for Shortcut sheet – Target: is highlighted and shows the command for running S-Plus • After the existing text that is already in the box for Target:, add S PROJ=pathname – Use the path where you saved the Puromycin data. – The pathname must be enclosed with double quotes if there is a space in it. • Start in: should be blank (see Figure 2.9) • Choose OK

You can now start S-Plus by double-clicking on the new icon, which will have the effect of working in the directory specified in the path name. If this is the first time S-Plus has used this directory, you will get the message, “Invalid Path - The path found for s proj ‘pathname’ does not contain folders ‘Data’ and/or ‘Prefs’ for project related files. Specify another path or accept this one and the necessary folders will be created.” At this point, you can specify a new path, or accept the choice and these two folders will be created for you (Choose OK). Open the Object Explorer and verify that the new S-Plus icon is using the database (empty) defined by the new path name (we did this earlier).

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Figure 2.9. Defining a working directory.

2.9 Data Import After this preparation, we are finally ready to look at how to read a data set into S-Plus. The procedure is essentially the same as with exporting data and works in the following manner.

Importing Data • Choose File from the main menu • Choose Import Data • Choose From File . . . • Define Files of type: to be SAS Files (*.sd2) – Check that location (path name) and file name appear as desired • Click on file name to select it (highlighted) – Import to Data Frame: has puro in it automatically so that the data set will be stored in the data frame puro

2.9. Data Import

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• Choose Open As soon as Open has been chosen, a data window should open containing the data set displayed in typical spreadsheet fashion as in Figure 2.10. It contains 23 rows and 4 columns (variables), 1 column more than the original data frame! What has happened? In addition, the variable names are now in all capitals and S-Plus is case-sensitive.

Figure 2.10. Data window from Importing Data.

You’ll note that the first column is called ROWNAMES and that the values go sequentially from 1 to 23. This column corresponds to the internal row counter in SAS ( N ) which was not part of the original data frame. The extra column we encountered is a special case of SAS, but there may be other file types which insert other special columns. This is a nice place

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to remind you that you should always check the data to make sure that there are no unpleasant surprises. We now want to remove the extraneous column.

Deleting Data Column(s) • Click on the first column of the Data Sheet (the one to be deleted) • Choose Data from main menu • Choose Remove • Choose Column . . . • Define the column(s) to be removed using the pull-down menu for Columns: – Remember to use Ctrl-Click to select noncontiguous column names • Choose ROWNAMES (see Figure 2.11) • Choose OK • Close Data Window

Figure 2.11. Deleting a data column.

Use the Object Explorer to open the data frame again to verify that only the three original variables are in it. Data frame is the term used by S-Plus to denote a rectangular collection data, possibly including variables with different data types (e.g., character and integer). Many data sets typically encountered are actually data frames.

2.10 Data Summaries Now that we have a data frame in the Object Explorer, we want to “look” at the data in some sense. One of the easiest ways of getting a sense

2.10. Data Summaries

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for your data is to generate simple summary statistics (mean, standard deviation, etc.). The following command box shows how this is done.

Generating Summary Statistics • Click on puro in left-hand pane of Object Explorer to select it • Choose Statistics from the main menu • Choose Data Summaries – Options are Summary Statistics, Crosstabulations, and Correlations • Choose Summary Statistics . . . – Summary statistics dialog opens to a data window. – Puro should be chosen as the default data frame (see Figure 2.12).

Figure 2.12. Summary Statistics data dialog.

• Choose All Variables (default) • Choose No Grouping Variables (default) – Can tick or untick various statistics to customize output. – Can save results by supplying a name using Save As.

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• Choose Statistics tab at the top of dialog window. – Can choose specific statistics to be calculated (see Figure 2.13).

Figure 2.13. Summary Statistics dialog.

• Choose OK A Report window is opened (Report1) containing the count of each level of the variable STATE and the summary statistics for the continuous variables CONC and VEL, as shown in Figure 2.14. We see, for example, that there are 12 treated and 11 untreated patients and that the mean CONC is 0.3122. The contents of the Report window can be saved into a file by choosing File-Save As . . . and specifying a path and file name. The default file type is .srp (S-Plus report file), which can be opened by any standard editor or from within S-Plus.

2.11 Graphs The problem with data summaries is that you don’t actually “see” the data itself. For a more visual summary of our data, we need to construct a

2.11. Graphs

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Figure 2.14. Summary Statistics Report window.

graph. There are actually several different ways of constructing a graph in S-Plus using the GUI (menu buttons), but perhaps the easiest and most flexible method is to use the palettes.

Creating a Graph Using a Palette • Open the Object Explorer if it is not already open • Expand the list in the left-hand pane of the Object Explorer such that the three variables of puro are visible in the right-hand pane • Click on CONC to highlight it • CTRL-Click on VEL so that both variables are highlighted – The order in which the variables are chosen (clicked) is important. The first will be used for the x-(horizontal) axis and the second for the y-(vertical) axis. • Click on the 2D palette button on the main toolbar

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• Click on the button for scatterplot (upper left) – A new graph sheet will open, containing a scatterplot of the two variables CONC and VEL. – If you don’t like the type of graph that you have chosen, simply click on a different one (e.g., line scatter, loess) and a new graph sheet will be opened with the same data but using a different plot type. Be adventurous, try several.

We can’t show you the graph very well at this point because the default graph uses a light color that doesn’t show up well in a black-and-white book, so we’ll wait until after explaining some editing techniques before actually showing the graph. The default graph styles used by S-Plus typically look very good, but they can always be modified to match your exact needs and desired. This leads us to the topic of how to edit a graph once it has been created. In particular, we need to change the color and symbol for the points on the scatterplot so that they will be more visible.

Changing Color and Symbol of Points • Click on any point in the graph – A green knob will appear in the lower left of the graph. • Double-click on the green knob – A dialog window named Line/Scatter Plot [1] will appear as in Figure 2.15. – The first tab for the window shows the Data to Plot panel. The data that are used in the plot could be redefined here. • Click on the Symbol tab • Using the pull-down menu, change the Style: to • Circle, Solid • Using the pull-down menu, change the Color: to Black (see Figure 2.15) • Choose OK – The symbol has changed both style and color. Notice that the green knob is still present (see Figure 2.16).

Making the graph looking “nicer” (depends on individual tastes) is usually of interest, particularly when it comes to labeling different parts of the graph. We will investigate how to change the axis labels, add a title, and add general text to the graph.

2.11. Graphs

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Figure 2.15. 2D Plot dialog (symbol).

Changing Axis Labels • Click on the default x-axis label (CONC) – It is now surrounded by green knobs. • Choose one of the following methods: – (A) One more Click allows you to change the label by simply typing over the default (Default is automatically taken as the column name and is denoted ‘@Auto’) – (B1) Right-click on the default label (dialog appears) – (B2) Choose Edit In-place. . . (see Figure 2.17) • Change text to “Concentration” • Click outside of box to finish without hitting Return The label has now been changed and the text is better now; however, the font is rather small. We can easily change this.

Changing the Font • Right-click on axis label text

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Figure 2.16. Scatterplot of Puromycin data.

Figure 2.17. Axis dialog box.

2.11. Graphs

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• Choose Font. . . • Define font size by varying Size: – Try 20 (see Figure 2.18) • Choose OK

Figure 2.18. Axis font dialog box.

We have now changed the label and font size for the x-axis and leave it up to you to repeat the procedure for the y-axis. Adding a title to a graph is most easily done using Insert from the main menu. The exact usage is explained below.

Adding a Title • Click on Insert in the main menu • Click on Titles • Choose Main... • Type in the desired title (Use ‘Scatterplot’) • Click outside the title box to end • Change the font as desired (see Changing the Font) • Change the box as desired

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Graphs in S-Plus are designed to be interactive in more than one sense; that is, we created the graph to learn something about the data and want to get information out of it. One of the nicest features about S-Plus is that the graphs are not static; they are not merely a nice picture to look at and study. Try positioning the cursor over a point and holding it there for a second. What happens? A small box appears that contains three pieces of information: the index number and the x and y coordinates. The index refers to the number of the entry in the data sheet. Choose some points and look at their values and indices. There are two very useful buttons on the Graph Sheet toolbar that open the graph tools palette and the annotations palette. By holding the cursor over the individual buttons in the palettes, you can see what their functions are. Explore the possibilities open to you with these buttons. We created the graph in Figure 2.19 with just a few clicks of the mouse.

Figure 2.19. Annotated scatterplot.

Now that the graph looks exactly the way we want, we’re ready to print it and save it for possible future use.

2.11. Graphs

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Printing a Graph • Click anywhere on the Graph Sheet to make it the active window • Choose File from the main menu • Choose Print Graph Sheet. . . – Make sure settings for printer are correct • Choose OK

Saving a Graph (Sheet) • Click anywhere on the Graph Sheet to make it the active window • Choose File from the main menu • Choose Save As. . . • Specify location and file name for the graph (see Figure 2.20) • Choose Save

Figure 2.20. Saving a Graph Sheet.

We have just saved the graph into a file, so we should be able to read the graph back into S-Plus. We leave it up to you to open the graph from the file that you just created.

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Another useful task that is often done is to create a graph in S-Plus for use in another application (e.g., MS Word). There are two ways of exporting a graph from S-Plus: the direct method and the indirect method.

Exporting a Graph - Direct Method • Open a Graph Sheet and make sure it is the active window • Press CTRL-C (or, Edit - Copy) to Copy • Move to MS Word (or other Windows application) • Press CTRL-V or, Edit - Paste) to Paste – Graph is now in MS Word

Exporting a Graph - Indirect Method • Open a Graph Sheet and make sure it is the active window • Choose File from main menu • Choose Export Graph. . . – From this window, you can choose from many different file types. MS Word, for example, can easily read a Windows Metafile (*.WMF). • Specify file type, file name, and location • Choose Export – A Windows Metafile can be inserted into MS Word using Insert Picture - From File, and then specifying the file name and location.

According to what software you have available on your system, you should try saving a graph using different file types and importing it into the appropriate package.

2.12 Trellis Graphs Trellis graphs are a convenient way of exploring relationships in data. They work by graphing one or more variables while conditioning on one or more others. Panels are constructed for each level of the conditioning set, and a plot of the variables of interest is made in each panel. An example of their use might be to plot height and weight, but to have one panel for males

2.12. Trellis Graphs

37

and one for females. In this manner, it is then easy to see if the relationship between height and weight is the same between the sexes. We will now show how to make a Trellis graph, although we have devoted an entire chapter (Chapter 6) to it later.

Trellis Graphs – Drag and Drop

Figure 2.21. Trellis Graphs – Drag and Drop.

• Create a scatterplot from the Puromycin data as we have done before • Left-click on STATE in the right-hand pane of the Object Explorer and keep the mouse key depressed

• Drag the mouse to the upper portion of the graph sheet until a dashed rectangular box appears, as in Figure 2.21 (a box appears next to the mouse as well – not shown) • Release mouse

• The graph is automatically transformed into a Trellis graph, as in Figure 2.22

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There is a small trick to producing Trellis graphs using the menu; we will show that approach separately.

Figure 2.22. A Trellis graph conditioning on State.

Trellis Graphs – Using the Menu • In the Object Explorer, click on the variable to be used on the x-axis (CONC). In the Object Explorer, click on the variable to be used on the y-axis (VEL). In the Object Explorer, click on the conditioning variable (STATE). • Click on Graph-2D-Plot in the main menu • An Insert Graph dialog appears • Change the axes type to Panel (the default is Linear) • Proceed as before to create a graph

2.13. Linear Regression

39

The only difference in the final graphs between the two approaches is that the same symbol is used in all panels with the drag and drop method, whereas each panel gets a different color/symbol using the menu approach.

2.13 Linear Regression We have already looked at the summary statistics for the variables in our data set and at a scatterplot of concentration versus velocity. We are now ready to try to fit a linear regression model to our data.

Performing Linear Regression

Figure 2.23. Linear Regression dialog.

• In Object Explorer, click on data frame puro • Choose Statistics • Choose Regression • Choose Linear . . . – Puro is the default data frame.

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– Sheets for specifying Model, Results, Plot, and Predict (see Figure 2.23) • Choose Create Formula • In the Variable section, click on VEL • In Add option, choose Response – Velocity will be the response (y) variable • Click on CONC • In Add option, choose Main Effect: (+) – Concentration will be the predictor (x) variable. – Can transform variables, add quadratics, and so forth. – In the Remove section, can tick box for removing y-intercept (see Figure 2.24)

Figure 2.24. Regression formula dialog.

• Choose OK • Try the following options: • In the Save Model Object section, specify a name for the regression object to be stored under (use the name my.reg). • Click on Results tab to move to that sheet • Tick ANOVA table (as an option) – Can also save fitted values and residuals


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• Click on Plot tab to move to that sheet – Can choose which plots to look at. None are ticked by default, but this doesn’t matter, as we’ll see later. • Click on Predict tab to move to that sheet – Can save predicted values and confidence intervals • Choose OK The Linear Regression opens a new Report window shown in Figure 2.25.

Figure 2.25. Regression Report window.

Looking in the Report window at the ANOVA table, we see that CONC has an F-value of 38.8 and a corresponding p-value of 3.5 × 10−6 . At first glance, it appears as though the model provides an excellent fit to the data. Recall from the scatterplot, however, that the relationship between the two variables does not look particularly linear. We should explore the fit of the linear regression model further by looking at certain graphical representations (i.e., the residuals plots). You will recall that we did not specify that any of the residual plots should be created, but we do not have to start from the beginning. Notice

42


that the object my.reg now appears in the right-hand side of the Object Explorer. Right-click on my.reg and choose Plot.... The original menu for the residual plots appears and we can now specify which plots we want to explore. Choose Residual vs Fit, Response vs Fit, Residuals Normal QQ, and Cook’s Distance as a start by ticking the appropriate boxes and then choose OK. A Graph Sheet is created with four pages, one for each plot (not shown). Note It should be mentioned at this point that graphs produced by statistical dialogs can be either editable or noneditable. The noneditable versions have the advantages of being quicker to produce and require less space for storage, but the disadvantages that they cannot be edited to change labels, add annotations, and so forth. Since most computers have ample computing speed and space, we recommend the use of editable graphs. The desired type of graph can be defined using Options - Graphs Options... and ticking (or unticking) the desired default. The next section will not be possible to perform unless the residual plots were created as editable graphs (i.e., change the options and redo the graphs). ⊳ If we look at page 1 of the Graph Sheet, however, notice that the plot of residuals (y-axis) versus fitted values (x-axis) does not look like a cloud of points randomly distributed about the line y=0. Furthermore, three points have been indicated as “suspect”; points 2, 10, and 23. A smoothed line has been added to make it all that much more apparant that there is a pattern to the points. Page 2 of the Graph Sheet shows the response variable (VEL) versus the fitted values (x-axis) with an overlaid regression line (dotted) and a smoothed line (solid). Notice that the points, from left to right, are mostly below the line, mostly above the line, and then mostly below the line again. This trend indicates a poor fit of the model. Page 3 shows the residuals (y-axis) versus the quantiles of the standard normal (x-axis) which also indicates the points 2, 10, and 23. The fourth page contains Cook’s distance (y-axis) versus an index (count of the order of the points) on the x-axis. Here, we see that points 2, 14, and 23 have the largest values of Cook’s distance. The graphs suggest that the simple linear regression provides a poor fit to the data, but you will perform a better analysis in the exercises. This is a good place to mention that we can put multiple plots on a single graph page by copying and pasting. This feature is particularly relevant and helpful when trying to deal with a multipage Graph Sheet.

Placing Multiple Graphs on a Single Graph Sheet • Click New on toolbar • Click on Graph Sheet


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Figure 2.26. Combining graphs onto one Graph Sheet.

• Choose OK • Click on Window from the main menu and then Tile Vertical (Just to simplify the copying process) • Click on page 1 of residual plots to highlight it • Choose Edit from the main menu and then Copy Graph Sheet Page • Click on header of target (empty) Graph Sheet to highlight it • Choose Edit from the main menu and then Paste – Graph is now copied in the target Graph Sheet. • Repeat the previous three steps to copy pages 2-4 onto the target Graph Sheet. – Notice how space is made at each step to accommodate each new graph. • Save the new Graph Sheet (calling it residuals). The result is shown in Figure 2.26.

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2.14 Powerpoint If you intend to make an MS Powerpoint presentation that includes graphs you have created in S-Plus, then your task has been made easy for you by an S-Plus wizard designed to set this up for you.

Putting Graphs into Powerpoint • Click on the Powerpoint Presentation button on the toolbar (upper right) – Creates Powerpoint presentation of selected, saved Graph Sheets – Opens S-Plus to Powerpoint Presentation wizard • Choose Next • Choose Add Graph. . . • Choose scatter.sgr • CTRL-click on residuals.sgr such that both are highlighted • Choose Open – Could add additional graphs using the same process • Click on Next • Choose Finish – Powerpoint is opened and creates the slides. • When complete, exit the wizard • Asks to save the lists of graphs - Can, but choose No • Powerpoint is open and is the active window • Choose File from main menu • Choose Save As. . . • Specify file name and location for Powerpoint file If several Graph Sheets have been saved, they can be put into the same Powerpoint presentation, or one per presentation it you prefer.

2.15 Excel The Windows-based version of S-Plus also installs add-ins for MS Excel. You should notice a floating toolbar in Excel with three buttons as well as

2.16. Script Window

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a new entry for S-Plus in the main menu. These tools give you S-Plus graphics capabilities from within Excel. If you have a data set in Excel and want to produce S-Plus graphs, you don’t have to transfer your data to S-Plus, but can stay in Excel and use the add-ins to produce your S-Plus graphs. The three buttons in the S-Plus toolbar (within Excel) are for creating a graph, modifying the layout of a graph, and modifying a plot. An explanation of how to use them is provided in the entry About S-Plus Add-In in the S-Plus menu entry. There are essentially four steps to create S-Plus graphs from within Excel. In Step 1, you are asked for the range of the data to be plotted. Keep in mind that you should specify the upper-left and lower-right corners of your block of data as in A1:B10. In Step 2, you are asked to specify details for conditioning the data if you don’t to plot certain data items. In Step 3, you are asked to specify the position where the graph is to be located. Only the cell for the upper left-hand corner of the graph needs to be given, along with the Sheet as in Sheet1, F15. The last step is where you specify the graph and plot type and uses the same dialog boxes as in S-Plus. The best way to see this is to open Excel, quickly type in some data, and give it a try.

2.16 Script Window The Script window offers an easy way to build and run a program or to keep track of what has been done so far. The idea behind this type of window is that it is a place where windows can be entered, run, and saved with the output appearing in a separate section of the window. It can also be used with GUI-style commands, but, as these can be quite cryptic, its greatest use is to be run in conjunction with line commands covered in the rest of the book.

Opening a Script Window • Choose New from main toolbar • Choose Script File • Choose OK There are two panes in the window: the upper pane is used for entering and running commands and the lower pane is for showing output. In the example to follow, you should open the Scatterplot Graph Sheet and choose Window (main menu) - Tile Vertical so that both the Graph

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Sheet (containing the scatterplot created earlier) and Script window can be accessed at the same time.

Viewing GUI Commands in the Script Window

Figure 2.27. The Script window.

• Click and hold mouse button on the scatterplot • Drag the mouse to the upper pane of the Script window • Release mouse button – All graph options used for the residual plots appear (see Figure 2.27)

2.16. Script Window

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Saving GUI Commands from the Script Window • Click in upper pane of Script window to make it active • Choose File from main menu • Choose Save As. . . – Default file type is *.ssc • Specify file name and location • Choose Save The GUI options and commands that have been shown here are sometimes difficult to interpret and are beyond the scope of this book. It can be helpful, however, to simply view what the current settings are at any given point in time. The real power and usefulness of the Script window in the context of this book is to edit and submit segments of code. Up until now all the “commands” we have entered have been in the form of mouse movements and interacting with the GUI. The commands to be entered in the Script window are more akin to programming language commands on which the S-Plus processor then acts. Suppose you have a data set x consisting of one vector and you wanted to calculate the mean of x. You would choose Statistics, Data Summaries, then Summary Statistics.... However, you could also ask for the mean to be calculated directly by specifying mean(x) in either the Commands window (executed with the <Enter> key) or in the Script window (executed with the Run button). You may have noticed when you opened the Script window that two buttons were added to the toolbar, as in Figure 2.28. The left button has a right arrow for running scripts and the right button is for finding specified text.

Figure 2.28. Script window toolbar.

Running Commands in a Script Window • Clear the existing text in the Script window. – Choose Edit from main menu – Choose Select All – Delete

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• In the upper pane, enter mean(1:10) – Calculates the mean of the numbers from 1 to 10 • Click on the Run button – Command and output appear in lower pane

In this manner, you could work on groups of commands to fine-tune a certain process and submit them as a group. It is a good way to work when building a function or longer program, tasks that will appear toward the end of the book.

2.17 Summary The GUI you just looked at is relatively new. S-Plus has incorporated most of the drag and drop philosophy of modern software. As mentioned before, the origins of S-Plus lie in the late seventies and early eighties. S-Plus is very popular because of its ability to process data in a flexible language. The modern concept and the easy-to-use library are still the core of the system and the whole power is in a way hidden behind the simple clicks required to use the GUI. Only by learning more about the entire language and its concepts will you get an idea of the possibilities S-Plus has to offer for data analysis. The remainder of the book will primarily focus on the language S-Plus. You will complete several sessions of analyzing data and encounter a variety of statistics routines along the way. You will see the commands involved in running some of the procedures that you have already learned in this chapter. We strongly recommend that you do not stop here after seeing the “easy” point-and-click approach to S-Plus. By continuing with the remaining chapters, you will not only understand what S-Plus is doing and how it works, but you will learn lots of things that you will really want to be able to do that you simply cannot do with just the GUI.

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2.18 Exercises Exercise 2.1 Perform a more complete analysis of the Puromycin data, including such exploration as transformations of the predictor variable (square root and log are typical), the effect of STATE, and the interaction of STATE and the transformed variable. Be sure to use only the GUI to do your analyses and visually check the fit of the model.

Exercise 2.2 Use the filtering capability on the Examples folder to locate the S-Plus built-in data set geyser. (Hint: Filter for Data Objects “Data” and “Models.”) Use the Help to find out something about the data and create a scatterplot. What do you notice about the points? Are they randomly distributed, linear, or grouped? Use the annotation palette and graph tools to group the points. Label the groups of points. Calculate the means of each group (use the Grouping variable option in the Summary Statistics menu). Write the means of each group onto the scatterplot. Add a linear regression line to the graph. Save the Graph Sheet and then make it into a Powerpoint presentation.

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2.19 Solutions Solution to Exercise 2.1 We know from our scatterplot that the relationship between the variables is not particularly linear and look perhaps quadratic. We shall try two transformations of the data and redo scatterplots to make a first check of which model might fit best. We will try 2 transformations, square root and the natural logarithm (ln - although the function in S-Plus is log), but first we need to add these variables to our data frame. Double-click on puro in the Object Explorer to open the data frame.

Inserting a Calculated Column of Data • Right-click on the top of second column (VEL) – New column will go before this one • Choose Insert Column... • Define a name for Name: – We chose sqrtconc • Define the transformation in Fill Expression: – Use sqrt(CONC) (see Figure 2.29) • Choose OK

Figure 2.29. Inserting a column of data.

Repeat the procedure to create the variable logconc using the log function. We want to replot the data using our transformed variables to look for a linear relationship. This time, however, we will add a smoothed line (curve) to the data that will help us visualize any trends that might exist.

2.19. Solutions

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Smoothed Line Plot

Figure 2.30. Loess smoother curves.

• Use the palette as before, OR... • Click on top of column for sqrtconc – Defines x-axis variable • CTRL-click on top of column for VEL – Defines y-axis variable • Choose Graph from main menu • Choose 2D Plot. . . – Leave Axes Type: as Linear • Change Plot Type: to Smoothing - Loess Plot (x, y1, y2, ...) – Example of layout appears • Choose OK (see Figure 2.30)

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Notice that the general shape of the data and the smoothed line are much more linear than with the original data but are still curved. Repeat the above process using logconc and you will find that the log transformation of the data is the one that produces what appears to be a linear relationship. Anyone who has worked with concentration data could probably have guessed this right from the start. Now that we are satisfied with a transformation on CONC, we can go back to the task of actually performing a better regression analysis.

Regression with Transformed Variables

Figure 2.31. Variable transformation for regression.

• Choose Statistics from main menu • Choose Regression • Choose Linear. . . • Choose Create Formula • Add VEL as response • Click on CONC • Choose Transformation – We already have the transformed variable in our data frame, but this is shown for information. • In the Function section, Choose Log: log(x) – New Variable(s) shows log(CONC) (see Figure 2.31)

2.19. Solutions

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• Choose Add • Choose OK • Click on log(CONC) in Variable section of Formula dialog • Add Main Effect: (+) • Choose OK for Formula • Choose OK for Linear Regression

Figure 2.32. Regression output using log(CONC).

The F-statistic is still highly significant (F=146.9 on 1, 21 df, p=6.0 × 10−11 ) (see Figure 2.32), but the graphs look much better this time (see Figure 2.33). Note To run a linear regression model, we could choose Apply instead of OK. Choosing Apply has the effect of producing the same output windows (Report and Graph Sheet), but now the Linear Regression window remains open and we can modify the formula rather than restarting our definition of it. ⊳

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Figure 2.33. Residual plots using log(CONC).

Reenter the commands from above to set up the model with log(CONC) as the predictor variable. Add STATE as a second predictor variable (as a Main Effect) (see Figure 2.34) and choose OK for the formula. This time, choose Apply. A new Graph Sheet is opened, but the the results are printed in the same Report window. The graphs still look good and we notice that STATE is significant on its own as a main effect. The output from this regression model appear in Figure 2.35, but the residual plots have not been shown. The last point we will cover is the interaction between log(CONC) and STATE. We will leave that to the reader to perform [i.e., the solution stops here (but you can refer to Figures 2.36 and 2.37 to see if you’re on track)]. We will simply note that the interaction term is significant, the interpretation of which is also left up to the reader (do scatterplots by STATE).

2.19. Solutions

Figure 2.34. Regression formula using STATE.

Figure 2.35. Regression output using STATE.

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Figure 2.36. Regression formula using interaction term.

Figure 2.37. Regression output using interaction term.

2.19. Solutions

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Solution to Exercise 2.2 The biggest problem with using the geyser data is locating it. If you look at the Examples folder created earlier, you will notice that the geyser data set is not there. However, you can look for it by changing the filter specifications for the Examples folder.

Changing Filter Specifications • Right-click on Examples folder • Choose Folder... • Tick Data, Models, and Functions (see Figure 2.38). The specification used earlier (only Data) is not sufficient because, although the geyser data are in a simple matrix format, they were created as a type list (see Chapter 9). • Click on Advanced tab – There are only three databases that end in Dataset. Try them one after another until you find the right one (see Figure 2.39). • Choose OK

Figure 2.38. Available data objects.

The Geyser data set was in the S-Plus database and using the Object Explorer, you can see that it has two variables: waiting and duration (see Figure 2.40). To find out what these two variables contain, invoke the Help system.

Help Entry for Geyser Data Set • Click on Help button on toolbar • Click on S-Plus Help • Choose Index tab

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Figure 2.39. Database filter.

• Type gey and hit Return – The help entry now appears

The Help entry states that these are the successive waiting times and durations of eruptions for the Old Faithful Geyser in Yellowstone National Park. With scatterplots, it is necessary to choose one variable for the x-axis andone for the y-axis. For this kind of exploratory analysis where no hypothesis is being examined, it doesn’t really matter which variable goes where, but we chose waiting time for the horizontal axis and duration for the vertical.

Scatterplot for the Geyser Data • Click on waiting in the right-hand pane of the Object Explorer • CTRL-Click on duration • Click on 2D Plots button in the toolbar • Choose scatter plot

2.19. Solutions

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Figure 2.40. Object Explorer.

The axis labels are not ideal and the plot symbol is a bit hard to see on the printed page, so we will change these features on our graph.

Changing the Axis Labels on the Geyser Scatterplot • Click on x-axis label • Right-click and choose Edit In-Place. . . • Change text to Waiting Time • Click outside of box to finish the text • Repeat to change font or font size as desired Repeat the above procedure to change the y-axis label to Duration of Eruption. The points are quite large so that they overlap and they are blue (but the print in the book is black and white).

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Changing the Plot Symbol • Click on any point • Double-click on green knob • Click on Symbol tab • Change Style to • Circle, Solid • Change Color to black • Change Height to 0.10 (see Figure 2.41) • Choose OK

Figure 2.41. Plot Symbol dialog.

The result of these modifications appears in Figure 2.42. There are many ways to “group” the data here. Speaking in terms of the way the data are distributed, you could easily say that the distribution of durations changes with waiting time. Consider, for example, a waiting time of 65. The durations corresponding to these shorter waiting times (65) range from roughly 1 to 4.8. You could also group the data by looking at the distribution of waiting times for short durations as opposed to long durations.

2.19. Solutions

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Figure 2.42. Geyser scatterplot.

We chose to split the data according to a waiting time of 60 minutes such that all points to the left of this defining line have short waiting times and those to the right have long waiting times. The remaining task is to record all of this onto the scatterplot using the Annotation palette in Figure 2.43.

Adding Annotations to a Graph • Click on Annotation palette in toolbar • Using the Line Tool, draw a vertical line starting at the waiting time of 60 minutes. Click on Select Tool to turn off the Line Tool. • Double-click on the line and change the Style, Color, and Weight, as in Figure 2.44, and then choose OK. • Use the Rectangle Tool to draw a box around the points on either side of the reference (vertical) line. What do you notice? Click on Select Tool to turn off Rectangle Tool.

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Figure 2.43. The Annotate palette.

Figure 2.44. Plot Line dialog.

• Use the Comment Tool to label the rectangular boxes just drawn. Click on Select Tool and then change the font, text, and scale as desired. Experiment with the various fonts (choosing Apply, rather than OK, simplifies the experimentation process). Choose OK when satisfied with the results.

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• Use the Arrow Tool to connect a label to its corresponding box and click on Select Tool to end. Change color of arrow (and arrow head) to black and choose OK.

If you drew and labeled in a manner similar to us, you should now have a graph that looks something like the one we obtained in Figure 2.45.

Figure 2.45. Geyser scatterplot with annotation.

The take home message here is that if the waiting time is short (60 minutes), there is no saying how long the duration might be.

3 A First Session

The remainder of the book is devoted to the use of S-Plus via the command line. We will look at how to use S-Plus as a calculator for simple and complex calculations. We will look at different data structures and how to handle data flexibly. We will generate graphs displaying data and mathematical functions, explore data sets in detail, and use statistical models. Finally, we will import data in several ways, automatize tasks by writing functions, and investigate many details that can help a lot in practice. Our first session begins with some general information on using S-Plus, including how to start and quit and how to access the Help system. This is followed by some useful commands and how to do basic arithmetic. Logical variables are covered and the chapter ends with a review of the material from the chapter, followed by hands-on exercises. To greatly enhance your learning of S-Plus, you should be seated at your computer while completing the exercises and also while reading each chapter. The exercises offer a good opportunity to try out the commands yourself and get acquainted with the system.

3.1 General Information We begin with general information on how to start and quit the system and how to access the online help facility.

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3. A First Session

3.1.1 Starting and Quitting The very first thing you need to know is how to start S-Plus. If you have S-Plus on Windows, all you have to do is double-click on the S-Plus icon (or use the task bar) and a new S-Plus window will be opened that should look approximately like the one in Figure 2.1 (page 8). If you’re running S-Plus under UNIX, you need to create an S-Plus chapter. A chapter is basically a directory tree containing all objects you are going to create: data, functions, and more. S-Plus should always be started from a chapter. Log on to your UNIX system and create an S-Plus directory. mkdir S move to the new directory, cd S initialize it as a new S-Plus chapter, Splus5 CHAPTER and start up S-Plus from there. Splus5 You don’t have to repeat the creation and initialization of the S-Plus chapter again. Just change to the directory and start up S-Plus next time. In the Commands window, the S-Plus command prompt > (the “greater than” sign) appears and indicates that S-Plus is ready to accept your commands. Now that you have started S-Plus, you will, at some point, want to quit. You can quit S-Plus for Windows by double-clicking on the upper left-hand corner of the window or clicking once on File and once on Exit. The standard Windows combination works as well. Another alternative is to type > q() at the command prompt. The latter works under S-Plus for UNIX as well. A quicker exit is to use (to quit the S-Plus shell).

3.1.2 The Help System If you get stuck, don’t panic. S-Plus has an online Help system. 1 To start the Help system, simply type help() at the S-Plus command prompt (or 1 There

is one note, although S-Plus 5.0 does provide online help, the system is rather basic. This is strengthened in S-Plus 5.1 with the reintroduction of help.start().

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67

help.start() under UNIX). The Help system is designed such that you do not have to know exactly what you’re looking for when you access it. It has a table of contents with commands categorized by function, and a search system that operates by typing in the command or subject on which you want information. Based on the subject, rather than the specific command, you can get information on your topic of interest. If you prefer to use the mouse, the Help system can as well be started by clicking on Help in the main menu or on the help button on the main toolbar. Depending on the version of S-Plus being used, the Help system might be set up differently. If you know the name of a specific S-Plus command, entering help(commandname) will open a new Help window with instructions on the functionality of commandname. If you are using Windows, you have the option of starting the Help system by double-clicking on the Help icon in the S-Plus folder. This enables one to use the Help system without actually starting the entire system. The help(commandname) form of accessing the Help system is especially useful when you know the name of a command but have forgotten the specific options it has, the input it requires, or the output it produces. The general help() command is useful if you don’t know an actual S-Plus command name, but only a particular subject.

3.1.3 Before Beginning We already covered notational conventions in the introductory chapter. Here, we present a few hints to keep in mind before you continue, a couple of which repeat the notational conventions. We present all S-Plus commands in typewriter font and they may appear either within the body of the text, as in help(), or on a separate line with the S-Plus command prompt, as with > help() Most commands in S-Plus contain opening and closing parentheses. Options to the commands are quite often included between the parentheses to further specify how a command should be used. The parentheses delimit the command and signify that an execution is to be made. If the parentheses are omitted, the contents of the variable (if data) or the definition of the command will be printed. Try typing help with no parentheses, and the internal commands associated with it are printed on the screen. It will sometimes be necessary to add a comment to either an S-Plus command or the output from one. The comment will be denoted with the number sign (#). Note This convention of writing notes is to catch your eye and draw your attention to something that is important - a helpful hint, or an exception

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3. A First Session

to something just mentioned. The “⊳” symbol is used to mark the end of a Note. ⊳

3.2 Simple Structures In this section, we show how variables can be assigned and simple data structures may be generated. We begin, however, by explaining how S-Plus handles simple arithmetic operations.

3.2.1 Arithmetic Operators Arithmetic operators in S-Plus function in the same fashion as in most programming languages and like most calculators. For example, the plus sign (+) is used for addition, the minus sign (-) for subtraction, the asterisk (*) for multiplication, and the slash (/) for division. To raise values to a certain power, use the caret (^) or, alternatively, two asterisks (**). For simple expressions, multiplication and division are evaluated before addition and subtraction. However, raising to a power takes precedence over all of them. You can experiment with the above symbols and rules by using S-Plus as a calculator. Suppose you want to multiply 7 by 3. Using the asterisk as mentioned above, enter > 7*3 [1] 21 and you are immediately shown the answer, 21. Note The answer in the example above includes the counter, [1]. For each line of output from S-Plus, a counter is included to number the first entry on that line of output. Thus, when variables with lots of entries are printed, it is easier to find a specific entry. ⊳ Note Now that you understand the appearance of the counter in the SPlus output, it will be omitted from (most of) the rest of the examples in the book for ease of presentation. ⊳ Practicing a little more with arithmetic operators, you see that you must pay attention to the order of operators. Thus, the example > 7+2*3 13 yields an answer of 13 because multiplication is evaluated before addition. If you really wanted the 2 to be added to the 7 before multiplying by 3,

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parentheses must be included to demarcate which will be evaluated first. Thus, in the example > (7+2)*3 27 the parentheses separate the operation that has to be evaluated before multiplying by 3. We present additional simple examples below to illustrate the above rules. > 12/2+4 10 > 12/(2+4) 2 > 3^2 9 > 2*3^2 18 These examples already show how S-Plus works; that is, commands are entered and the response follows immediately. You do not have to write a program before you start (but you may).

3.2.2 Assignments The act of assigning a value or a function to a variable name occurs all the time in S-Plus, and it is important that you are familiar with it. In mathematics, you would write x=3 to specify that a variable x equals 3. In S-Plus, there are several alternatives. The statement > x x3 Note For this reason, the underscore cannot be used as part of a variable name, as in my data. Use my.data for these cases. ⊳ Throughout the book, we use “ x=3 is perfectly valid. Normally, you also need to see the value stored in a variable. This is done in one of two ways: either enter the name of the variable or explicitly print the variable with the print function. > x 2 > print(x) 2 The first method is easier and quicker, but the latter may have to be used in certain programming situations. Note All assignments in S-Plus remain until explicitly removed or overwritten. The rm command may be used to explicitly remove a variable. For example, rm(x) removes the variable x. ⊳ Note The value of a variable may be changed at any time.

⊳

As an example of these concepts, we change the value of x and then remove it with the following: > x x 4 > rm(x) > x Error: Object "x" not found

# overwrite the previous value

# remove x

Note All assigned variables are written to disk and stored in the current working directory. This means that you can quit S-Plus at any time without having to remember to save your data, as it is done automatically. When you restart S-Plus, all data from previous sessions are still intact and ready to be used. How about trying it out now? ⊳ The previous note brings up the point that, after a while, you may begin to forget the variable names you have already used. In this case, it is very easy to use the objects function to get a list of all the variable names currently being used in the data directory. If you used the command at this point, you would find the following:


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> x objects() "x" The function has a few optional parameters whereby you can search a different database (and look at all the S-Plus functions, for example) or search for only a subset of variable names. It would be good practice with the Help system for you to find out about these options on your own.

3.2.3 The Concatenate Command: c Up to this point, you have worked only with single numbers (called scalars). Although it is possible to work in this fashion, it is often advantageous to work with vectors, which are simply ordered collections of numbers. Suppose you have the numbers 1.5, 2, and 2.5, and you want to work with the square of each value. You could save each number into a variable and square each variable, or you could put all three numbers into a vector and square the vector. The latter is much quicker if you have an easy way to combine the numbers into a vector. The concatenate command (c) can be used to create vectors of any length. You simply have to list the elements to be combined, separating each element by a comma. To perform the simple task above, we create x and square it. > x x 1.5 2.0 2.5 > x^2 2.25 4.00 6.25 The c command can be used with variables. If we forgot a number, it could be added easily. > x x 1.5 2.0 2.5 3.0 Note The c command can be used with any type of data, like the character values below. > y y "This" "is" "an" "example" The c command works with a mixture of numeric and character values. On combining different types of data, S-Plus converts them all to the most general type. Therefore, the result is a character vector. > z z "1.5" "2" "2.5" "3" "x" The consequence of converting all elements to character strings is that numeric operators may no longer be applied to the vector. ⊳ If you are running S-Plus under Windows and have had the Object Explorer open, you should have noticed that the list of variables in the data folder increased with each new variable.

3.2.4 The Sequence Command: seq It is often useful to work with regular sequences of numbers. This could include the numbers from 1 to 10, the years 1991 to 2000, or the numbers from 0 to 1 using an increment of 0.1 (0.0, 0.1, 0.2, etc.). All of these examples are easily done with the seq command. The general syntax of this function is seq(lower, upper, increment). Our three examples are then > seq(1, 10, 1) 1 2 3 4 5 6 7 8 9 10 > seq(1991, 2000, 1) 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 > seq(0, 1, 0.1) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 It is helpful to know that the default increment used by the seq function is 1. Hence, we can shorten the first two examples slightly by > seq(1, 10) 1 2 3 4 5 6 7 8 9 10 > seq(1991, 2000) 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 There is a lower default value of 1 used by the seq function, so the first example can be further simplified to > a a 1 2 3 4 5 6 7 8 9 10 The best shorthand for the seq function is to use the colon (:). Simply putting the lower value, a colon, and the upper value is equivalent to the seq function with an increment of 1. > b b 1 2 3 4 5 6 7 8 9 10


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When using the colon operator, the lack of an increment is not a problem if you are clever. > 0:10/10 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Compare the output obtained here to that from the third example above. Note Sequences can be constructed in descending order. > seq(5, 1, -1) 5 4 3 2 1 > 5:1 5 4 3 2 1 ⊳ The seq function has an option that is sometimes useful. If you know how long the sequence should be but don’t know the exact upper value, use the length option as below. > seq(0, 1, 0.03, length=15) Thus, this sequence starts at 0 and goes up in increments of 0.03 until the vector contains 15 elements. The length option can be used to obtain equally spaced intervals. Consider our example above, but suppose that we want 15 values to be equally spaced between 0 and 1 (creating 16 intervals). What would you do?

3.2.5 The Replicate Command: rep It is useful to generate data that follow a regular pattern. The replicate function takes a pattern and replicates it. The general form of the command is rep(pattern, number of times). > rep(1, 5) 1 1 1 1 1 > rep(c(0, 6), 3) 0 6 0 6 0 6 > rep(c(0, "x"), 3) "0" "x" "0" "x" "0" "x" > rep(1:3, 4) 1 2 3 1 2 3 1 2 3 1 2 3 The second option for the number of times to replicate the pattern need not be a scalar. If it is a vector, it should have the same number of elements as the pattern and then each element is replicated the corresponding number of times. > rep(1:3, 1:3) 1 2 2 3 3 3

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Another interesting example, left to the reader to try and interpret, can be constructed by invoking the rep function twice. > rep(1:3, rep(4, 3)) Note Sometimes, the length of the desired vector is known rather than the number of times a pattern should be replicated. In this case, the length option can be used in place of the number of times option. Suppose we want a vector with 10 elements using the pattern 1, 3, 2. To do this, we would use the length option as in the following example. > rep(c(1, 3, 2), length=10) 1 3 2 1 3 2 1 3 2 1 ⊳

3.3 Mathematical Operations We have already seen how S-Plus can be used as an expensive calculator by simply entering the numbers and the desired operator(s). The example shown was to multiply 7 by 3 as in > 7*3 21 In Section 3.2.2, we learned how to save numbers of interest into variables for possible future use. The above example can be redone with this idea in mind. > > > >

a x y x 2 4 6 8 10 > y 1 2 3 4 5 We now define the function, f, in terms of the vectors x and y. This will have the effect of evaluating f at each combination of x and y. Thus, the vector z - which we create - will have the same number of elements as both x and y. > z z 2.160247 2.081666 2.054805 2.041241 2.033060

3.4 Use of Brackets There are three types of brackets in S-Plus: round (), square [ ], and curly {}. More conventionally, these are referred to as parentheses, brackets, and braces, respectively. We have already seen two uses of parentheses (for function calls and for grouping in arithmetic expressions). We will investigate the use of square brackets here, and braces are covered in Chapter 9 on programming. The different types of brackets used in S-Plus are summarized in Table 3.2. Table 3.2. Different brackets in S-Plus

Bracket ( ) [ ] { }

Function For function calls as in f(x), and to set priorities like in 3*(4+2) Index brackets as in x[3] Block delimiter for grouping sequences of commands. Example: if { ... } else { ... }

Square brackets are used to index a particular vector or matrix entry. Suppose we have a vector with three elements and we want to look at or use just the first element. We simply put the number of the element of interest within square brackets, as in the following examples: > x x 0 10 20 > x[1] 0

# First element of x

3.5. Logical Values

> x[2] 10 > x[3] 20 > x[4] NA

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# Second element of x # Third element of x # Doesn’t exist # Indicates missing

The index reference that is enclosed within the square brackets can be a vector itself, indicating that more than one value is desired. > x[1:2] 0 10 > x[c(1, 3)] 0 20 This type of notation may also be used to exclude values. Suppose we want all values except the first. This is done in the same way as above, except that the minus sign is placed before the index. > x[-1] 10 20 > x[-c(1:2)] 20 > y x[-y] 20 > x[2] x 0 100 20

# Use y as index

3.5 Logical Values There are two logical values (also called Boolean values), TRUE and FALSE, which are used extensively for logic. On computers, these variable types are used a great deal for making comparisons. To appreciate the usefulness of logical variables, consider the following examples. Does 3 equal 4? > 3==4 F Obviously 3 does not equal 4, which gives us an answer of F (FALSE). Notice that the comparison was made with two equal signs. This and other comparison symbols are summarized in Table 3.3. Additional examples follow. > 3 = == !=

Function Less than Greater than Less than or equal to Greater than or equal to Equal to Not equal to

> 3!=4 T The above two lines show us that 3 is less than 4 and that 3 is not equal to 4. The exclamation point (!) is used for to negate the equals sign, so ! = gives us “is not equal to.” Now consider using logical variables with vectors. > x x < 3 T T F We see that 1 and 2 are less than 3, but that 3 itself is not. Note Logical values are coded 1 for TRUE and 0 for FALSE. As with most other languages, S-Plus stores logical variables as 1/0 values with an attached flag indicating that this variable is of type logical. With the flag and 1/0 coding, it is possible to perform both logical comparisons and numeric computation. ⊳ > x x sum(x height[height weight[weight weight[height>65] 155 160 135 Or conversely, we can look at the heights of people whose weight is less than 130.

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> height[weight x[x seq(1, 6, 2) 1 3 5 > rep(1, 6) 1 1 1 1 1 1 > rep(6, 1) 6 > rep(1:3, 2) 1 2 3 1 2 3 > rep(1:3, rep(2, 3)) 1 1 2 2 3 3

# 1 to 6, increment of 1 # 1 to 6, increment of 2 # Repeat 1, 6 times # Repeat 6, 1 time # Repeat (1:3), twice # Repeat 1 twice,...

Here are more examples of the principles from our first session using the Geyser data set of S-Plus. The Geyser data set comes with S-Plus as an example and contains data collected about the Old Faithful Geyser in the Yellowstone National Park. There are two variables: the waiting times (in minutes) and the corresponding duration (also in minutes) of the eruption. We can use the attach function to enable use of the data. We simply attach a data set and then can directly access the components contained in it.

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> attach(geyser) We are now ready to do some work and start by creating a vector with the first 10 waiting times. > wait.first10 wait.first10 80 71 57 80 75 77 60 86 77 56 We follow that by creating another vector with the next five waiting times. > wait.next5 wait.next5 81 50 89 54 90 Concatenate the vector with the first 10 waiting times with the vector containing the next 5 waiting times to create a new vector with the first 15 waiting times. It might not be the most exciting example, but it is a nice example of vector concatenation. > wait.first15 wait.first15 80 71 57 80 75 77 60 86 77 56 81 50 89 54 90 To get some practice with data editing, insert the value 100 at the beginning of the wait.first10 vector. > c(100, wait.first10) 100 80 71 57 80 75 77 60 86 77 56 Do the same type of thing by putting the value 100 between the vectors wait.first10 and wait.next5. > c(wait.first10, 100, wait.next5) 80 71 57 80 75 77 60 86 77 56 100 81 50 89 54 90 For a little practice with subsetting, we will create vectors containing the waiting times less than 50 and the corresponding durations. > short.wait mean(short.wait) 47.8125 > mean(waiting) 72.31438 > mean(short.wait.duration) 4.432292 > mean(duration) 3.460814 > short.wait.duration 4.600000 4.466667 4.333333 4.000000 4.600000 4.916667 4.583333 4.000000 4.000000 4.466667 4.450000 4.416667 4.983333 4.000000 4.333333 4.766667 Logical operators can be used to create subsets as we saw above, but remember that a double equal sign (==) is used for the logical “equal to.” > short.wait.duration==4 F F F T F F F T T F F F F T F F Replacing one value with another is something that you most often do. Such replacements can easily be done in S-Plus by working on the lefthand side of the assignment statement ( adj.short.wait.dur adj.short.wait.dur[adj.short.wait.dur==4] adj.short.wait.dur 4.600000 4.466667 4.333333 4.010000 4.600000 4.916667 4.583333 4.010000 4.010000 4.466667 4.450000 4.416667 4.983333 4.010000 4.333333 4.766667 > detach(geyser) When you’re done with a data set, it is a good idea to detach it using the detach function. We have seen in this chapter how we can use just a few simple commands and ideas to calculate some rather sophisticated results.

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3.7 Exercises Exercise 3.1 Calculate the first 50 powers of 2, i.e. 2, 2∗2, 2∗2∗2, and so on. Calculate the squares of the integer numbers from 1 to 50. Which pairs are equal (i.e., which integer numbers fulfill the condition 2n = n2 )? How many pairs are there? (Use S-Plus to solve all these questions!)

Exercise 3.2 Calculate the sine, cosine, and the tangent for numbers ranging from 0 to 2∗π (with distance 0.1 between them). The constant π is available in S-Plus and is denoted by pi. Remember that tan(x) = sin(x)/cos(x). Now calculate the difference between tan(x) and sin(x)/cos(x) for the values above. Which values are exactly equal? What is the maximum difference? What is the cause of the differences?

Exercise 3.3 Use the S-Plus help routines (not the manuals) to find out how to use the functions round, trunc, floor, and ceiling, and what they do. Predict what each of these functions will give as an answer for the numbers −3.7 and +3.8. Use S-Plus to test your predictions.

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3.8 Solutions Solution to Exercise 3.1 We first create a vector (ints) containing the integers from 1 to 50. > ints x x 2 4 8 16 32 64 128 256 512 1024 2048 ... > y y 1 4 9 16 25 36 49 64 81 100 121 ... We create a T/F vector (being the same length as both x and y) which contains a T when elements of x and y are equal and an F when they are not equal. > equal equal F T F T F ... F Now we print the values that are common in both x and y. > x[equal] 4 16 > ints[equal] 2 4 We can see which entries were the ones in common. (The second and the fourth entries of x were equal to the second and fourth entries of y.) The length function is used to determine how many elements are contained in a vector. > length(x[equal]) 2 We can do the same calculation using the sum function on the logical vector equal. > sum(equal) 2

Solution to Exercise 3.2 It helps to notice that π is a built-in constant in S-Plus, denoted by pi. You need to generate the set of numbers using the seq function and then

3.8. Solutions

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calculate the difference between the tan and sin/cos results. Note that the values below may differ on different machines. > > > > >

x matrix(data, nrow, ncol, byrow=F) Note that there are four arguments. Only the first is needed. The others are optional. If neither nrow (the number of rows) nor ncol (the number of columns) is specified, as in > matrix(1:6) 1 2 3 4 5 6 the returned argument is a one-dimensional matrix. If only one argument of nrow or ncol is given, as in > matrix(1:6, nrow=2) 1 3 5 2 4 6 the other value is determined (by simple division) and set. If both dimensions are supplied, the matrix is simply filled up. In all these situations, a dimension that does not fit the data can be specified. This is usually not what you want to do, but a matrix will be created nonetheless, and a warning will be issued. Let’s look at how this simple matrix was created and some of its characteristics. We wanted to put the integers 1 to 6 into the matrix using two rows. With six values and two rows, S-Plus does the simple division (6/2=3) to calculate that the resulting matrix will have three columns. SPlus works in terms of column vectors; hence, it fills the first column first, then the second, and, finally, the third. In addition, the matrix can be filled rowwise instead of columnwise by setting the byrow argument to T. > matrix(1:6, nrow=2, byrow=T) 1 2 3 4 5 6 You will find that this will be necessary to do when reading in your own data.

4.1. Constructing and Manipulating Data

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Try to guess what the following commands produce, then try it out. > > > > >

x data country.data country.data 227 8 1.3 1534 58 1.2 2365 82 1.8 Our matrix is not terribly useful to us in this state because we don’t know which column corresponds to which variable, or which row pertains to which country. To make our lives easier, we can give names to the rows and/or columns by using the dimnames function. This is a useful way to remember what is stored in which fields. Dimnames are used to obtain a better display and can also be used to access field entries. Since a matrix has two dimensions, let’s say n rows and p columns, the two vectors containing the dimnames that need to be attached to the matrix should have n and p elements, respectively. Each element is a character string describing the corresponding contents. Example 4.1. Attaching dimnames to a matrix In the following code, we assign row and column headings, using dimnames, to a matrix. # NULL means empty > dimnames(country.data) NULL > dim(country.data) 3 3 > countries variables dimnames(country.data) country.data Austria 227 8 1.3 France 1534 58 1.2 Germany 2365 82 1.8 1 The

data are taken from The Economist Pocket World in Figures, 1999 Edition.

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> dimnames(country.data) country.data GDP Pop Inflation 227 8 1.3 1534 58 1.2 2365 82 1.8 > dimnames(country.data) country.data GDP Pop Inflation Austria 227 8 1.3 France 1534 58 1.2 Germany 2365 82 1.8 > dimnames(country.data) [[1]]: "Austria" "France" "Germany" [[2]]: "GDP" "Pop" "Inflation" Notice that with dimnames, the actual row and column names are entered using a list structure (covered later in this chapter). However, the important thing is that, for a matrix, there are two parts to the dimnames specification: the row names and then the column names separated by a comma. The value NULL is used either to remove the dimnames of the rows or columns, or to simply specify that none will be defined. Accessing Elements of Matrices To access a piece of information in a matrix, you need to know the corresponding coordinates from the rectangular layout. In other words, we need to know the row and column numbers of the cell of interest. We know that Austria is in the first row and that population is in the second column. To find the population of Austria, we specify the following code. > country.data[1, 2] 8 We discover that the population of Austria is 8 million. Note Data are accessed from a matrix by first specifying the row number(s) and then the column number(s). ⊳ To access an entire row (column), put the number of the desired row (column) and all the corresponding column (row) numbers. For France, this becomes > country.data[2, 1:3] GDP Pop Inflation 1534 58 1.2


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A shorthand notation for this operation is to leave out the specification of all the column (row) numbers, as this is implied. > country.data[2, ] GDP Pop Inflation 1534 58 1.2 As you can see, this alternate notation leads to the same answer. You can also access data through the dimnames rather than the row and column numbers. The numbers need only be replaced by the appropriate names (surrounded by quotation marks). Some examples follow. > country.data["Germany", "Inflation"] 1.8 > country.data[, "Inflation"] Austria France Germany 1.3 1.2 1.8 Using these few simple principles, you should be able to access any part of your matrix. The same methods of accessing elements or sections of a matrix can also be used to assign or overwrite. Simply specify the elements you want to redefine and assign the new value(s). We could change the population of Austria to 10 million with the following command: > country.data["Austria", "Pop"] 8 > country.data["Austria", "Pop"] country.data["Austria", "Pop"] 10 We should set it back to its correct value though: > country.data["Austria", "Pop"] A B A+B 0 3

matrix(0:5, 2, 3) matrix(seq(0, 10, 2), 2, 3) 6 9

12 15

In S-Plus, there is one big exception to the rule of having matrices’ dimensions match. If you have a single number (scalar), it can be added to, subtracted from, multiplied by, or divided into a matrix of any dimension. Take the matrix, A, above, and define a scalar, d, to be any number. Try adding d+A. Is the result what you expected? Now, try A/d. Given what you saw with the addition, did the division give you what you expected? Matrix multiplication is more complicated and is not elementwise. The S-Plus syntax to multiply two matrices is as follows: > A %∗% B The trick to matrix multiplication is that the number of columns of the first matrix has to be equal to the number of rows of the second. Hence, in the example above, if matrix A has two rows and three columns, matrix B must have three rows but can have any number of columns. Note Matrix multiplication can be done elementwise as well. To do this, we simply use the normal multiplication sign (*). > A*B 0 8 2 18

32 50

For the resulting matrix, the first element of A is multiplied by the first element of B and so on. ⊳ In general, division is not possible with matrices. However, a common matrix computation is to take the inverse, which is possible in S-Plus using the solve function. More advanced work with matrices may be possible with the singular value decomposition. This function is appropriately named svd in S-Plus. Recall that column vectors may be multiplied (or divided), even if they both come from the same matrix. From our example with GDP, we can divide GDP by the population in the following way: > country.data[, "GDP"]/country.data[, "Pop"] Austria France Germany 24.625 23.36207 25.61728 Mathematical operations on vectors work in an elementwise fashion, unlike matrix multiplication.


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Merging Matrices Frequently, you may discover that you have forgotten a row or column (or several), and would like to add it at a later stage. To add rows or columns to matrices, simply create a new matrix containing the missing data and merge it to the existing matrix. To add extra rows, use the rbind function, and to add extra columns, use the cbind function. Suppose we forgot the variable, area (×1000 square kilometers), for the countries in our example. We could add this information to our matrix in the following way: > Area country.data country.data GDP Pop Inflation Area Austria 227 8 1.3 84 France 1534 58 1.2 544 Germany 2365 82 1.8 358 Notice that if you create a (column) vector, you can add it with no special manipulation. Also, if you give the vector the appropriate name, the dimnames are automatically updated as desired. You add a row in exactly the same manner as with column vectors, except that you use the rbind function instead of cbind. To demonstrate how this works, we will add a row of data for Switzerland. > Switzerland country.data country.data GDP Pop Inflation Area Austria 227 8 1.3 84 France 1534 58 1.2 544 Germany 2365 82 1.8 358 Switzerland 314 7 0.5 41 You can add multiple rows/columns in the same manner, but pay careful attention to the dimensions of both the existing matrix and the one to be added.

4.1.2 Arrays In S-Plus, an array is a data construct that can be thought of as a multidimensional matrix (up to eight dimensions). It is difficult to conceive of anything beyond three dimensions, but arrays may sometimes be useful to hold multiple two- or three-dimensional matrices. One nice feature of this construct is that the rules and procedures introduced for matrices hold for arrays as well.

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An array is defined as follows: > array(data, dim) The data must be a single expression like c(1, 4, 11, 18, 12, 6) and the same for the dimensions. A simple array is created by the following command: > array(1:24, c(3, 4, 2)) This produces a three-dimensional array: the first dimension has three levels, the second has four levels, and the third has two levels. When printing an array, S-Plus starts with the highest dimension and works toward the lowest dimension, printing two-dimensional matrices at each stage. The easiest explanation of how this process works is to look at an example for yourself. Example 4.2. Handling arrays Here is a short example to show how arrays work. > x x , , 1 [,1] [,2] [,3] [,4] [1,] 1 4 7 10 [2,] 2 5 8 11 [3,] 3 6 9 12 , , 2 [,1] [,2] [,3] [,4] [1,] 13 16 19 22 [2,] 14 17 20 23 [3,] 15 18 21 24 If we fix the second index at the value 2, we get a 3×2 matrix. > x[, 2, ] [,1] [1,] 4 [2,] 5 [3,] 6

[,2] 16 17 18

Notice that S-Plus prints one matrix for each level of the third dimension (highest dimension) of the array. The three-dimensional array is shown in two-dimensional slices. The above matrix of dimension 3×4×2 is shown in two slices, each of dimension 3×4. Row and Column Functions Due to the matrix layout of data that one commonly encounters, it is often required to perform certain functions on each row or on each column


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of the data. The most commonly used functions on rows and/or columns have been made into functions in S-Plus. The operations of mean, sum, variance, and standard deviation can be calculated for each column of a data matrix or for each row (not the standard deviation). The names of the functions are self-explanatory and are denoted by; colMeans, colSums, colVars, colStdevs, rowMeans, rowSums, and rowVars. The simplest way to see what the functions do is to look at a simple example. We will use the country.data data set that we have already created, and compute the mean of each of the variables (columns). > colMeans(country.data) GDP Pop Inflation Area 973 38.5 1.875 256.75 We see that the mean population of our four countries is 38.5 million. Care must be taken about the interpretation of the output from these functions. If we had used the function rowMeans instead of colMeans, we would have obtained results, but what is the mean of “Austria?” In this example, it is the mean of the GDP, population, inflation, and area of Austria, and has little meaning. In other situations, the row functions are appropriate. In the next subsection, we will explore a more general way of using any function and applying it to either the rows or columns. Apply The apply function offers a very elegant way of handling arrays and matrices. It works by successively applying the function of your choice to each row (first dimension), each column (second dimension), or each level of a higher dimension. The syntax is > apply(data, dim, function, ...) data is the name of your matrix or array, and function is the name of any S-Plus function that you want to apply to your data. For a matrix of two dimensions, the option dim can take the integer value of 1 or 2 to refer to the rows or columns, respectively. The option, ..., can be filled in with options to be passed on to the function being specified. We can use such a function to compute the maximum value of each variable from our European Countries example. Example 4.3. Using apply on arrays Here, we use the apply function to calculate the maximum of each column in our country data set. > apply(country.data, 2, max) GDP Pop Inflation Area 2365 82 1.3 544

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The max function simply returns the maximum value of the values supplied to it. Remember that any function can be given to apply so that you can use it for many purposes.

4.1.3 Data Frames One of the most used data constructs in S-Plus is the data frame. Its popularity stems from its flexibility, as it allows you to bind data vectors of different types together, such that the data can be accessed like a matrix (similar to data sets in SAS). The syntax for creating a data frame is as follows: > data.frame(data1, data2, ...) Here, the ... notation simply shows the fact that you can specify as many data sets as you want. This functionality fits in nicely with the next step in our European Countries example. You may have noticed a major difference between the countries (other than language), which we will indicate with a new character variable, namely all the countries belong to the European Union (EU) except Switzerland. We could try to use the principles from the previous section on merging matrices, but the different data types cause the following problem: > EU country.data1 country.data1 GDP Pop Inflation Area Austria "227" "8" "1.3" "84" France "1534" "58" "1.2" "544" Germany "2365" "82" "1.8" "358" Switzerland "314" "7" "0.5" "41"

EU "EU" "EU" "EU" "non-EU"

> apply(country.data1, 2, max) Error in FUN(x): Numeric summary undefined for mode "character" Dumped The matrix was created, but all entries were converted to character values, and numeric calculations are no longer possible. This may be what you want to do on a rare occasion, but more often than not, you will want to preserve the numeric nature of your matrix by creating a data frame as shown in the following example. Example 4.4. Creating data frames Here, we make a data frame out of our European Countries data set so that we can add a character variable.


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> country.frame country.frame GDP Pop Inflation Area EU Austria 227 8 1.3 84 EU France 1534 58 1.2 544 EU Germany 2365 82 1.8 358 EU Switzerland 314 7 0.5 41 non-EU For the most part, data frames can be used as if they were matrices. For instance, we can calculate the maximum values of all of the numeric components of our data frame. > apply(country.frame[, 1:4], 2, max) GDP Pop Inflation Area 2365 82 1.3 544 Note You can only apply numeric computations to the numeric variables in a data frame. ⊳ The individual variables in the above example are then referenced by one of three different notations; country.frame$GDP, country.frame[, "GDP"] (as with matrices), or country.frame["GDP"]. These forms of referencing are rather long and tedious, especially if your typing is not up to par. However, if you know that you are going to be working with a particular data frame, there is some relief in the form of the attach command. If you “attach” a data frame, you can work with the simple variable names (without the country.frame$) until the data frame is detached with the detach function. Example 4.5. Attaching and detaching a data frame When attaching a data frame, you must be aware of variables already existing in your session that might conflict with the name of a variable from an attached data frame. We will see this problem below. > attach(country.frame) > Pop Austria France Germany Switzerland 8 58 82 7 > EU "EU" "EU" "EU" "non-EU" Note that the double quotes are included when this variable is printed. This is the local version of the variable that was originally created to add to the data frame. To see the data frame version, detach, remove the local version, and attach again. > detach() > rm(EU)

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> attach(country.frame) > EU EU EU EU non-EU > detach() > EU Error: Object "EU" not found Note The local version of a variable always has preference rather than a version from the attach function. ⊳ In the example above, the local version of the variable EU was printed after the attach was made. We know this because double quotes were printed, explicitly showing us that the original character vector was used. After we deleted the local version and reattached, the double quotes no longer appear. The data frame version of the variable EU doesn’t contain double quotes because the data frame itself recognizes that the variable is of type character and dispenses with them. The attach function is much like the “set” command in a SAS data step.

4.1.4 Lists Lists are a construct that allow you to tie together related data that do not share the same structure. Recall that with data frames, the data could be of different types, but had to have the same structure. Consider again the European Countries example. Suppose we want to record all the European countries in which German is an official or major language and then associate this information with the rest of our European data. First, you need to know that there are six European countries in which German is an official or major language: Austria, Belgium, Germany, Liechtenstein, Luxemburg, and Switzerland. We now continue the European Countries example by showing how to create and work with lists. Example 4.6. Creating lists We start by creating a vector to add to our European countries data to make a list. > Ger.Lang country.list country.list [[1]]: Austria France Germany Switzerland

GDP "227" "1534" "2365" "314"

[[2]]: "Austria" "Liechtenstein"

Pop "8" "58" "82" "7"

Inflation "1.3" "1.2" "1.8" "0.5"

"Belgium" "Luxemburg"

Area "84" "544" "358" "41"

EU "EU" "EU" "EU" "non-EU"

"Germany" "Switzerland"

We now try to access some specific information. > country.list$Ger.Lang NULL > country.list[[1]]["Austria", "Pop"] 8 > country.list[[1]]["Austria", "EU"] EU Levels: "EU" "non-EU" > apply(country.list[[1]][, 1:4], 2, max) GDP Pop Inflation Area 2365 82 1.3 544 The list function works by simply “pasting” together all the pieces supplied to it. It creates one item for each supplied argument, and these are sequentially referenced using double square brackets. Notice that the syntax used with data frames (country.list$Ger.Lang) does not work with lists, and a value of NULL is returned. With some modification, however, this systax works as well. We must first assign labels as follows and then the syntax is recognized: > list(country.frame=country.frame, Ger.Lang=Ger.Lang) > country.list$Ger.Lang Individual elements in a list may be referenced by considering the double brackets as part of the name of the structure of interest. The command country.list[[1]] extracts the first list element. The referencing is then like any matrix. If you want to see the entry in the first row, second column, add [1, 2] (population of Austria). The entire reference is in the example. When this is used with a character variable, S-Plus gives you the additional information of the different levels for that variable (only two levels here). You can also gives names to the various elements in a list and then the referencing looks more like that from a data frame (e.g., country.list$Countries[1,1]). Use the names function to add names to

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a list. (A more thorough discussion of lists and how they work is covered in Section 9.1.)

4.2 Introduction to Functions One of the most valuable resources of S-Plus, and what distinguishes it from many other statistical software, is the ability to write your own functions. Functions that are built into S-Plus are called by stating the name of the function, the parentheses, and the parameter(s) being passed to the function. We have seen this often already, as in seq(1, 10, 1). The principle of defining your own functions is the same as above and can most easily be seen through a simple example of calculating the standard deviation which is the square root of the variance. Example 4.7. Creating your own function > SD y SD(y) 3.02765 Here, we see that the function has one parameter (x) that is used to define the data for which we want to calculate the standard deviation. The actual calculations of the function are contained within curly brackets and can be extended over several lines with several commands. The result of the function that should be returned to the user should be contained within the return function. By default, the result of the calculation on the last line of the function will be returned (made visible) to the user. We could have defined the contents of the function as either return(sqrt(var(x))) or y attach(country.frame) > country.frame$GDP[1] country.frame$GDP NA 1534 2365 314 The whole process seems quite straightforward. However, the problem is that certain mathematical functions in S-Plus either don’t work or they give unexpected values when applied to vectors containing missing values (NAs). Take, for example, the max function used several times in the European Countries example. When applied to the vector above, we don’t get the value of 2365 that you might have expected. > max(country.frame$GDP) NA S-Plus returns the value NA as the maximum of the GDP values. In a sense, it is being conservative because the value NA might arise as a product of other mathematical operations. In this case, the value of NA being returned as the maximum of your vector would warn you that something strange happened in your calculations. In other cases though (like ours), you don’t care about the missing value because you put it there to represent unknown information. This problem can be circumvented by determining which values are not missing, and keeping only the valid values to perform the desired mathematical function. We use the attach function here to avoid some typing. > attach(country.frame) > gdp.na gdp.na T F F F Austria France Germany Switzerland T F F F > max(GDP[!gdp.na]) 2365 > max(GDP[-gdp.na]) 2365 > mean(GDP[!is.na(GDP)]) 1404.333 > mean(GDP,na.rm=T) 1404.333 > detach() We have used the is.na function to determine which GDP values are missing (only the one for Austria). Then, using index subsetting to select certain elements, we have excluded the missing values first with the exclamation mark (used for negation), and then with the minus sign. (A more in-depth look at missing values will be presented in Section 7.5.)

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4.4 Merging Data If you have two data sets that contain a common variable, it is often desired to merge them based on the value of the common variable. The function that performs this task is appropriately named merge. We show its use below by creating two small data sets: one with names and telephone numbers, and one with names and addresses. Naturally, we want a database that has all three entries: names, telephone numbers, and addresses. To get this new database, we have to merge the original two data sets. > > > > > > > >

friends short.wait m.short.wait waiting duration 43 4.333333 45 4.416667 47 4.983333 47 4.766667 > m.long.wait 95, ] > dim(m.long.wait) 4 2

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> m.long.wait waiting duration 108 1.950000 96 1.833333 98 1.816667 96 1.800000 You can place matrices on top of one another with the rbind function. > join join waiting duration 108 1.950000 96 1.833333 98 1.816667 96 1.800000 43 4.333333 45 4.416667 47 4.983333 47 4.766667 Notice that the items in the matrix join are not sorted. Values in a single vector may be sorted directly using the sort function, as in > sort(join[, 1]) 43 45 47 47 96 96 98 108 However, if you want to sort on more than one variable, it is best to use the order function to get a permutation of the indices that will give the desired ordering when used. The following is a short example of this. > order(join[, 1], join[, 2]) 5 6 8 7 4 2 3 1 > join[order(join[, 1], join[, 2]), ] waiting duration 43 4.333333 45 4.416667 47 4.766667 47 4.983333 96 1.800000 96 1.833333 98 1.816667 108 1.950000 We specified the waiting times first and then the durations to the order function so that our new list would be sorted first by waiting time and then, in case of a tie, by duration. Look at the third and fourth entries, which both have a waiting time of 47 minutes. The row with the shorter duration comes first in the list. The variable ord contains the ways the rows should be ordered, and the join function puts them in this order.

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4.6 Exercises Exercise 4.1 Work with the European Countries data used throughout most of this chapter. • Compute the GDP per capita and add it to the data frame. • Compute the maximum of each of the variables in the data frame. • Which country has the largest GDP per capita? • Assume that the GDP is not known for France (set it to “missing”). Recalculate the GDP per capita and the maximum value for each variable. • What has now happened to the maximum of the GDP variable? Suppress the row containing the missing value and again recalculate the maximum of each variable. • Look carefully at the maximum area. To which country does this area belong? Is this really the largest country in our small sample? What happened when we excluded the row with the missing GDP for France? Determine how to correctly adjust for the missing GDP and still get correct results for the maximum values. (Hint: Use the help facility.)

Exercise 4.2 This exercise is really a continuation of the previous one. Once you have added the variable of GDP/capita to the data frame from the European Countries example, sort the data frame according to GDP/capita. Notice that it puts the smallest GDP/capita first and the largest at the bottom. Use the help utility to change the sort so that the largest GDP/capita is first and smallest is at the bottom (reverse sort).

Exercise 4.3 Create a list of European countries whose official or major language is French. Add this vector to the list from the European Countries example. Use S-Plus to produce a list of countries where both German and French are either an official or a major language. To do this, you will need to create a list where either German or French is spoken and find the countries that are duplicated in this list. (Hint 1: Use the help utility.) What are

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the physical characterics of these countries? (Hint 2: The French-speaking countries in Europe are Belgium, France, Luxemburg, and Switzerland.)

Exercise 4.4 Write a function with two parameters that offers a solution for the famous Y2K problem. The first parameter will be a two-digit year and the second will be a two-digit cutoff. The cutoff works such that if 30 is specified, then all years below (or equal to) 30 are considered to be in the 21st century (20xx) and years above 30 are considered to be in the 20th century (19xx). You will want to look at the syntax of the ifelse function.

Exercise 4.5 Continue the example on merging data shown in Section 4.4. In particular, merge the phone.numbers and addresses data frames such that: • All entries in both data frames appear • Only entries in the phone.numbers data frame appear • Only entries in the addresses data frame appear Not included in the solution is the extension that you should rename the names variable in the addresses data frame to friends, and then perform a simple merge.

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4.7 Solutions Solution to Exercise 4.1 We begin by creating a new data frame by adding a new variable to an existing data frame. Notice that we have used the round function to specify that one place to the right of the decimal point will be printed. > country.frame1 country.frame1 GDP Pop Inflation Area EU GDPcap Austria 227 8 1.3 84 EU 28.4 France 1534 58 1.2 544 EU 26.4 Germany 2365 82 1.8 358 EU 28.8 Switzerland 314 7 0.5 41 non-EU 44.9 Use the apply and max functions to calculate the maximum of each of the variables (columns). Remember that the fifth column must be excluded from the calculations because it is not numeric. > apply(country.frame1[, -5], 2, max) GDP Pop Inflation Area GDPcap 2365 82 1.3 544 44.9 We set the GDP for France equal to NA and recalculate the maximum values. > country.frame1[2, 1] apply(country.frame1[, -5], 2, max) GDP Pop Inflation Area GDPcap NA 82 1.3 544 44.9 Now we determine which rows contain a missing value and exclude them from the new maximum calculations. > gdp.na apply(country.frame1[!gdp.na, -5], 2, max) GDP Pop Inflation Area GDPcap 2365 82 1.3 358 44.9 The problem we have encountered here is that by searching for rows with missing values, we have excluded the entire row. In our example, this corresponds to all data on France. Hence, when we calculate the maximums, we find the largest area to be 358 which is that of Germany. What we really want to do is to exclude only that one missing value and not the whole row. The simple solution is found by looking at the help documentation for the max function. We find that the additional parameter, na.rm, excludes individual missing values from analyses.

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> help(max) > apply(country.frame1[, -5], 2, max, na.rm=T) GDP Pop Inflation Area GDPcap 2365 82 1.3 544 44.9

Solution to Exercise 4.2 The trick here is that we want to sort our rows according to the GDP per capita. Hence, we have to specify this variable to the order function. The order given is the order in which the rows must be printed to be ordered. > ord.gdpcap ord.gdpcap 2 1 3 4 By simply putting the new ordering into the row indices of the data frame, we get the data frame with our desired sorting. > pr.sort pr.sort GDP Pop Inflation Area EU GDPcap France NA 58 1.2 544 EU 26.4 Austria 227 8 1.3 84 EU 28.4 Germany 2365 82 1.8 358 EU 28.8 Switzerland 314 7 0.5 41 non-EU 44.9 We can’t stress enough that you must make use of S-Plus’s help facility. If you look up a related command, it often contains a cross-reference to the command you are actually interested in. Here, for example, we look at the help for the order function and it has a reference to the rev function. Using a little imagination, we might think that this will give us the reverse (descending) ordering. If you follow the reference, you will see that rev is the reverse ordering, which works in the same way as the order function, except that it orders in descending order. > help(order) > pr.rev pr.rev GDP Pop Inflation Area EU GDPcap Switzerland 314 7 0.5 41 non-EU 44.9 Germany 2365 82 1.8 358 EU 28.8 Austria 227 8 1.3 84 EU 28.4 France NA 58 1.2 544 EU 26.4 We could also look at other orderings if we were so inclined.

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Solution to Exercise 4.3 There are four European countries where French is an official or a major language. Using the list function, create the new vector and combine it with the original data frame and the vector of German-speaking countries. If you were to combine the vector of French-speaking countries with the country.list already created in the chapter, the numbering of the components would be rather strange, but it could be done. > + > >

Fr.Lang country.list1[[1]][dup, ] GDP Pop Inflation Area EU NA NA NA NA NA NA NA NA NA NA Switzerland 314 7 0.5 41 non-EU Notice that the first two lines of our list containing the characteristics of the French- and German-speaking countries is filled with NAs. This is simply because we don’t have any country characteristics for Belgium and Luxemburg.

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Solution to Exercise 4.4 The idea is that if a date of 00 arises, chances are that it refers to a current date of the year 2000 and not for a date of birth of 1900. We set up a window of uncertainty by defining the cutoff year high enough that it captures all current years but low enough so that birth dates from the beginning of the 1900s are treated correctly. A good cutoff to use might be 10. Y2K merge(phone.numbers, addresses, all=T) friends tele address 1 a 1 Street 2 b 2 NA 3 c 3 Road 4 d NA Lane Note that all the entries from both data sets appear. A value of NA is assigned to variables that have no value as a result of the merge. To include only entries from the phone.numbers data frame, we need to use the all.x option in the same manner as we did above. > merge(phone.numbers, addresses, all.x=T) friends tele address 1 a 1 Street 2 b 2 NA 3 c 3 Road Here, we have an entry for a, b, and c, but no entry for d because it doesn’t exist in the phone.numbers data frame.

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In the same manner as above, if we want to include only the entries in the addresses data frame, we use the all.y option. > merge(phone.numbers, addresses, all.y=T) friends tele address 1 a 1 Street 2 c 3 Road 3 d NA Lane Here, we have no entry for b because it doesn’t exist in the addresses data frame.

5 Graphics

Graphs are one of S-Plus’s strongest capabilities and most attractive features. This section gives an overview of how to create graphs and how to use the different parameters to modify elements and layouts of graphs. There are many options and functions available for working with graphs and graphical elements. The most important commands with their functionality are treated, and many other possibilities are mentioned. If you are interested in more details, this chapter will enable you to quickly find and understand the relevant information in the manuals provided with S-Plus or from the online help. Windows users can switch back and forth between the graphical user interface and the command line. Even if the Windows GUI is used, all actions are mapped to commands, as has been explained earlier.

5.1 Basic Graphics Commands You can start to enter graphics commands right away. S-Plus determines if a graphics device (a window on the screen or a file) is already opened. If this is not the case, it opens a window on the screen and displays the graph in it. Graphics windows are opened by using the command > graphsheet() on Windows systems. On UNIX systems the corresponding command is > motif()

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If we have a variable x and a variable y, we can plot the two variables against each other simply by entering > plot(x, y) If you don’t have two variables at hand, directly pass the data to plot. > plot(1:100, sin(1:100/10)) It’s as easy as that. The plot should now appear in the graph sheet. In the same way, a histogram of a variable x can be generated by entering > hist(x) Simple plotting using only the elementary plotting functions becomes insufficient very quickly. You might want to add lines, change the colors of the boxplot, have different symbols to plot the data, or change labels. Let us start systematically by specifying first where the output (the graph) should go. The next step is to create the graph.

5.2 Graphics Devices The graphics output depends on the device to which you send commands. Generating a line on the screen requires a different action internally than generating a line and storing it in a file. Therefore, you can either use the default graphics device (a window on the screen) or specify explicitly the device where the graph should be created. You do this by calling the appropriate S-Plus function (you can also call it the “device driver”). From now on, all commands producing graphical output send their result to the active device. The active device, if not specified otherwise, is typically the last device opened. In general, the standard graphics device on Windows system is the graph sheet, opened with the graphsheet function. A graph sheet can be used to open a window on the screen or to create a graphics file of specified format. On UNIX systems, the standard graphics window is opened by using the motif function. For opening printer devices, the standard MS Windows printer device is started with > graphsheet(format="printer") Under UNIX, you typically create a PostScript file and send it to a PostScript printer. The command to use is > postscript("filename") Of course a standard way of printing graphs is to create the graph on the screen and, once it is finished, to click the Print button in the menu bar. The graph is automatically converted into the printer format and sent off

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to the printer. Knowing about the printer device is useful, especially if a series of graphs is to be created and sent to the printer. Instead of creating a graph and clicking the Print button, creating the next graph, printing it, and so forth, all graphs can be sent to the printer directly. We list the most common devices in Table 5.1. Table 5.1. The most common output devices

System

Device function

Description

UNIX

motif postscript hpgl hplj pdf.graph graphsheet graphsheet(format="printer") postscript

Motif graphics window PostScript file HPGL file HP LaserJet file PDF file Graphics window Default printer PostScript file

Windows

Note: Many of the printer functions need a file name as argument, in order to save the output to a printer file, like in postscript (file="graphics.ps"). The functions are called by adding parentheses to their names, as in graphsheet(). After opening the graphics device, S-Plus is ready to receive graphics commands. The device is initialized, some starting values are set, and the user can create graphics elements for that device by calling the appropriate functions. An easy example is to plot the Geyser data we examined earlier. > plot(geyser$waiting, geyser$duration) If you want to close the last device opened, enter > dev.off() To close all open devices, enter > graphics.off() Note Do not forget to close the device. If you quit S-Plus, this is done for you. Closing the device is especially important if you are in the process of creating a graphics file or if you are sending graphics commands directly to a printer. Only if you close the device will the last page be ejected to the printer; otherwise, the eject page command will not be sent to the printer or added to the file, respectively. The reason for this is that S-Plus is still ready to receive graphics commands for the same page. ⊳ Note Typically, there is a menu bar, which includes the Print button. If you click on this button, S-Plus generates a file from the graph on

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the screen and sends it to the system printer. Make sure that the correct printer is installed. On UNIX systems, S-Plus keeps a file with the name ps.out.xxxx.ps in the current directory (it can be configured to remove it after printing though). ⊳ Options Device parameters can be supplied with the function call, like specifying orientation (portrait or landscape) or the height and width of the plot. All the devices have several options that are set to a default value. You can override the default values by supplying the parameter to the function. For example, if the PostScript graphics should be in portrait mode (not horizontal but vertical) and stored in a file smallgraph.ps, you need to enter > postscript(file="smallgraph.ps", horizontal=F)

5.2.1 Working with Multiple Graphics Devices S-Plus offers the possibility of working with more than one graphics device at a time. For interactive data analysis and presentations, it is very useful to have two or more graphics windows on the screen and to show interesting details by switching between them. Table 5.2 shows an overview of the most commonly used functions and provides a brief explanation of their functionality. The functions offer great flexibility of having several graphics windows on the screen while analyzing a data set. They also offer a nice toolkit for creating an impressive demonstration of self-developed applications.

5.3 Plotting Data After learning now to prepare S-Plus for receiving graphics commands, we proceed now to generating graphical output. In the following section, we present the most important commands to produce S-Plus graphs. After acquiring this basic knowledge, you should be able to create graphs with more sophisticated functions.

5.3.1 The plot Command The plot command is the most elementary graphics command. It is used to create a new figure. The plot function calculates the width and height of the figure and acts appropriately. The most elementary usage is, as shown earlier,

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Table 5.2. Commands for graphics devices

Command

Effect

dev.list() dev.cur() dev.next()

Lists all currently open devices Displays name and number of the active device Returns the number and name of the next device on the list without activating it Returns the number and name of the previous device on the list without activating it Sets the active device to the device number n. Can be combined to dev.set(dev.next()) Copies a graph to a new device, as in dev.copy(motif) or dev.copy(graphsheet) Copies the current graph to device number n. Example: dev.copy(which=2) Closes the current device Sends the current graph to the printer device Closes all currently active devices Prompts the user to confirm erasing the contents of the currently active graphics device before displaying the next graph

dev.prev() dev.set(n) dev.copy(device) dev.copy(which=n) dev.off() dev.print() graphics.off() dev.ask(ask=T)

> plot(x, y) to plot two variables x and y of equal length against each other. In this usage, the variable in the first position (x) is plotted along the horizontal axis and the one in the second position (y) is plotted along the vertical axis. You can also specify x only and omit y. If x is a vector, S-Plus plots an index vector against x (1..n, n being the length of x). If x is a matrix with two columns, S-Plus will plot the first column against the second. If x is a list containing the elements x and y, these elements are used for plotting. Every single point of the graph is displayed in the type option point, as this is the default. (We will come back to this in the next section.) You can change the character for the points by specifying the parameter pch. For example, > plot(x, y, pch="P") displays the data points with the character P.

5.3.2 Modifying the Data Display If you want to display data points in a graph, there are many ways of doing it. You might display the data as dots, stars, or circles, simply connect them by a line, or both. The data might be displayed as height bars or

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made invisible to set up the axes only and add more elements later. For this purpose, the function plot has a parameter type. Figure 5.1 shows examples of the most commonly used type options. Options for the type mode in which to plot are summarized in Table 5.3. The statements used to generate the example graph are listed for reference.

• •

•

•

0 5 10 15 20 25

-4

•

-2

• 0 x

•

• 2

4

•

• •

• •

• •

-4 0 5 10 15 20 25

•

•

-2

• 0 x

•

• 2

4

•

• •

• •

• •

-4

-2

•

• 0 x

•

• 2

4

-4

-2

0 x

2

4

-4

-2

0 x

2

4

-4

-2

0 x

2

4

0 5 10 15 20 25

•

0 5 10 15 20 25

•

0 5 10 15 20 25

0 5 10 15 20 25

Different options for the plot parameter type

Figure 5.1. Examples of different options for the type parameter of the plot function.

Table 5.3. Options for the parameter type in the plot command

type option

Plot style

type="p" type="l" type="b" type="h" type="o" type="s" type="n"

Points (the default) Lines connecting the data Both (points and lines between points) Height bars (vertical) Overlaid points and connected lines Stairsteps Nothing

See Figure 5.1 for a display of the type options.

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These are the commands used to generate Figure 5.1. > > > > > > > > > > +

x > + +

x axis(1, at=c(0, 1, 2, pi, 4, 5, 2*pi), + labels=c(0, 1, 2, "Pi", 4, 5, "2 Pi"), pos=0) > axis(2, at=c(-1, -0.5, 0, 0.25, 0.5, 0.75, 1), adj=1)

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> abline(h=c(-1, -0.5, 0.5, 1), lty=3) Finally, we add text to the graph in a specified position and adjusted to the left. > text(pi, 0.1, " sin(pi)=0", adj=0) > title("The sine function\nfrom 0 to 2 Pi") The commands used above (and more) are summarized in Table 5.5. Table 5.5. Commands that add to existing graphs

Function

Description

abline(a, b) arrows(x1, y1, x2, y2) axes() axis(n)

Line with intercept a and slope b Arrows from (x1, y1) to (x2, y2) x- and y-axes Axis at a specified side. 1=x-axis, 2=y-axis, etc. Outer box (frame) of a graph Points at the coordinates given by x and y Lines through the points given by x and y Shaded polygon figure Disconnected line segments from (x1, y1) to (x2, y2) text at the specified locations Title and/or subtitle

box() points(x, y) lines(x, y) polygon(x, y) segments(x1, y1, x2, y2) text(xpos, ypos, text) title("title", "subtitle")

Note Sometimes you will want to plot line segments. This is easy to do if you know that S-Plus interrupts a line where a missing value (NA) is encountered. The line does not connect from and to the missing value. We will have a detailed look at missing values later. Alternatively, you can use the segments function. ⊳

5.4.2 More About abline As abline is a very useful function for enhancing a graph with one or more lines, we give some application examples in Table 5.6 and have, in fact, already used it in Figure 5.3. The function abline is especially useful for drawing gridlines. If you want to have the gridlines more in the background of the picture, make them gray instead of black, such that they disturb the impression of the whole picture less. For this purpose, you need to figure out what the color coding of the gray tones on your output device is. Another approach is to add dotted lines instead of solid ones. As usual, there are

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many ways to achieve a result. You need to try out different possibilities to find the one most suitable for your application. Table 5.6. Some examples for the use of abline

Function

Description

abline(a,b) abline(l)

Line with axis intercept a and slope b If l is a list and contains an element whose name is coefficients, it is used as argument to abline Parallel horizontal lines through all the points (0, x). Can be used to add grids, as in abline(h=0:5) The analogon for vertical lines. Parallel vertical lines through all points x on the x-axis

abline(h=x) abline(v=x)

Of course, horizontal and vertical parameters can be combined. Can you think of how to create a set of parallel diagonal lines with a single S-Plus command?

5.4.3 More on Adding Axes Sometimes, it is important to have the tick marks sitting on the axes in the right position. S-Plus tries to do this in a general and almost always satisfying way, but, sometimes, you might want to have nonequidistant tick marks or to add a specific point on the axis. It is also possible to have the y-axis on the right-hand side, and to change the labels to anything desired. Basically, there are two ways of achieving the desired graph. The first possibility is to supply the corresponding parameter setting to the plot function. The possibilities for setting axis styles in the call to a plot function (not just plot) are summarized in Table 5.7. The second possibility of obtaining specific axes is to create a graph without axes and add the axes later, using the axis or axes function. In the latter case, one would enter > plot(x, y, axes=F) Now, you can go ahead and add axes in the way you want. The standard routine for doing this is axis to add a single axis, or axes to add all axes and the surrounding box. The box only can be added by using box. > box() > axis(4) The surrounding box and an axis on side number 4 is added, which is the right-hand side of the graph (see Table 5.9 for definition of side numbers). Some examples for adding specific axes are given in Table 5.8.

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5. Graphics Table 5.7. Optional parameters for plotting axes

Parameter

Description

lab=c(5,5,7) Number of ticks on x- and y-axis and label length las=0 Label style for axes. 0=parallel to axes, 1=horizontal, 2=perpendicular to axes mgp=c(3,1,0) Line of margin to place axis title, label, and axis line xaxt="s" x-Axis type. xaxt="n" omits the x-axis yaxt="s" y-Axis type. yaxt="n" omits the y-axis tck=-0.02 Tick mark length. positive numbers point into the picture xaxp= The current tick parameters start, end, and intervals lty=1 Line type. 1=solid, 2=small breaks, etc. lwd=1 Line width. 1=default, 2=twice as thick, etc. Note: The values above refer to the preset values (defaults) used if nothing else is specified. Table 5.8. Optional parameters for plotting axes using the axis function

Parameter

Description

side=n

Axis at side n, where n=1 denotes the bottom, n=2 the left, n=3 the top, and n=4 the right side Labels at position x Specification of what to put in place of the labels Shifted axis to go through the coordinate y las=0/1/2 adds labels parallel to the axis/horizontal/ rotated by 45◦

at=x labels=s pos=y las=m

Note: Of course, these parameters can be combined as in the following example: axis(3, at=(1:3)*pi, labels=c("pi", "2 pi", "3 pi")). Note that an axis typically consists of lines. Therefore, all settings for lines, like changing the thickness or the color, also apply to axes. For example, > axis(4, lwd=2, col=2) adds an axis on the right-hand side twice as thick as the standard style, and in color number 2.

5.4.4 Adding Text to Graphs Adding some text to a graph is useful if specific details of a graph should be pointed out. The standard functions text and mtext do a good job. To obtain the desired result, supply additional parameters to these functions.

5.5. Setting Options

125

Table 5.9. Optional parameters for adding text

Parameter

Description

text and mtext: Text in a figure adj=0.5 Text adjustment. 0=left justified, 0.5=centered, 1=right justified cex=1 Character expansion relative to standard size cex=2 draws characters twice as big col=1 Color to plot in. 0 is the background color crt=0 Character rotation in degrees, counterclockwise srt=0 String rotation in degrees, counterclockwise font=1 Font for plotting characters, device dependent mtext only: Text in the margin of the figure side=n Side to add text to. 1=bottom, 2=left, 3=top, 4=right outer=F Text not on the outer margin of the whole figure. If outer=T, text can be placed on top of a layout figure. For an example, see Figure 7.15 (page 213) line=n Places the text on line n toward the margin of the figure. Negative numbers lie inside the figure Note: Text is typically added by using the functions text or mtext. Some devices like text terminals cannot provide all of the functionality, like rotating text. A standard example for adding the text “extreme value” to a graph at the coordinates (1, 2), such that the text begins on the right side of the point, is > text(1, 2, "extreme value", adj=0) Sometimes, the text overlaps or touches the point on the graph just slightly. In such a case, add an extra space to the beginning of the text. The optional parameters of text and mtext offer more possibilities. They are listed in Table 5.9.

5.5 Setting Options We have already seen that almost all graphics functions accept parameters for setting options like colors, character size, or string adjustment. There are many more parameters and almost all of the functions generating graphics output accept them as arguments. They can also be set generally, such that they become the system default for all succeeding graphics commmands. For doing this, we can use the par function. Check the different options that can be set by par. Enter the command

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5. Graphics

> par() to see all options and how they are currently set. The help pages provide a very detailed explanation. We supply a brief overview of the most common parameters and their possibilities in Figure 5.4 and Table 5.10. If changed using the par function, these settings become global from then on. Nevertheless, if a new session is started they are set back to the original system defaults. 63 margin ?

*

* *

margin 2

-

*

margin 4

-

Plot 61 margin

Figure

?

Figure 5.4. The S-Plus definition of figure and plot regions.

Table 5.10. Layout parameters

Parameter

Description

fin=c(m,n) pin=c(m,n) mar=c(5,4,4,2)+0.1 mai=c(1.41,1.13,1.13,0.58) oma=c(0,0,0,0) omi=c(0,0,0,0) plt=c(0.11,0.94,0.18,0.86)

Figure size in inches. m width, n height Picture in inches, as in fin All margins in lines All margins in inches Outer margin lines Outer margin in inches Plot region coordinates as fraction of figure region x-Axis and y-axis min. and max. Useful for querying a graph’s boundaries Multiple figure layout, rowwise plotting, to generate a (m, n) matrix of pictures Multiple figures, columnwise plotting

usr mfrow=c(m,n) mfcol=c(m,n)

Note: In the left column, the letters m and n are to be replaced by integer numbers. Settings given are the system default values. To query settings, use (for example) par("usr").

5.6. Figure Layouts

127

The parameters referenced in Table 5.10 can be set by using the par function, as in the following example where the outer margin is set to be equal to 1 line on all sides, for all successive graphs. > par(mar=c(1, 1, 1, 1)) A setting can be retrieved by entering > par("mar") or > par()$mar If you run S-Plus under Windows, most of these settings can alternatively be changed by using the menu bars on top of the S-Plus window. Note You can add your own parameters to the system settings using an unused variable. The command > options() shows all settings not related to the graphics subsystem. A new variable my.personal.setting can be set to TRUE by entering > options(my.personal.setting=T) at the prompt. Your own functions can then use this parameter setting by asking for its setting as follows: > options("my.personal.setting") T ⊳

5.6 Figure Layouts The graphical parameters we looked at in this chapter are all preset with meaningful values. Nevertheless, you might want to change the default values. This section deals with the most common cases to demonstrate how to achieve satisfying results quickly. If you use a multiple-figure layout, the system may not always choose the layout as you prefer (the size of the characters, the white space between figures, etc.). We now show, for the standard layouts, how parameters are set to generate the graphics wanted. Note Creating a multiple figure graph can sometimes be cumbersome. In some situations, you need to adjust the margins to be smaller or the font size to be larger. A graph’s exact layout is still to some extent device dependent, so if you plan on creating a PostScript graph, don’t do the graph on the screen until the layout satisfies you. Adjust everything on the final output device.

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For PostScript graphs, specific options can be set using ps.options. For example, the overall font size can be set to 14 pt (1 mm is approximately 3 pt or 1 inch is approximately 72 pt) with > ps.options(pointsize=14) Alternatively, you can set the character expansion. Enter > par(cex=1.5) to make the characters 50% larger.

⊳

5.6.1 Layouts Using Trellis Graphs If you are planning to create an advanced figure layout, you should read the following sections and the next chapter on Trellis displays. If Trellis displays fulfill your needs, they can easily be composed into a multifigure graph of almost any layout. If you are already familiar with Trellis displays, jump to Section 6.4.7 (page 151). After having obtained an overview on what methods are available and how they can be used, you can take your pick. Select the method which is most convenient for getting the desired graph.

5.6.2 Matrices of Graphs A very straightforward way of adding several figures to the same graph is to declare the graph as a matrix of figures. The graph can be a 2 × 2 matrix of figures, containing 4 figures. It can either be filled rowwise or columnwise. To illustrate, a 2 × 2 figure layout can be created with > > > > > >

par(mfrow=c(2, 2)) x par(mar=c(2, 2, 1, 1)) such that there is a space of two lines on the lower and the left side of the graph, and a single line on top and on the right of the graph. Note Once the graph page is filled with figures, a new page is opened (the screen is cleared) on creating another figure. One can jump to the next figure by using > frame() Entering frame() twice leaves a field blank. Restore the default settings by allowing for a single figure per page: > par(mfrow=c(1,1)) ⊳

5.6.3 Multiple Screens Graphs Another way of creating a set of figures in the same graph is to use the split.screen function. In its simplest use, it works like the par(mfrow) command. One would enter > split.screen(c(2, 2)) to generate a 2 × 2 matrix of figures. In comparison to the par(mfrow) variant, splitting the screen provides the ability to hop around between areas. On the other hand, it does not automatically plot the next graph in the next plot area, but waits for explicit specification of another plot area. > screen(2) will activate the second screen in the display. The matrix of screens can be filled either horizontally (which is the default) or vertically by setting byrow=F in the call to split.screen. It is possible to have a matrix of differently sized graphs or even any layout. If a new graph is started, its initial coordinates range from 0 to 1, similar to 0 - 100%). Each screen in the graph is defined by its lower left and top right corner. The following command defines a layout of a 2 × 1 matrix, where the first row stretches from 0 to 1 on the x-axis and from 0 to 3/4 on the y-axis. The second row stretches from 0 to 1 on the x-axis and from 3/4 to 1 on the y-axis. > split.screen(figs=matrix(c(0, 1, 0, 3/4, 0, 1, 3/4, 1), + 2, 4, byrow=T))

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Note that the matrix argument to figs must have four columns. Each row defines the corners of one of the screens. The screen layout is closed by using close.screen. > close.screen(all=T)

5.6.4 Figures of Specified Size An alternative to what we have looked at so far is to fill an empty graph successively with figures and define the size of each figure as we go (i.e., right before the figure is created). We need to open a new graphics page, which is done by entering the command > frame()

# open next graph

After having opened a new graphics page, the coordinates of the plot are set to (0,0) for the lower left corner and (1,1) for the top right corner. Let us now plot the Geyser data set that comes with S-Plus. We want to set up a figure such that a scatterplot of the data appears as the main plot with the addition of a histogram of the x data on top of the scatterplot and a boxplot of the y data to the right-hand side of it. The layout we have in mind looks like the one in Figure 5.5. The plot itself should have the lower left corner coordinates (0, 0) and the top right corner coordinates (0.7, 0.7). Therefore, the figure on top of the data plot has the lower left corner (0.0, 0.7), and the upper right corner is (0.7, 1.0). For the vertical boxplot on the right side, we get the lower left corner (0.7,0) and the upper right corner (1.0, 0.7). (0, 1)

(0.7, 1)

(1, 1)

Histogram (0, 0.7)

Graph of (x,y)

(0, 0)

(0.7, 0.7) (1, 0.7)

Boxplot

(0.7, 0)

(1, 0)

Figure 5.5. A customized graph layout.

By experimenting with the different sizes and the borders, we discover that it is better to have the figures overlapping a little bit.

5.6. Figure Layouts

> > > + > > > > >

131

frame() par(fig=c(0, 0.7, 0, 0.7)) plot(geyser$waiting, geyser$duration, pch="*", xlab="Waiting Time", ylab="Duration of Eruption") title("\nOld Faithful Geyser Data Set")1 par(fig=c(0, 0.7, 0.65, 1)) hist(geyser$waiting) par(fig=c(0.65, 1, 0, 0.7)) boxplot(geyser$duration)

0

40 80

Finally, we obtain Figure 5.6.

40

60

80 geyser$waiting

100

5 4 3

• 50

60

70 80 Waiting Time

2

•

1

• • • • • •• ••• •• •• • • •••• • • • •••• •• • ••• • ••• • • • • • •••• •• • • •• •••• ••• •• ••• • • • • • ••••• ••• ••••••••• • • •• •• ••••••• •••••• ••• •••• ••••••••••••••• ••••••• • • • ••• •• • • • • • • • • • • • • ••• •• • •• • •• • • • • •••••••••••••••••••••••••••••• ••• • •••• •••••• ••• • •

1

Duration of Eruption 2 3 4

5

Old Faithful Geyser Data Set

90

100

110

Figure 5.6. A customized display of the Geyser data set.

1 The

control character \n is a “linefeed” character, advancing to the next line. In this way, the title gets closer to the graph.

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5. Graphics

5.7 Exercises Exercise 5.1 In a single figure, plot the functions sin, cos, and sin + cos with different colors and line styles. Use 1000 points in the interval [−10, 10] and label the figure with a title, subtitle, and axis labels.

Exercise 5.2 Construct two vectors, x and y, such that the S-Plus command > plot(x, y, type="l") creates the following figures: a) a rectangle or square b) a circle c) a spiral Hint: Do not think of y as being a function of x (in a mathematical sense), but think of how the trace of the figure can be created. Hint for drawing circles: you might use the property that for any x, a point (sin(x), cos(x)) lies on the unit circle with origin (0, 0) and radius 1.

Exercise 5.3 We consider the so-called Lissajous figures. They are defined as z(x) =

sin(ax) z1 = , sin(bx) z2

or, in different notation, z1 (x) = sin(ax) z2 (x) = sin(bx) a, b positive integers, x between 0 and 2 π. Plot z1 against z2 , using lines to connect the points, and choose different pairs of values for a and b. Plot several figures on a single sheet. On what do the forms of the curves depend? Compare, for example, the figures for (a, b) = (3, 4), (3, 6), and (6, 8).

5.8. Solutions

133

5.8 Solutions Solution to Exercise 5.1 What we need to create this graph is a sequence of x values, for which we calculate the values sin(x) and cos(x). If several functions are to be plotted within one picture, the ordering of plotting the figures is important. The arguments to the plot function, usually the first curve, determine the boundaries of the figure. The following curves are then added to the existing picture within the existing limits. If you want to see more of the sin(x) + cos(x) curve than that which fits within the boundaries of sin(x), extend the y limits by using the option ylim of the plot function or - and this is easier - plot the curve sin(x) + cos(x) first.

-1.5

-1.0

-0.5

y values 0.0

0.5

1.0

1.5

Trigonometric Functions

-10

-5

0 x values sin(x), cos(x), and sin(x)+cos(x)

5

10

Figure 5.7. Trigonometric functions.

> > > > > +

x > > > > >

x > > > +

plot(xlimits, ylimits, type="n", xlab="x values", ylab="y values") lines(x, y1) lines(x, y2, lty=2, col=2) lines(x, y3, lty=3, col=3) title("Trigonometric Functions", "sin(x), cos(x), and sin(x)+cos(x)")

5.8. Solutions

135

Solution to Exercise 5.2 The idea of this exercise is to learn how to think in terms of geometric figures and not in terms of mathematical functions, where y is a function of x. (a) A rectangle is a simple figure consisting of four lines. In order to draw a rectangle, we need to create two vectors x and y containing the coordinates of the corners. These vectors must have five elements, as the line has to return to the first point to complete the figure.

A rectangle

Figure 5.8. Geometric figures (a): A rectangle

> > > >

x > > > >

z > +

z > >

0.0 sin(3 * x)

0.5

1.0

-1.0

-0.5

0.0 sin(3 * x)

0.5

1.0

-1.0

-0.5

0.0 sin(3 * x)

0.5

1.0

-1.0

-0.5

0.0 sin(7 * x)

0.5

1.0

sin(8 * x) -1.0 -0.5 0.0 0.5 1.0

-0.5

sin(11 * x) -1.0 -0.5 0.0 0.5 1.0

-1.0

Figure 5.11. Lissajous figures for different values of a and b.

6 Trellis Graphics

Cleveland (1993) introduced the idea of Trellis displays as a graphical way of examining high-dimensional structure in data by means of conditional one-, two-, and three-dimensional graphs. The problem addressed is how observations of one or more variables depend on the observation of other variables. To answer this question, the data are subset (split up into groups) according to the observations of the conditioning variables – the ones that are supposed to have an influence on the others. Take an example. We want to determine how the relation between body weight and height of a set of people depends on their sex. To inspect this relationship graphically, we can split up the data into the groups male and female. Each of the two groups is displayed in a separate graph instead of blurring the sex effect in a single graph. If we had another variable available, say age group, observed as young, intermediate, and old, we could subset the data again. For each combination of observations for age and sex, six altogether, we generate a separate display of body weight and height. Arranging the displays in a reasonable way, there is a lot of information that can be derived. A Trellis graph layout is such an array of graphs, arranged as sketched in Figure 6.1. We can get the basic idea by looking at Figure 6.1. If the relationship between body height and weight for males is to be compared for different age groups, we need to compare the graphs in the first row of the graph matrix. How would we answer if there is - for young men and women - a different relationship between height and weight? Well, simply by comparing the two graphs in the leftmost column.

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6. Trellis Graphics

sex=male age=young

sex=male age=intermediate

sex=male age=old

graph of weight, height



sex=female age=young

sex=female age=intermediate

sex=female age=old




Figure 6.1. An example of a Trellis graph layout.

If we switch to the middle column, we can graphically compare the relation between height and weight for the middle age group. And, finally, the rightmost column gives us an idea of how different the height/weight relation is for old men and women. As an aside, Figure 6.1 displays the origin of the name Trellis, based on its layout of graphs. Altogether, a graph as sketched in Figure 6.1 displays the four-dimensional structure in the variables height, weight, sex, and age. Note Trellis graphs are implemented somewhat differently from the other graphics in S-Plus and this is why we treat them in a separate section. All Trellis graphs follow the same structure, have a consistent interface and parameter setting, and offer features like saving graphs in variables. You might find Trellis graphs more appealing than the traditional graph routines, just because of the layout, rendering, and default coloring. Therefore you may want to use Trellis graphs as your default graph type. ⊳

6.1 An Example The data set singer contains measurements of the heights of singers in the New York Choral Society together with their assigned voice part. Entering > singer displays the entire data set. You see that it consists of two variables of name height and voice.part. We are interested in figuring out if the height of the singer differs between voice part groups. This is exactly the type of question for which Trellis has appropriate displays. We can get a set of histograms, one for each voice part’s height, by entering

6.1. An Example 60 Alto 2

65

70 Alto 1

75

60 Soprano 2

65 70 Soprano 1

141

75

40

30

Percent of Total

20

10

0 Bass 2

Bass 1

Tenor 2

Tenor 1

40

30

20

10

0 60

65

70

75

60

65

70

75

height

Figure 6.2. Heights of singers for the voice parts in the New York Choral Society.

> histogram(∼height | voice.part, data=singer)

The histogram is shown in Figure 6.2.1 The graph easily reveals the basic structure of how the height of the singers differs for different voice parts. The height of the bass and tenor voice parts is typically larger than the height of the alto and soprano parts. Of course, this leads directly to the suspicion that the voice part is just an indicator for the sex of the singer. One would want to obtain the data for sex to condition on male and female and redo the graph. Unfortunately, the data are not available. Note some specificities of the call to the histogram and other Trellis graph functions. At first, you might have realized already that we called a function histogram to generate these histograms, although we have used the function hist before to generate a single histogram. histogram is an extension of hist, allowing for Trellis-type displays. 1 The strip background color is modified to make the labels more legible. This will be explained shortly. The commands used are the following.

> > > > +

stripcol xyplot(y∼x) is sufficient to display the graph on the graphics device. Note that the command xyplot(y∼x) on the command line calls the print function, when entered on the command line. The following note points out a pitfall when the graph is not created from the command line. Note If a Trellis graph function is not called from the top level (the interactive environment), a statement like histogram(∼x) calls the function histogram. The function histogram generates a histogram figure and returns it to the calling function without displaying it. In those cases, calling the print function explicitly is necessary to display the graph, as in print(histogram(∼x)) Otherwise, there will be no display of the graph calculated. This is important to keep in mind when writing functions and scripts. You might want to note that this is consistent with S-Plus’ behavior in other cases. Entering a statement like 3*5 on the command line returns the result; the same statement embedded in a block (like a function, as we will see later) forces S-Plus to calculate the result, but it will not be displayed unless you use print. Here is an example: > x xyplot(sin(x)∼x) > title("Sine Curve") Now, enter the following and observe closely what happens. > { xyplot(cos(x)∼x); title("Cosine Curve") } Try the same and omit the curly brackets.

⊳

6.3 Output Devices We have seen that Trellis graphs can be created without actually printing them. Mostly though, graphs are created to be displayed either on the screen or on a printout. Trellis uses the standard S-Plus output devices with slight modifications to their settings. In order to print a Trellis graph, open an output device (a graphics window on the screen or a file on disk), and send the Trellis graph to the device. The general way of opening a device is to call trellis.device with the device to open as a parameter. If a query to show a graph is submitted to S-Plus and no device is open,

6.3. Output Devices

145

S-Plus opens the standard device, a graphics window on the screen. For example, > trellis.device(motif) opens a motif graphics window on a UNIX workstation. > trellis.device(postscript, file="trellis1.ps", color=F) opens a PostScript file, and > trellis.device(graphsheet) opens a graphsheet window on Windows systems. An overview of some Trellis devices available is given in Table 6.2. Table 6.2. Trellis graphics devices

System

Trellis call

Device

Windows

trellis.device(graphsheet) trellis.device("printer", format="EPS")1,2 trellis.device("printer", format="clipboard")1,2 trellis.device(motif) trellis.device(postscript)1

Graph Sheet window PostScript printer

UNIX

Windows clipboard Motif window PostScript device

Note: 1 To specify the file to write to, add file="filename" to the function call. 2 Windows has many more file formats it can write, like GIF and JPEG. See the documentation on graphsheet. After having started a graphics device, we can generate a new Trellis graph and display it right away; for example, by calling > histogram(∼height | voice.part, data=singer) We may also display a figure that has been created beforehand and saved in a variable. > graph.singers print(graph.singers) Note The function trellis.device with a device as argument does nothing but start the device with settings that are specifically customized for Trellis displays. These include margins, colors, and other styles. One may also display Trellis graphs on the noncustomized devices that are opened (for example) by calling motif(), but the output will not be as nice as on a Trellis display. ⊳

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6. Trellis Graphics

All comments for other devices hold for Trellis devices as well. You can open a set of devices, switch between them using dev.list() and dev.set(), and copy from one device to another using dev.copy. If you want to close the last device opened, enter > dev.off() To close all open devices, enter > graphics.off() If the current device is a graphsheet, it will be cleared before a new function call generates new graphs. Multiple graphs created during a single function call are all shown on separate pages of the graphsheet. When the next graphics generating function is called, the graphsheet is reset to an empty state. If you want to keep the graphs, just open another graphsheet. The last device opened is automatically the current device. Note Our warning about closing devices before printing the graphics is valid for Trellis devices as well. See the note on page 115. In case you have entered graphics commands and do not get a full graph, check if you closed the device (enter dev.list()). If the device was closed, check if you put a print statement around the call to the graphics function. ⊳

6.4 Customizing Trellis Graphs 6.4.1 Setting Options To see the current device settings in form of a graphical display, use the function show.settings. > show.settings()

# show device settings

A graph is displayed where each plot shows the settings for one specific option list. The option names are shown below the graph. They are listed in Table 6.3. Retrieve the specific component you are interested in and look at its settings. In the following example, we retrieve the settings for the line width in Trellis graphs, change its value, and store the settings back to the system settings such that they become active for all lines in Trellis displays. > trellis.plot.line trellis.plot.line$lwd trellis.par.set("plot.line", trellis.plot.line) For all subsequent graphs of the current session, line widths will be “3.”

6.4. Customizing Trellis Graphs

147

Table 6.3. Trellis plot options

Option name add.line box.dot dot.line plot.line strip.background wires

add.text box.rectangle dot.symbol plot.symbol strip.shingle

bar.fill box.umbrella fill reference.line superpose.line

box.3d cloud.3d pie.fill regions superpose.symbol

Note: The settings above are retrieved and set using trellis.par.get("option") and trellis.par.set("option", ...).

6.4.2 Arranging the Layout of a Trellis Graph If the Trellis standard graph layout needs to be modified, there are several options. All following descriptions refer to arguments to Trellis plot functions that can be passed when calling the function. They are not separate functions.

6.4.3 Layout If the conditioning variable has eight categories, for example, one might not want to plot them on a 2 × 4 or 4 × 2 arrangement, but rather in a 3 × 3 layout with one of the fields left blank. The layout is specified by using the layout parameter to define the overall layout, in this case by setting layout=c(3,3) in the call to the Trellis function. Where the graphs are positioned or if a field is left blank is indicated by a vector of logical values. A T (TRUE) in the skip argument indicates that the field is to be left blank. We use the singer data to generate the layout described. > histogram(∼height | voice.part, data=singer, + layout=c(3,3), skip=c(rep(F,4),T,rep(F,4)))2 The result is shown in Figure 6.3. The first value of layout determines the number of columns in the graph, the second the number of rows (just in contrast to the specification of a matrix dimension). As a 3 × 3 matrix consists of 9 fields, the parameter skip is a logical vector of length 9 with a T in position 5 to indicate that field 5 is to be left blank. If layout is a vector of length 3, the third value defines the number of pages to generate. In the same way, one can change the aspect ratio of a graph. The aspect is the ratio of the height and width of a graph, or to be precise, aspect ratio = y-axis length/x-axis length. 2 The

footnote about changing the strip color (page 141) holds here as well.

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6. Trellis Graphics

60 Alto 1

65

70 Soprano 2

75 Soprano 1

40 30 20 10 0

Alto 2

Percent of Total

Tenor 1

40 30 20 10 Bass 2

Bass 1

0

Tenor 2

40 30 20 10 0 60

65

70

75

60

65

70

75

height

Figure 6.3. Customizing a Trellis graph arrangement.

The aspect ratio is important in the perception of a graph. Setting it to 1 creates a graph where the x-axis and y-axis have the same length. Setting it to "xy" generates a graph with a 45◦ banking. To make the x-axis 25% longer than the y-axis, set aspect=1/1.25. For 3D figures, you need an aspect argument of two values, defining height and depth relative to the width of the graph. Note that you don’t need to care about the size of the graph in inches or pixels; this is still determined by S-Plus to use the space on the page optimally.

6.4.4 Ordering of Graphs You might have noticed that S-Plus has a standard way of arranging the graphs belonging to certain levels of the conditioning variables. If there is only a single conditioning variable, the graph that conditions on the first level of the conditioning variable shows up in the lower left corner. The second graph is to the right of it, until the bottom row is filled up. The second row from the bottom is filled next, and so it continues until all rows are filled. Check out the arrangement of levels in the original data by


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querying the levels of the conditioning variable. For the Barley data set, the levels of the variable site are obtained as follows: > levels(singer$voice.part) "Bass 2" "Bass 1" "Tenor 2" "Alto 2" "Alto 1" "Soprano 2"

"Tenor 1" "Soprano 1"

The ordering of the graphs – how the matrix of graphs is filled up – is shown in Figure 6.4 (left).

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Figure 6.4. Default ordering of graphs in a 3 × 3 Trellis arrangement with a single conditioning variable (left) and ordering of graphs setting as.table=T (right).

Note You might wonder why the ordering is not in the way one reads a book - from the top left to the lower right. You can arrange the graphs in this way by setting as.table=T. The default arrangement is as described, because it creates the analogy to numeric x-axis and y-axis in putting the first (the smallest) value to the left on the x-axis and to the bottom on the y-axis. This is especially important if the categories are ordered. If the categories represent ordered factors like low, medium, and high education or income, placing the “lowest” category lowest (physically) in a graph makes the reading of the graph much more intuitive. For example, the group with low education and low income is in the bottom left corner of the graph, whereas the group with high education and high income is in the top right corner, where the high x and y values are found on a numeric scale. ⊳

6.4.5 Changing Graph Elements Changing of specific elements of a graph can be done by setting the corresponding parameters for all Trellis graphs created during a session using

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trellis.par.get and trellis.par.set (as discussed earlier). If the graph element should only be modified for a single or a few graphs, it is easier to supply the settings with the call to the function. Look up the documentation on trellis.args. Suppose we want to have larger characters on the axis ticks in a scatterplot. > xyplot(duration∼waiting, data=geyser, + scales=list(cex=1.5, tick.number=10)) generates the graph with the axis tick labels 50% larger than the standard size (1.5 times the size) and approximately 10 tick marks. S-Plus might alter the specification slightly if the number of ticks scales better. If we wanted to set different specifications for the x-axis and y-axis, we could enter > xyplot(duration∼waiting, data=geyser, + scales=list(x=list(cex=1.5, tick.number=10), + y=list(cex=1, tick.number=5)))

6.4.6 Modifying Panel Strips The panel strips are the horizontal bars on top of each graph in a Trellis layout. They indicate the conditioning variable’s values for the graph. For the 2D Trellis functions, there is an argument strip: an optional specification of a function that draws the strips across the panels. It is preset with a call to the function strip.default. Take the example of calling dotplot. A call to dotplot like > dotplot(year∼yield | site, data=barley)

is equivalent to

> dotplot(year∼yield | site, data=barley, + strip=function(...) strip.default(...) ) If you are interested in all details, have a look at strip.default or check only the arguments of the function using > args(strip.default) You will see that strip.default is pretty long, but we can make our life simple. We can define a new strip function that simply calls strip.default with different settings. By looking at the documentation for strip.default, we find that the setting we are interested in for this example is the style setting. We want to set it to the value 1 to achieve that “the full strip label is colored in background color and the text string for the current factor level is centered in it” (documentation excerpt). > dotplot(year∼yield | site, data=barley, + strip=function(...) strip.default(..., style=1) ) does the job for us.


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6.4.7 Arranging Several Graphs on a Single Page

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If you use Trellis graphs, you can experiment with the arrangement of the graphs by creating the graphs and storing them in variables. Once the graphs are generated, they can be moved around on the page. There are basically two ways of getting a multigraph page. First, a matrix of graphs with all graphs in rectangular equally sized areas can be generated. Second, we can give the separate graphs different sizes and any location on the page. Let us start with the first layout type.

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Figure 6.5. A multigraph page of Trellis displays.

> > > > > + > > > >

graph1 >

x key(text=list(c("Curve 1", "Curve 2")), + lines=list(col=2:3, lty=1:2)) does just the opposite.

⊳

Look up the manuals for a series of further examples. The manual and the online help provide a full list of all available options, among them title for putting a title string on top of the key, border for framing the key in a specified color, background for specifying different background colors for the key or making it transparent, and columns for specifying an arrangement of the key entries in multiple columns.

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6.5.4 The subscripts Option in Panel Functions All panel functions can have an additional parameter called subscripts. If we specify the panel function as > panel=function(x, y, subscripts, ...) { ... }

the indices of the data used for the current panel will be passed to the panel function. For a data frame, the indices correspond to the row numbers. In this way, further variables can be passed to the panel function, and the subscripts can serve to extract the data for the current panel. For example, if a data set consists of age and body weight for a group of people, we might also have the names of the people and their sex. We can plot age against body weight for the two sexes. If we want to add the names of the people to the graph, we need the subscripts for the male and female people in graphs 1 and 2, respectively, in order to extract the names.

6.6. Exercises

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6.6 Exercises Exercise 6.1 Examine the data set singer. The variable voice.part is a factor with eight levels, Soprano 1 and 2, Alto 1 and 2, Tenor 1 and 2, and Bass 1 and 2. Reduce the levels in voice.part to four levels only by putting Soprano 1 and Soprano 2 into a Soprano category, Alto 1 and Alto 2 into an Alto category, and so on. Graph the data in a Trellis histogram. Hint: Check for the levels of the factor variable by using the levels function. Convert the factor variable that is internally coded as integer to integer numbers and use cut for regrouping the data.

Exercise 6.2 Create Trellis histograms for 50 random Normal and random uniform numbers. Add a key to the graph that displays the number of observations, mean, and variance for each group (each panel). Hint: Use the function make.groups to generate the data together with a conditioning variable. The functions split and sapply can be used for calculating the summary statistics.

Exercise 6.3 Write a panel function that shows the following characteristics of a data set graphically: • Minimum and maximum as color bars • Mean and standard deviations centered around the mean as lines within the bars To be more precise, Figure 6.8 shows what we want to achieve.

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i h g f e d c b a -3

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Figure 6.8. Trellis display of data spread, mean, and standard deviations.

6.7. Solutions

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6.7 Solutions Solution to Exercise 6.1 In order to categorize the factor variable voice.part into new categories, let us first look at what levels the variable currently has. > levels(singer[,"voice.part"]) "Bass 2" "Bass 1" "Tenor 2" "Alto 2" "Alto 1" "Soprano 2"

"Tenor 1" "Soprano 1"

Factor variables are stored as integer values with assigned descriptions. We can display the integer values by converting the factor to an integer. > as.integer(singer[,"voice.part"]) 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

8 7 6 6 5 3 2 2 1

8 7 6 6 5 3 2 2 1

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We group the data into new classes by using the cut function. We specify the breaks in between the integer values, such that it is clear to what group a value is assigned. The function cut takes a labels argument for assigning tags to the groups. It does not generate a factor though, such that we must convert the grouped data to factors explicitly. > + + >

voicepart.4gr singer singer[, "voicepart.4gr"] rm(voicepart.4gr) # remove global variable The last statement, removal of the variable voicepart.4gr as separate variable, is important. You can easily generate confusion if there is a variable of the same name (voicepart.4gr) in the interactive environment and in a data frame (singer) of the same name.

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Solution to Exercise 6.2 We first generate the data we want to display. We then derive the statistics (number of observations, mean, and variance) we want to display and add them to the graph as a key later. Let us start with the creation of the data. x > > >

par(mfrow=c(2,3)) attach(barley) hist(yield[site=="Morris"], main="Morris") hist(yield[site=="Crookston"], main="Crookston") ...

7.2. Graphical Exploration

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Figure 7.1. Histogram of the variable yield in the Barley data set.

Alternatively, we could use the split function that splits a variable according to values of another variable. > yield.split par(mfrow=c(2,3)) > lapply(yield.split, hist) With both solutions, we did not make sure that we get the same histogram bins. The bins need to be calculated beforehand and supplied as a parameter to each call of the hist function. We better stop here and be happy that we can use the histogram function that comes as part of the Trellis library. Continuing our investigation, we condition the yield on year and site. We create histograms for each combination of site and year. As there are 6 sites in the data sets, each having two yields for 1931 and 1932, we obtain 6 × 2 = 12 histograms. They show 10 values for the 10 different varieties planted. We use the Trellis function histogram again to condition the yield data according to the values of year and site. Figure 7.3 shows the histogram display by year and site. The figure was obtained by entering > histogram(∼yield | year+site)

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Figure 7.2. Histogram of the barley yield by site.

In this way, we are able to review our data set in a concise form. The histograms can be compared columnwise, observing the yields for the different sites in 1931 and 1932, respectively. If they are viewed rowwise, we can see the different yields for the 2 years for a given site. Let us now graphically compare the sites’ yields for the two harvests in 1931 and 1932. We choose the boxplot display to show all sites for a specific year in the same plot. By reading the online documentation or the manuals, we find that if we want to display more than one data set in the same picture, the boxplot function expects a list of variables to plot. What we need to do is create a list containing one vector of yields for each of the sites. We have learned that the S-Plus function split splits a vector according to another variable, which we are going to use now. We split the yield by site, as we have done before, and supply the outcome to the boxplot function. Since we want to produce two plots, one for 1931 and one for 1932, we extract the data for the year first, then split the extracted data. > > + >

# True or False for 1931/1932 is.1931 > > >

yield.split.1931 bwplot(year∼yield | site, data=barley)

Using these Trellis displays, Cleveland (1993) and Becker, Cleveland, and Shyu (1996) conclude that the years 1931 and 1932 must have been swapped for Morris. Of course, this can no longer be proven, but there is some clear

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Figure 7.6. The Barley data in boxplot displays, categorized by site.

evidence for it. Although this data set has undergone many analyses in the statistical literature, this obscureness had not been detected, which shows how graphical methods can help provide more insight into a data set. To illustrate further techniques, we examine the Geyser data set in detail. This data set was previously introduced in Section 3.6. The Geyser Data Set The Geyser data consist of continuous measurements of the eruption length and the waiting time between two eruptions of the Old Faithful Geyser in Yellowstone National Park in 1985 in minutes. Some duration measurements, taken at night, were originally recorded as S (short), M (medium), and L (long). These values have been coded as 2, 3, and 4 minutes, respectively. The original publication by Azzalini and Bowman (1990) provides more details. Having a look at the geyser data (have a look!) shows that they are stored in a list with two components: waiting and duration. Lists can be attached, such that we can use waiting instead of typing geyser$waiting all the time. > attach(geyser)


189

We might want to begin by examining the basic data characteristics. > summary(waiting) Min. 1st Qu. 43.00 59.00 > summary(duration) Min. 1st Qu. 0.8333 2.0000

Median 76.00

Mean 3rd Qu. Max. 72.31 83.00 108.00

Median Mean 3rd Qu. Max. 4.0000 3.4610 4.3830 5.4500

This gives us a basic impression. The duration of an eruption lies between less than a minute and almost 6 minutes, and the waiting time for the next eruption lies between a little more than 40 minutes and more than 100 minutes. On average, if you just missed the last eruption, then you would have to wait for more than an hour to see an eruption of approximately 4 minutes. Next we are interested in a graphical data display. We display the two variables in a simple scatterplot. > + + > +

plot(waiting, duration, xlab="Waiting Time for the Eruption", ylab="Eruption Length") title("Old Faithful Geyser Data\n Waiting Time and Eruption Length")2

Figure 7.7 shows what the data look like. We can see some surprising details from the graph. We already know that parts of the data were set to 2, 3, and 4 minutes, such that quite a few points lie on these lines parallel to the x-axis. Second, it seems that the data also lie on lines parallel to the y-axis. Examining the data confirms that waiting time was measured in integer minutes. Next, we see three clusters of point clouds. Interestingly enough, they are more or less clearly separable. We can use an interactive facility to determine the cluster’s boundaries. Plot the data and use the mouse to click on some points in the graph. Calling the function locator, S-Plus will return the coordinates of the points you click on with the left mouse button. Stop recording the clicks by clicking on the right mouse button. > plot(waiting, duration) > locator() If we cut the x-axis at x=67 and the y-axis at y=3.1, the plot is divided into four areas, three of which contain a cluster of data points, and the fourth, the one in the lower left position, contains no data at all. This might require questioning the Park Ranger who collected the data. To show it to the ranger, we plot the graph again and add the dividing lines to it. Furthermore, we choose a different plotting character for each 2 The

title command should be on a single line with no space around the \n, which is impossible here due to space constraints.

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Old Faithful Geyser Data Waiting Time and Eruption Length • •

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Figure 7.7. The Old Faithful Geyser data.

of the four subsets, and label the extreme value in the lower right corner (the only one with a waiting time of more than 100 minutes). The following commands produced Figure 7.8. We first plot a graph with no points, which contains everything except for the points (the axes, labels, and other layout parameters). We then add the horizontal and vertical lines and, finally, the points. > + > + > >

plot(x, y, type="n", ylab="Eruption Length", xlab="Waiting Time for the Eruption") title("Old Faithful Geyser Data\nWaiting Time and Eruption Length")3 abline(h=3.1) abline(v=67)

The data are divided into three subcategories, which we determine by the logical expressions subset1, . . ., subset3. These vectors of logical TRUE and FALSE elements determine whether or not a point belongs to the subset.

3 The

title command should be on a single line, with no space around the \n, which is impossible here due to space constraints.


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Old Faithful Geyser Data Waiting Time and Eruption Length C C CC CC C C C CC C C C C C C CC C C C C CC C CC C C C C C C C C C C C C C C CCC CC C C CC CC CC CC C C CC C CC CCCC CCCCC CC C C

B B BB B B B B B B BB BB B B B BB B B B BB BBB BB B BB B B B B BB BBBB BB BB BBBB BBBBBB B B B B BB B BBB B B B B B B A A A A A A A A A A AA AAA AAA AA A A A A A A A A A A A A A A AA A A A A AA AAA AAA AAA A AA A A A AAA A AA AAAAAA AA A A AA A A

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Figure 7.8. The Old Faithful Geyser data in subsets.

> subset1 = 67) & (duration < 3.1) > subset2 = 67) & (duration >= 3.1) > subset3 = 3.1) The next step is to plot the groups separately by using the points function, but displaying different characters (instead of points) and using different colors to demarcate the groups. > points(waiting[subset1],duration[subset1],pch="A",col=1) > points(waiting[subset2],duration[subset2],pch="B",col=2) > points(waiting[subset3],duration[subset3],pch="C",col=3) Finally, we add an arrow pointing to the extreme point with a waiting time of about 110 minutes. Note that we do not determine the end point of the arrow, but go to its waiting time (which we know is the maximum value) and the corresponding duration (by selecting the index of the maximum waiting time). We want a V-shaped arrow; therefore, open=T is set. Finally, we place the text “extreme value” on the bottom line of the plot, ending where the arrow starts (adj=1 means the text is right-justified). The text we want to add has two extra spaces at the end, such that there is a little gap between the text and the arrow’s end point.

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> arrows(105, 1, max(waiting), + duration[waiting==max(waiting)], open=T) > text(105, 1, "extreme value ", adj=1) We will now continue with graphical functions that can deal with multiple variables: multivariate graph types. We start with an overview of what is available in Table 7.4. Table 7.4. Multivariate graphical data exploration functions

S-Plus Function

Description

Barplot (bars with subbars stacked) Plot principal components and factor analysis results, show data and variables in a two-dimensional coordinate system boxplot Boxplot(s) of one or more variables in a single graph contour Contour lines of a two-dimensional distribution to access the common distribution function coplot Plot matrix of two variables conditioned on the values of a third variable dotchart Dotchart (values on parallel lines stacked) for a vector with optional variable grouping faces Chernoff faces illustrating by means of face elements a highdimensional data set (up to 15 variables) hexbin Hexagonal binning, a display technique for spatial data using hexagonally shaped bins image Image plots, color scales for (geographical) heights or density values on a two-dimensional grid matplot Plotting columns of matrices against other matrix columns by using different symbols for each combination pairs Two-dimensional x-y scatterplots in a matrix of scatterplots for each combination of variables persp Perspective three-dimensional surface plots on a grid for density estimates stars Star plots of multivariate data in which each observation is displayed as a star. Each point is further away from the middle the larger the value is barplot biplot

See also Table 6.1 (page 143) for Trellis graphics functions. We have already seen that the function hist2d is useful for putting continuous data like the Geyser data into a table frame. If we do it with our Geyser data set, we get back a list with five components, telling us the mid-values of the chosen intervals and the end points, together with the tabulated data.


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> hist2d(waiting, duration) $x: [1] 45 55 65 75 85 95 105 $y: [1] 0.5 1.5 2.5 3.5 4.5 5.5 $z: 40 50 60 70 80 90 100

to 50 to 60 to 70 to 80 to 90 to 100 to 110

$xbreaks: [1] 40 50

0 to 1 1 to 2 2 to 3 3 to 4 4 to 5 5 to 6 0 0 0 0 16 0 0 0 0 1 56 3 0 0 1 2 28 1 0 13 19 9 40 0 1 31 25 9 24 0 0 11 3 2 3 0 0 1 0 0 0 0

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$ybreaks: [1] 0 1 2 3 4 5 6 The table is put into the component z, and we see that the visibility of effects depends heavily on the intervals chosen. The effect of the three point clouds is not that clearly visible, although you might guess at them by studying the values. Experimenting with different choices of interval bounds might give more insights. You can use the hist2d function introduced before to set up data to be used as input to other functions. The persp function produces a 3D-type plot that accepts input from hist2d. > persp(hist2d(waiting, duration)) Figure 7.9 shows a perspective plot of the surface of the empirical distribution. The x-axis and y-axis are set up by taking the variables xbreaks and ybreaks of the hist2d output, and the matrix of values in the z component determines surface height. Try to reveal the three peaks more clearly by choosing another set of intervals for hist2d to display the perspective surface plot. You can use the same data from hist2d to get an image display of the surface. The image display and the corresponding S-Plus function image are often used to display geographic data, using latitude and longitude as axes, and the geographical height of the area is color coded. This is the principle of every contour map. We use the color display to look at the tabulated Geyser data “from the top,” getting the heights in different colors.

7. Exploring Data

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Figure 7.9. A perspective plot of the Old Faithful Geyser data.

> image(hist2d(waiting, duration, + xbreaks=seq(40, 110, by=5), ybreaks=seq(0, 6, by=0.5)) > title("An image plot of the geyser data") For a nicer display, we chose different classes than the default settings used above. The result is shown in Figure 7.10, and although the loss of the coloring in this picture is a disadvantage, we can still see the three “mountains” as well as the “desert area” in the lower left corner. Other options available are different color schemes, like a “heat color scheme” ranging from cold (blue) to hot (red). It becomes obvious that we need to experiment with parameters if we examine Figure 7.11. We show two pictures of the same data, but the underlying categorizations into tables are different. From the picture on the left-hand side we see the three groups in the data, whereas the picture on the right-hand side leads us to conclude that we have only two mountain tops. This figure shows contour lines or, in other words, height lines of a data set. The contour function also accepts input from hist2d. > > + >

par(mfrow=c(1,2)) contour(hist2d(waiting, duration), xlab="waiting time", ylab="eruption time") title("Contour plot (1)")


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Figure 7.10. An image plot of the Old Faithful Geyser data.

> + + >

contour(hist2d(waiting, duration, xbreaks=seq(40, 110, by=5), ybreaks=seq(0, 6, by=1.5)), xlab="waiting time", ylab="eruption time") title("Contour plot (2)")

This excursion into data analysis using graphics for exploring structure reveals that the term “exploratory data analysis” describes a search for effects discovered by using many different displays and summaries of the same data set. A single number or display can never describe the whole complexity of a data set. If you want to explore the Barley or Geyser data sets with interactive techniques and rotating point clouds, try out the functions listed in Table 7.5. Use the mouse to define a brush and move it over point clouds to mark (“highlight”) them. If you get stuck, consult the manuals. Just do it. Table 7.5. Multivariate dynamic graphics functions

S-Plus Function

Description

brush spin

Interactive marking of subsets on 2D scatterplots Rotating data, marking and highlighting of subsets

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Figure 7.11. A contour plot of the Old Faithful Geyser data.

7.3 Distributions and Related Functions This section examines a variety of distributions covered by built-in S-Plus functions. As we will see later, adding further distributions to the existing set is also very straightforward. We divide the distributions themselves into continuous and discrete distributions. Statistical distributions occur in many practical data analysis situations. Most models to describe data behavior are based on distributional assumptions and derive the estimates for unknown model parameters from the underlying (assumed) distributions. The most important functions related to a distribution are as follows. – The density function, which specifies a random variate distribution (with fixed parameters). – The probability function, which is the integral over the distribution function or, in the discrete case, the sum up to the specified point. In other words, the probability function at x gives the probability of the random variable X being less than or equal to x.

7.3. Distributions and Related Functions

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– The quantile function, which is the inverse of the probability function. – The random number generating function, which generates numbers distributed according to the specified distribution function. For a large set of random numbers, the distribution of the random numbers (visualized, for example, by a histogram), looks approximately like the density function. S-Plus has a systematic naming of functions related to distributions. The functions related to the same distribution have the same name except for the first letter. The first letter indicates what type of function it is. Table 7.6 explains the system. Table 7.6. Categorization of distribution-related functions in S-Plus 1

Type Character

Function Type

d p q r

Distribution function Probability function or cumulated density function Quantile function (inverse probability function) Random number generation

1

The significant letter for identifying the functionality of the distribution function is either d, p, q, or r. See also Table 7.7. To identify a specific function, we need to know the abbreviations for the distributions available to us, which are listed in Table 7.7 and Table 7.8. To determine the function’s name, combine the characteristic letter from Table 7.6 with the abbreviation in Table 7.7 or 7.8. For example, to generate random numbers from the Normal distribution, the function to use is rnorm (“r” + “norm”). Table 7.7. Distribution-related functions in S-Plus(1): Discrete distributions

Distribution

S-Plus Name1

Binomial Geometric Hypergeometric Neg. binomial Poisson Wilcoxon (rank sum)

binom geom hyper nbinom pois wilcox

1

Parameters2 size prob m, n, k size lambda m

prob

prob n

The characteristic letter, one of d, p, q, r, plus the listed name, gives the name of the S-Plus function. Use the sample function to generate random numbers from a discrete distribution.

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Table 7.8. Distribution-related functions in S-Plus(2): Continuous distributions

Distribution

S-Plus Name1

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Beta Cauchy Chi-square Exponential F Gamma Logistic Lognormal Normal Normal, multivariate3 Stable Student t Uniform Weibull

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The characteristic letter, one of d, p, q, r, plus the listed name, gives the name of the S-Plus function. 2 If parameters are preset with a default value, like in mean=0, they do not need to be specified unless another parameter value than the default is desired. 3 There is no quantile function for the multivariate Normal distribution. There are four functions related to the Normal distribution dnorm for calculating the value of the distribution function pnorm for calculating the value of the probability function qnorm for calculating the inverse probability function rnorm for generating random numbers from the Normal distribution These four functions all have the same parameters, as they refer to the same distribution, namely mean and sd, the mean and the standard deviation, respectively. Both arguments are optional. They are preset to define the standard Normal distribution with mean 0 and standard deviation 1. Note The standard deviation, not the variance, is the argument used for dispersion by the functions for the Normal distribution. The standard deviation is simply the square root of the variance. ⊳ All these functions take numbers and vectors as input arguments. For a vector, the corresponding value for each element is computed. For example, > dnorm(-3:3, 0, 1:7)


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calculates dnorm(-3, 0, 1), dnorm(-2, 0, 2), ..., dnorm(3, 0, 7). You can check out to see the numerical accuracy of the distributional functions. Calculate the difference between some values x and p(q(x)) or q(p(x)), which should be 0 in theory (p stands for the probability and q for the quantile function of the distribution). Note As you might know, the random numbers generated by a computer are never really random. They consist of a sequence of numbers, where each number depends somehow on the previous number generated. S-Plus uses the variable .Random.seed to store the state of the random number generator. You can use set.seed(n), where n is an integer number, to produce the same sequence of random numbers several times, maybe to test a self-written routine. ⊳

Graphing Distributions You will often want to see what a distribution looks like, which usually means looking at the density function. Here, we need to make a distinction between continuous and discrete distributions. Recall that a continuous density has a continuous support; that is, all values of the density function in a given interval are greater than zero. This interval usually ranges from minus infinity to plus infinity. For this reason, we cannot graph the “whole” distribution from minus to plus infinity, but have to restrict ourselves to a part of it. A good idea is to plot, for example, 90% of the density support. To calculate these 90% limits, we cut 5% off each side and get the boundaries by calculating the 5% and 95% quantiles. As an example, we plot the standard Normal(0, 1), t(5), and standard Cauchy distribution and calculate the boundaries as described. > > > >

x >

yrange + + + >

key(min(bounds), max(yrange), corner=c(0,1), lines=list(lty=1:3, col=1:3), text=list(paste(c("Normal(0,1)", "t(5)", "Cauchy"), "Distribution"), adj=0, cex=0.8)) title("A graphical comparison of distributions")

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The result is shown in Figure 7.12. What we see is that the Normal distribution is more peaked at x=0 than the other two distributions and has shorter tails. For a discrete distribution, the density can only be calculated at discrete points. For example, consider the distribution function of the Binomial distribution - the function describing the probability of x heads out of n coin tosses, where the probability of a head for a single toss is p. The distribution function is greater than zero only at points 0, ..., n. To graph this adequately, we plot vertical height bars at the discrete points. The result is shown in Figure 7.13.

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Figure 7.13. A graph of the Binomial distribution (n=20, p=0.6).

> > > > > >

n qnorm(c(alpha/2, 1-alpha/2), m, s) ⊳ Table 7.9. Statistical tests in S-Plus

S-Plus Function Description of the Test binom.test chisq.gof chisq.test cor.test fisher.test friedman.test kruskal.test ks.gof mantelhaen.test mcnemar.test prop.test t.test var.test wilcox.test

Exact binomial test Chi-square (χ2 ) goodness-of-fit test Chi-square (χ2 ) test on a two-dim. contingency table Test for zero correlation of two samples Fisher’s exact test on a two-dimensional table Friedman rank sum test Kruskal–Wallis rank sum test Kolmogorov–Smirnov test for one or two samples Mantel–Haenszel test McNemar test on a two-dimensional contingency table Test for proportions of success Student t-test for one and two samples F-test for comparison of variances of two samples Wilcoxon/Mann–Whitney rank sum and signed rank test

7.4. Confirmatory Statistics and Hypothesis Testing

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Creation of a Graph Illustrating a Statistical Test The following describes how Figure 7.14 was created. It combines what we have learned about plotting distributions with graphical illustration of statistical testing. The combination of many functions shows how you can easily create even complex graphics like Figure 7.14. We store the two yield subsets in variables with shorter names, just for convenience. We calculate the degrees of freedom and set the significance level in a variable. > > > >

x >

bound.left > >

critical.left >

test.stat > > + > + > >

text(bound.left , 0.08, "rejection\nregion", adj=0) text(bound.right, 0.08, "rejection\nregion", adj=1) text((bound.left+bound.right)/2, 0.16, "acceptance region") xaxis x.no.na x mean(x) NA > mean(x, na.rm=T) 2.333333

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In the latter expression, the S-Plus function removes all missing data and performs the action on the remaining data. The result is that the mean function divides the sum of data in the above example by 3 and not by 4, the total length of x. The same effect is obtained by removing the missing data “by hand,” as in > mean(x[!is.na(x)])

7.5.3 Missing Values in Graphs Missing values are ignored in graphical routines except for special cases such as when you want to plot x, which consists only of missings (this would return an error message). So, for example, the plot and lines functions ignore all points where one of the coordinates is an NA. Note Using NAs can be useful if you want to plot an interrupted line. Example: > plot(1:10, c(2,3,2,4,NA,3,2,2,4,5), type="l") creates a line, which is not connected between the points (4, 4) and (6, 3). ⊳

7.5.4 Infinite Values There is another category of special values, the infinite values. Typically, infinite values are not part of a data set, but are sometimes created from a calculation. For example, dividing a number by zero leads to an infinite value. But since the number you divide by zero can have a sign, there is also plus and minus infinity. Dividing a positive number by zero leads to the “value” plus infinity, and dividing a negative number by zero leads to minus infinity. Infinity is displayed as Inf. Similar to missing values and the NA code, Inf can be entered from the keyboard or can be in a file. If S-Plus comes across Inf, it interprets it as infinity. You can check for infinite values by using is.inf. Sometimes, it might be necessary to check for missing and infinite values separately, as an infinite value is not missing and vice versa. See the following example. > x x -Inf NA Inf > is.na(x) F T F

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> is.inf(x) T F T One can check if a value is finite by using is.finite. > is.finite(x) F F F Have a look at the definition of is.finite. You will see that it uses other functions. Have a look at them as well. There are some simple arithmetic rules for infinite values. You can try exploring them on your own, or go through the exercises that follow.

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7.6 Exercises Exercise 7.1 Generate 100 and 1000 random numbers, both samples from a Normal distribution with mean value 3 and variance 5. Draw histograms with bin width 0.5, 1, and 2 for each of the two samples. Remember that all figures have exactly the same underlying distribution. Plot them all in a single graphics window and label them accordingly. What is visible ?

Exercise 7.2 Create a vector containing three real numbers, one missing value (NA), and one infinite value (Inf). Calculate the mean and variance of the nonmissing data. Figure out how the missing and infinite data are treated in a simple plot and histogram. What happens if you divide x by itself ? Can you explain this behavior ?

Exercise 7.3 Do an analysis of the car data set provided with S-Plus. The data set, stored under car.all, contains a set of cars plus a variety of their attributes, like fuel consumption, price, and many more. Before starting the analysis, determine which car models are contained in the data and what the variables given in the data set are. Extract the fuel consumption (miles per gallon) and determine for each type of car (small, medium, compact, large, van, sporty) the model with the lowest and the highest fuel consumption. On the basis of average fuel consumption and tank size, calculate the maximum travel distance without a gas refill. Take the variables maximum travel distance, tank size, mileage, and horsepower, and display them graphically. Try to determine, using only graphics, the two variables with the strongest dependency. Check your guess by calculating the correlation matrix for these variables.

Exercise 7.4 Test the S-Plus random number generator with the “dice test.” Simulate a number of die rolls and check if they all occur with approximately the

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same frequency. In other words, we need to simulate a die roll, where the six possible outcomes (1, 2, 3, 4, 5, and 6) occur with the same probability. 2 6 Calculate the test statistic T = i=1 (ni −n/6) which is known to have n/6 an asymptotic Chi-square distribution with 6 degrees of freedom. Use SPlus to calculate the critical value for the Chi-square distribution at the 5% level. Can you reject the hypothesis that all numbers occur with the same frequency? Repeat this exercise with different numbers of rolls of the die.

Exercise 7.5 In this exercise, we will use simulation to examine a statistical hypothesis test. We want to test the null hypothesis that the mean of a Normally distributed sample is 0, with known variance 1. Let the alternative be that the mean is not equal to 0 (two-sided). We know that if we set the significance level to α (alpha), the test will reject the null hypothesis in α% of the times, although the data actually come from a Normal distribution with mean 0 and variance 1. Prove this by generating 1000 samples of size 100, calculating for each sample whether or not the null hypothesis is rejected, and summarizing how often the null hypothesis was rejected. In a second step, examine the power of the test: the probability of rejecting the null hypothesis, given the alternative is correct. Simulate a data set coming from the alternative hypothesis that the mean is equal to 0.1 and estimate the power of the test based on the sample.

Exercise 7.6 Suppose we had forgotten the formula for computing the area of a circle. All is not lost. We can easily use the computer to approximate a circle’s area by simulation. For convenience, we want to determine the area of the unit circle with radius 1 and name this unknown constant π (pi). We know that the area covered by a square with the corner points (−1, 1), (1, 1), (1, −1) and (−1, −1) equals 4, as the length of each side is 2. This square completely contains the unit circle. If we had a random point somewhere in the square, we could calculate the probability that the point is also inside the unit circle. A random point Z is determined by two coordinates, say X and Y, which have independent uniform distributions on the interval [−1, 1]. The probability of Z being inside the circle is given by the ratio of the circle’s area and the square’s area or, more mathematically, letting

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P=probability: P(Z is inside the circle) = (circle area)/(square area) = π/4 Knowing this, we can estimate (the unknown) π by multiplying the estimated probability for Z being inside the circle, by 4. What’s left is to generate a large number of random points Z, calculate the proportion of points inside the circle, and multiply this by 4. This is our estimate of pi. Write an S-Plus program for this problem and estimate pi with 100, 1000, and 10,000 random points. Try to work with complete vectors only and avoid loops. How far is the approximation away from the pi we (fortunately) happen to know ? Hint: You need to know how to check if a point (x, y) is inside the circle. As the circle is defined by its origin and the radius, calculate how far the distance is from the circle’s origin to the point. Use Pythagoras’s rule to determine the distance between the middle of the circle and a point.

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7.7 Solutions Solution to Exercise 7.1 This exercise shows what different samples coming from the same distribution can look like. If you compare the graphs in the same column in Figure 7.15, the data are always exactly the same, but the parameter bin width for the histogram is changed. If you compare two graphs in the same row, the difference comes from the sample size (100 versus 1000), and, of course, from stochastic variation. The figure created is shown in Figure 7.15.

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We start by creating the two data sets. > x y > > > > > > >

min.xy max.xy br1 title.string mtext(title.string, side=3, outer=T, line=-1)

Solution to Exercise 7.2 In this example, we will work with missing data and infinite values, to understand how they are treated and how you can actually handle them in data sets. We begin by creating a vector by entering it directly from the keyboard. > x mean(x) NA

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As we have learned in this section, we can specify a parameter for removing the missing data before doing the calculation: > mean(x, na.rm=T) Inf We see that S-Plus removed the missing value, but not the infinite value. And, of course, the sum of data with an infinite value is infinite (it works vice versa with negative infinite data). To calculate the mean (and variance) over all real numbers, we need to remove the infinite value, too. > y mean(y) 2 > var(y) 1 Alternatively, > y x/x 1 1 NA 1 NA Note that a missing value divided by anything else results in a missing value again, and infinity divided by infinity also results in a missing value.

Solution to Exercise 7.3 We now perform the car data analysis interactively. The first thing we do is attach the data set car.all. > attach(car.all) To determine which variables are in the data set, the names function delivers the data set’s column labels or, in other words, the variable names. > names(car.all) "Length" "Wheel.base" "Front.Hd." "Rear.Hd" "Frt.Shld" "RearShld" "Tires" "Steering" "HP" "Trans1" "Tank" "Model2" "Pwr.Steer" ".empty." "Trans2" "Gear2" "Country" "Reliability"

"Width" "Frt.Leg.Room" "Luggage" "Turning" "Gear.Ratio" "Dist.n" "Disp2" "Eng.Rev2" "Mileage"

"Height" "Rear.Seating" "Weight" "Disp." "Eng.Rev" "Tires2" "HP.revs" "Price" "Type"

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Since we have already attached the data set, we can access the variables just by their names. For example, entering HP returns the HP data for all the cars in the data set. The names of the cars in the data set are returned by the names of the rows of the data matrix. Therefore, we enter > dimnames(car.all) to see the car names and the variable names, or just > dimnames(car.all)[[1]] to get only the first component of the list, the names of the cars. Now we save the names of the cars for later use. > car.names max.travel summary(Type) Compact Large 19 7

Medium 26

Small 22

Sporty 21

Van 10

Na’s 6

We see that there are only 7 cars classified as large in the data, 6 have no classification, and there are more than 20 cars in the groups Medium, Small, and Sporty. Determining the cars with the highest fuel consumption (or the lowest mileage) in the different categories is done by using min in combination with by as follows: > by(Mileage, Type, min, na.rm=T) Compact Large Medium Small Sporty 21 18 20 25 24

Van 18

Note that we need to supply the parameter na.rm=T to the min function, as there are missing values in the data. The minimum will be missing if any of the data are missing. In the same fashion, we obtain the maximum values. > by(Mileage, Type, max, na.rm=T) Compact Large Medium Small Sporty 27 23 23 37 33

Van 20

You might have produced slightly different output, because we used the command unclass(by(...)) to obtain this shortened output.

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Figure 7.16. Some components of the car data set (horse power, tank size, mileage, and maximum travel distance).

A first display of the data can be generated using pairs. To analyze the four variables HP, Tank, Mileage, max.travel, we create a new data set containing only these data. > car.subset pairs(car.subset) The display of pairwise scatterplots is shown in Figure 7.16. We immediately see strong relationships between many variables, such as between HP and tank size or between tank size and mileage in a negative way: The smaller the mileage, the larger the tank, and vice versa. We can confirm this by numbers by calculating the correlation matrix between the variables. If we just enter cor(car.subset), we get an error message telling us that S-Plus refuses to calculate the correlation for a data set with missings. Looking up the cor function in the Help system tells us that we need to set na.method="omit" to ignore the missing data for the correlation calculation. > cor(car.subset, na.method="omit")

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HP Mileage Tank max.travel

HP 1.000000 -0.666153 0.676410 -0.020566

Mileage -0.66615 1.00000 -0.80134 0.37271

Tank max.travel 0.67641 -0.020566 -0.80134 0.372705 1.00000 0.212167 0.21217 1.000000

The results of our first guess can be immediately confirmed, as the correlation between HP and tank size is 0.68. There is also a strong negative correlation between tank size and mileage, about −0.8. Some functions do not have options to remove the missing values before doing the calculation. In this case, we would need to remove all data (all rows) that contain one or more missings. The function is.na is useful for checking if any data are missing, and if you remember that TRUE values are coded with 1 and FALSE values with 0, then it becomes clear what to do. We sum up the TRUE and FALSE values obtained from is.na and see if the sum is 0 (indicating that none of the data are missing) or greater than 0 (indicating that one or more variables are missing). > problems problems problems car.without.missings n rolls rolls.count rolls.count # the tabulated outcome 1 2 3 4 5 6 988 1028 988 1062 944 990 The test statistic is calculated using the given formula. > test.stat test.stat # the test statistic T 8.152 Finally, we need to have the critical value to compare it to the test statistic. We can derive the critical value for three significance levels simultaneously. > alpha qchisq(alpha, 6) 10.64464 12.59159 16.8119

# the critical values

Our test statistic is smaller than all three critical values of the chi-square distribution, such that we cannot reject the hypothesis that we generated rolls with a fair set of dice. In fact, the lowest alpha level at which the hypothesis could be rejected (the p-value) is calculated as > 1-pchisq(test.stat, 6) 0.2271776 This result helps give us confidence that the S-Plus random number generator is doing a good job.

Solution to Exercise 7.5 You can simulate a statistical test’s rejection percentage by generating data from the null hypothesis several times, and by calculating how often the test would reject the null hypothesis, although the data come from the distribution specified in the null hypothesis. In this exercise, we will only work on matrices and vectors, as this is an elegant and efficient way to program in S-Plus. We first generate 1000 samples of size 100 from the null hypothesis, being a Normal distribution with mean 0 and standard deviation 1. The samples are stored in a matrix, each row representing one sample. > x m z qnorm(0.975, 0, 0.1) The result is a vector of TRUE/FALSE values, and summing them up yields the number of TRUE values, which in our case was 47. The mean value is the mean number of rejections over all 1000 samples, which should be around 0.05 (as we are testing on the 5% level). > mean(z) 0.047 You can also check what’s in the z variable by tabulating the data. > table(z) FALSE TRUE 953 47 To calculate the approximate power of the test when the data come from the alternative, or, in other words, the number of correct rejections of the null hypothesis, we generate samples from the alternative hypothesis and calculate how often the test rejects the null hypothesis. > > > >

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8 Statistical Modeling

We have now learned some elementary statistical techniques in S-Plus and the basics of graphical data analysis. The next step is to see what S-Plus has to offer in terms of modeling. Statistical modeling is one of the strongest S-Plus features because of its unified approach, wide variety of model types, and excellent diagnostic capabilities. We start with an example of how to fit a simple linear regression model and corresponding diagnostics. The example is presented with a minimum of technical explanation, designed as a quick introduction. We then formally explain the unified approach to model syntax and structure, along with comments on several of the more popular types of statistical models.

8.1 Introductory Examples We start with a simple linear regression model to show how to carry out the fit and what the output looks like.

8.1.1 Regression For our example, we will use the Swiss fertility data set that is directly accessible from within S-Plus by typing swiss.fertility and swiss.x. We chose this data set because it is easy to understand and work with. The data come from French-speaking provinces in Switzerland from about the year 1888. Table 8.1 describes the variables.

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8. Statistical Modeling Table 8.1. Swiss fertility data set

Variable

Description

swiss.x Agriculture Examination

% of population with this occupation % of soldiers with high marks on army examination % with education beyond primary school % of population with this religion % of live births who live less than 1 year Standardized fertility measure I[g]

Education Catholic Infant Mortality swiss.fertility

We will restrict our discussion here to only one predictor variable, education, so we will not use the entire swiss.x matrix. The outcome of interest is the vector swiss.fertility. Linear regression (simple or multiple) is done using the lm function. All you have to supply is the outcome variable followed by the predictor variable. For simplicity, we put the outcome and predictors together into a single data frame. This avoids having to copy individual vectors out of the swiss.x matrix. > swiss dimnames(swiss)[[2]][1] dimnames(swiss)[[2]] "Fertility" "Agriculture" "Examination" "Education" "Catholic" "Infant.Mortality" > fit summary(fit) Call: lm(formula = Fertility ∼ Education, data = swiss) Residuals: Min 1Q Median 3Q Max -17.04 -6.711 -1.011 9.526 19.69 Coefficients: (Intercept) Education

Value Std. Error 79.6101 2.1041 -0.8624 0.1448

t value Pr(>|t|) 37.8357 0.0000 -5.9536 0.0000

Residual standard error: 9.446 on 45 degrees of freedom Multiple R-Squared: 0.4406 F-statistic: 35.45 on 1 and 45 degrees of freedom, the p-value is 3.659e-007

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Correlation of Coefficients: (Intercept) Education -0.7558 It is convenient to put the fit of the model into a variable because, as we will see later, model building and diagnostic routines use the information contained in it. Notice that the first part of the output from the fit of the regression model contains basic summary information about the residuals. This is one simple way of checking the fit of the model. If the residuals are skewed in any way, then chances are that more formal diagnostic techniques will confirm that the model does not fit well. The next section of the output contains information on the coefficients (i.e., the specific terms included in the model). Included are the estimates of the coefficients, standard errors, t-statistics, and p-values with which you can assess the fit of any of the specific terms in the model. An intercept is included in the model by default, but this may be removed (add a “−1” as a predictor variable). Note that in the fit of the model, both the intercept and education are statistically significant. In practical terms, this means that the education level is significantly related to the fertility level. Further, the coefficient of education is negative (−0.8624), implying that an increase in education is associated with a decrease in fertility. With simple linear regression, you can take the square root of the R-squared value (here, 0.4406) to obtain a correlation of −0.6638. The cor function yields the same result: > cor(swiss$Education, swiss$Fertility) -0.6637889 The summary function also gives the residual standard error, R-squared, and the overall F-test. The residual standard error is used in the calculation of many regression tests, the R-squared can be helpful in assessing the degree of association between outcome and explanatory variables, and the overall F-test is used for the entire model, as opposed to each separate factor.

8.1.2 Regression Diagnostics At a first glance, the above regression model appears to fit the data quite well. However, as part of a proper and thorough regression analysis, you should check that the model assumptions are met. You can easily do this by examining the residuals that are contained in the object fit. Four easy graphs to assess the model fit and check the assumptions are a scatterplot of the data, a histogram of the residuals, a scatterplot of the residuals versus the fitted values, and a QQ-plot of the residuals. The commands for obtaining these graphs are shown below. The graphs appear in Figure 8.1.

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Figure 8.1. Regression diagnostic plots.

Example 8.1. Regression diagnostics We first set up the graphics layout to contain two rows and two columns of plots. Our first plot is a simple scatterplot of the data with the fitted regression line superimposed. > > > > >

par(mfrow=c(2, 2)) attach(swiss) plot(Education, Fertility) detach() abline(fit)

The second plot is a histogram of the residuals to see if they are approximately Normally distributed. > hist(resid(fit)) The third plot is of the residuals against the predicted values. To this we have put the reference line at y=0 and a lowess curve (a scatterplot smoother). This will tell us if the variance is a function of the covariate or some other problem with the fit of the model. > plot(fitted(fit), resid(fit)) > lines(lowess(fitted(fit), resid(fit)))

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> abline(h=0) The last plot is a QQ-plot with the expected line to tell us whether or not the residuals follow a Normal distribution. > qqnorm(resid(fit)) > qqline(resid(fit)) The scatterplot of the data shows that there is indeed an approximately linear relationship between the two variables, which is accentuated by the estimated regression line that has been superimposed. A histogram of the residuals would ideally look like the bell-shaped curve of the Normal distribution. The histogram here, with a little imagination, is not so far away from that ideal. The scatterplot of the residuals versus fitted values (values predicted from the regression equation) should show no discernable pattern. A lowess curve has been added to help detect whether or not hidden (or obvious) trends exist in the residuals. The flat curve indicates that perhaps the assumptions of the linear regression are met. A last check is to use a QQ-plot to assess whether the residuals follow an approximately Normal distribution. The superimposed straight line is what we would expect if the residuals followed a Normal distribution exactly. What we find is that our residuals, although not perfectly following a Normal distribution, are not that far away from the line.

8.2 Statistical Models S-Plus offers many possible ways of modeling data according to the assumptions made on the distribution and type of outcome (see Table 8.2). Simple linear regression is done with the lm function, as shown in the example in the previous two subsections, and analysis of variance (ANOVA) with the aov function. ANOVA is actually a regression model, and this can be run with the lm function, but the aov function makes it easier to define the classification variables used in these models. The glm function is based on exponential family theory and can be used to do logistic, Poisson, and other types of non-Gaussian regression. Such regression models are called generalized linear models (GLMs). The gam function extends the GLM methodology by allowing nonparametric functions to be estimated from the data using smoothing operations. A local regression model is done with the loess function, which is a very convenient way of gaining an understanding of the relationships in the data. Classification and regression tree models can be run with the tree function, and nonlinear models with the nls and ms functions. Classical multivariate models, which include multivariate ANOVA, factor analysis, and principal components analysis, can be fit with the respective functions of manova, factanal, and princomp. Loglinear modeling can be done with

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8. Statistical Modeling Table 8.2. Selected statistical models in S-Plus

Function

Type of Model

lm aov glm gam loess tree nls, ms manova factanal princomp crosstabs survfit coxph

Linear regression ANOVA Generalized linear models Generalized additive models Local regression smoother Classification and regression tree models Nonlinear models Multivariate ANOVA Factor analysis Principal components analysis Cross-classifications Kaplan–Meier survival curves Cox proportional hazards models

the crosstabs function. And, finally, for our short list, the most common survival analysis procedures can be done using the survfit and coxph functions.

8.3 Model Syntax A particularly nice feature of S-Plus is that it takes a unified approach to modeling, including the model syntax. All statistical models in S-Plus specify the response and explanatory variables in the same manner, namely Response∼Explanatory The tilde (∼) can be read as “is modeled by” or simply as “equals.” If we have the response variable, NumCars, and the explanatory variable, Income, and we want to model their relationship, we use the simple formula NumCars∼Income We discuss the slightly more complicated matter of how to use the formulas in the following sections. For the moment, we concern ourselves only with the syntax of the model formula. If you also have information on age and want to incorporate this into the model, simply add it with a plus sign (+) as in NumCars∼Income+Age Interactions are usually of interest and are specified using the colon (:) as shown in the following example: NumCars∼Income+Age+Income:Age

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S-Plus has several more advanced ways of specifying formulas for models. They are listed, along with those already discussed, in Table 8.3. Table 8.3. Model syntax

Terms Included

Functionality

A+B A:B A*B A%in%B A/B A−B A^m

Main effects of both A and B Interaction between A and B Shorthand for A + B + A:B A nested in B Shorthand for (A + B) %in% A All of A except what is in B All terms in A and m-order interactions

8.4 Regression We now show, very briefly, how to use elementary regression functions. We include in the descriptions some commonly used modeling and model-building techniques. The nonparametric regression functions are, in particular, one of S-Plus’s major strengths, and you should consult the manual for detailed explanations. There are, in principle, two types of model fits in S-Plus, categorized by what is returned. Functions like linear regression return the parameters of the fit; that is, they return what is needed to determine the functional form. On the other hand, the nonparametric functions like ksmooth (kernel smoothing) return a list of points, x and y, where y contains the smoothed data at points x, as there is no explicit functional form for these types of smoothers. For the first type of model, it is helpful to do the fit with, for example, lm, > fit plot(x, y) and then add the fitted line with the command > abline(fit) These commands were shown in Section 8.1.2. In the second case, where the return argument consists of a list with x and y component (and others), you enter the following commands instead: > fit plot(x, y) > lines(fit)

8.4.1 Linear Regression and Modeling Techniques As we already presented simple linear regression in Section 8.1.1, we use this section to review the unified modeling syntax presented in Section 8.3, and some of the model-building techniques available in S-Plus. The examples in this section use the Swiss fertility data used earlier. To begin, we simply repeat the model shown in Section 8.1.1. The model consisted of the response variable, Fertility, and the explanatory variable, Education, both contained in the data frame swiss. > fit1 summary(fit1) Call: lm(formula = Fertility ∼ Education, data = swiss) Residuals: Min 1Q Median 3Q Max -17.04 -6.711 -1.011 9.526 19.69 Coefficients: (Intercept) Education

Value Std. Error 79.6101 2.1041 -0.8624 0.1448

t value Pr(>|t|) 37.8357 0.0000 -5.9536 0.0000

Residual standard error: 9.446 on 45 degrees of freedom Multiple R-Squared: 0.4406 F-statistic: 35.45 on 1 and 45 degrees of freedom, the p-value is 3.659e-007 Correlation of Coefficients: (Intercept) Education -0.7558 Note that the output here is exactly the same as shown in the example in Section 8.1.1. We will now build on this simple model. With the unified approach to modeling, we can very easily make use of many sophisticated techniques. For example, suppose we decided that education was not adequate to describe fertility and we wanted to add more terms to the model. We want to update the previous model, and this is done, appropriately, with the update function as follows:

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> fit2 summary(fit2) Call: lm(formula = Fertility ∼ Education + Catholic, data = swiss) Residuals: Min 1Q Median 3Q Max -15.04 -6.576 -1.426 6.126 14.32 Coefficients: (Intercept) Education Catholic

Value Std. Error 74.2319 2.3518 -0.7882 0.1293 0.1109 0.0298

t value Pr(>|t|) 31.5636 0.0000 -6.0968 0.0000 3.7224 0.0006

Residual standard error: 8.331 on 44 degrees of freedom Multiple R-Squared: 0.5746 F-statistic: 29.71 on 2 and 44 degrees of freedom, the p-value is 6.822e-009 Correlation of Coefficients: (Intercept) Education Education -0.6838 Catholic -0.6144 0.1540 Note that in the updated model, fit2, both Education and Catholic are significantly related to Fertility. Our choice of adding the variable Catholic was more or less random, but we could have used built-in functions to make the choice a little less whimsical. The add1 function takes a list of explanatory variables and adds them one at a time to the current model. The resulting output shows statistics for each explanatory variable, had each been included together with the base model. > add1(fit1, .∼.+Agriculture+Examination+Catholic+ + Infant.Mortality) Single term additions Model: Fertility ∼ Education Df Sum of Sq <none> Agriculture 1 61.9657 Examination 1 465.6258 Catholic 1 961.6083 Infant.Mortality 1 891.2462

RSS 4015.236 3953.270 3549.610 3053.627 3123.989

Cp 4372.145 4488.635 4084.975 3588.992 3659.354

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Applied to our data, we see that with the largest sums of squares and the lowest Cp value, the variable Catholic actually does provide the best predictive power when one variable is added to a model that includes Education. On the other hand, we could have chosen to use stepwise regression with the step function or to add a lot of variables and remove them using the drop1 (opposite of add1) function. The choice between using add1, drop1, or step is left entirely up to you. In general, they will yield the same model, but often in practice, they do not. So, have fun. S-Plus offers a host of residual checking techniques. Feel free to use the simple guidelines shown in Section 8.1.2, but you can do some more advanced work by simply plotting the fit from the regression. If you simply plot the fit of the regression model, you obtain six graphs, one diagnostic plot per graph; residuals versus fitted values, square root of the absolute of the residuals versus fitted values, response versus fitted values, normal QQ-plot of residuals, r-f spread plot, and the Cook’s distances. If, for convenience, you prefer to view them all at once on one page, enter the mfrow option as follows to set up the graphical area for a matrix layout of graphs: > par(mfrow=c(3, 2)) > plot(fit1) S-Plus can calculate the predicted values, standard errors, and confidence intervals using such functions as predict and pointwise. A variable’s polynomial effects are obtained with the poly function. To include linear, quadratic, and cubic terms of X in a model, use poly(X, 3), but beware that the resulting parameter estimates are in terms of orthogonal polynomial contrasts and are not on the same scale as X. To see the parameter estimates on the original scale, use > poly.transform(poly(X, 3), coef(fit)) Indicator Variables S-Plus can automatically generate indicator variables using the I function. Entering the term I(age > 70) into your model is a way of looking at the effect of “young” versus “old” rather than the effect of age on a continuous scale. Factors Factors are related to the issue of indicator variables. You may often have variables like “political affiliation” or “treatment” that can assume one of several values. Giving numeric codes to these types of variables and entering them into your model as continuous variables makes little sense. To see the problem here, consider the following example. Suppose a questionnaire had been done collecting information on political affiliation, where

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1=Democratic, 2=Republican, 3=Independent, and 4=Other. If we entered the continuous variable “political affiliation” (values 1 to 4), we would be making assumptions like Democratic is less than Republican, the difference between Democratic and Republican is the same as the difference between Republican and Independent, and Other is the highest. These assumptions, however, have no meaning and are of little interest. By defining these categorical variables as factors, it is possible to view the effect of each level of the variable separately, or combined in certain ways using contrasts. Factor is a special data type in S-Plus and is defined with the factor function. The data set we have used so far doesn’t contain any categorical variables, but we can easily create one using the cut function as follows: > educ.lev educ.lev High Average Low Average ... The optional breaks specifies at which values of education to define the breaks for the levels, and labels defines the names for the new categories. We have created a new variable with three levels of education; low is from 0 to 5, average is from 5 to 10, and high is from 10 to 100 (i.e., over 10). The function is.factor can be used to determine if your variable is really a factor. As there is an inherent ordering to the levels of education, it makes more sense to think of the levels as being ordered. To define this data type, we use the function ordered. > educ.ord educ.ord 3 2 1 2 3 ... Notice that the values contained in educ.ord are numeric with 1 representing low education, 2 average education, and 3 high education. Try fitting a regression model with educ.lev and compare the results to the model with Education. What do you notice? Remember that Education is a continuous variable but that educ.lev is a categorical variable with three levels. Now try fitting a model with educ.ord. How does this compare to the other two?

8.4.2 ANOVA The analysis of variance (ANOVA) is a statistical method of analyzing data that assumes a continuous outcome variable and one or more classification variables. You might wonder how this differs from using a linear regression model with a factor variable. It doesn’t. Theoretically speaking, an

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ANOVA model is a regression model. The difference comes more with the presentation of the results, as we see in the following example. The aov function is used exactly in the same fashion as the lm function to fit the desired model. > swiss fit.ord summary(fit.ord) Df Sum of Sq Mean Sq F Value Pr(F) educ.ord 2 1350.891 675.4456 5.100272 0.01018307 Residuals 44 5827.064 132.4333 We see from the large F-value and very low p-value that education is related to fertility, but we already knew this from the regression model. Is there any information we can obtain from the ANOVA model that we haven’t already seen with the regression model? Of course! With a factor variable, you can split the degrees of freedom into contrasts of interest. Take the educ.ord variable from above, which has three levels and, hence, two degrees of freedom. We could use one degree of freedom for a linear test and one for a quadratic test. This breakdown of the degrees of freedom leads to results that are comparable to using polynomial regression, as mentioned in Section 8.4.1. The difference is that with contrasts, you aren’t restricted to polynomial transforms, but we show it here for simplicity. To obtain a polynomial breakdown of the degrees of freedom, use the split option together with lists: > summary(fit.ord, split=list(educ.ord=list(L=1, Df Sum of Sq Mean Sq F Value educ.ord 2 1350.891 675.446 5.100272 educ.ord: L 1 1173.055 1173.055 8.857705 educ.ord: Q 1 177.836 177.836 1.342838 Residuals 44 5827.064 132.433

Q=2))) Pr(F) 0.0101831 0.0047286 0.2527848

The linear and quadratic effects of education on fertility are displayed and tested, showing that it is actually the linear component of the relationship between education and fertility that is statistically significant. The main objective with many ANOVA models is merely to test for certain effects. Useful additional information includes the relative effect of each level of the term of interest. These are obtained using the dummy.coef function:

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> dummy.coef(fit.ord) $"(Intercept)": (Intercept) 70.70725 $educ.ord: 4.932745

2.662745 -7.595490

The coefficients for educ.ord show the relative effect on fertility of the education levels of low, average, and high, respectively. Note that fertility drops as education increases. The choice between the lm and aov functions is often a matter of preference. If you are used to using one of the two, you naturally continue to do so.

8.4.3 Logistic Regression So far, we have examined models for fitting data with a continuous outcome. Often though, the outcome is binary in nature, representing success/failure, 0/1, or presence/absence. In such cases, a logistic regression model is used to fit the data. Such models are fit in S-Plus using the glm (or gam) function. This function is a very general procedure, as it can be used to fit models in the exponential family, including the Normal, binomial, Poisson, and gamma distributions. To explore this type of regression, we will use the built-in data frame, kyphosis, which contains a suitable variable with a binary outcome. The data frame contains the four variables described in Table 8.4. Table 8.4. Kyphosis data frame

Variable

Description

Kyphosis Age Number Start

Presence/absence of the postoperative spinal deformity Age of child in months Number of vertebrae involved in surgery Beginning of the range of vertebrae involved

We want to see what relationship exists, if any, between the age, number, and start, and the subsequent presence of kyphosis. For example, are younger children more susceptible to kyphosis or more resilient? We use the glm function to fit our logistic model. All we need to specify is the model equation, the exponential family type, and the data frame as follows: > fit.kyph summary(fit.kyph) Call: glm(formula = Kyphosis ∼ Age + Number + Start, family = binomial, data = kyphosis) Deviance Residuals: Min 1Q Median 3Q Max -2.312363 -0.5484308 -0.3631876 -0.1658653 2.16133 Coefficients: Value Std. Error t value (Intercept) -2.03693225 1.44918287 -1.405573 Age 0.01093048 0.00644419 1.696175 Number 0.41060098 0.22478659 1.826626 Start -0.20651000 0.06768504 -3.051043 (Dispersion Parameter for Binomial family taken to be 1) Null Deviance: 83.23447 on 80 degrees of freedom Residual Deviance: 61.37993 on 77 degrees of freedom Number of Fisher Scoring Iterations: 5 Correlation of Coefficients: (Intercept) Age Age -0.4633715 Number -0.8480574 0.2321004 Start -0.3784028 -0.2849547

Number

0.1107516

The function call is repeated at the top, followed by some summary information on the residuals, the model coefficients, general fitting information, and the correlations between the model coefficients. The t-statistics given are partial t-tests, meaning that all other factors in the model have been adjusted for. Notice that p-values are not provided with the t-statistics, but this is an easy calculation to do in S-Plus. You learned how to do it in the previous chapter. The t-statistics indicate that age and the number of vertebrae do not seem to be related to kyphosis, but that the starting vertebra of the surgery is related to it. We know that the degrees of freedom to be used for the ttests are the same as for the residual deviance (77). In this case, the number of degrees of freedom is large, so the cutoff for the t-distribution is similar to that of the Normal (i.e., ± 1.96 for the usual significance level of 0.05).

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Only the t-statistic for the starting vertebra of the surgery has a value that is more extreme than −1.96, implying that it is associated with kyphosis. To be a little more precise and a lot more helpful, we have calculated both the cutoff level of this distribution and the p-value. > qt(.975, 77) 1.991254 > qt(.025, 77) -1.991254 > 2*pt(-3.05, 77) 0.003137395 > 2*(1-pt(abs(-3.05), 77)) 0.003137395 Using an overall level of significance of 0.05 (each tail gets 0.025), we have calculated both the upper and lower cutoffs for the t-distribution with 77 degrees of freedom, and the p-value using two slightly different methods. Note that this type of test is a form of Wald test and is asymptotically t-distributed (Venables and Ripley (1997), p. 237). The modeling techniques covered in the previous subsections can also be applied to logistic models. You may wish to try some of them now, or simply wait until the exercises.

8.4.4 Survival Data Analysis What do we mean by “survival data”? Quite loosely, we refer to data that record the time until the occurrence of an event. In cancer trials, for example, you measure the time (in days, months, etc.) until the patient’s death. In this case, the event is death. The problem is that, quite often, the patients either withdraw from the trial or are still alive at the end of it (rather good from the patients’ point of view). The problem raised is that the actual time of the event is not observed for these patients, so their survival time is “censored.” Special statistical techniques have been devised to analyze this type of data, including the famous Kaplan–Meier curves. Recently, there has been a lot of statistical research in this area on the topic of martingales and martingale residuals. S-Plus is one of the few software packages to include this functionality. This book is not, however, a statistics book, so we will stick to the very basics and it will be up to you to carry on with martingales should you so desire. With survival data analysis, the outcome “variable” consists of two pieces, the time to event and an indicator of whether the time refers to an event or is a censored time. It is then typical to have some sort of group variable such as a treatment indicator. One such data set included with S-Plus is the Leukemia data set, which contains three variables as described in Table 8.5. The data come from a trial to determine the efficacy of maintenance chemotherapy for acute myelogenous leukemia.

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8. Statistical Modeling Table 8.5. Leukemia data frame

Variable

Description

time status group

Time to remission after chemotherapy (in weeks) Indicator of status: 1=remission time; 0=censored time Treatment group: maintained or nonmaintained

The survfit function can be used to fit a Kaplan–Meier survival curve, or other types of survival curves according to the options chosen. You must use the Surv option to define the composite outcome variable, followed by the explanatory variables and the name of the data frame. > leuk.fit summary(leuk.fit) Call: survfit(formula = Surv(time, status) ∼ group, data = leukemia) group=Maintained lower upper time n.risk n.event survival std.err 95% CI 95% CI 9 11 1 0.909 0.0867 0.7541 1.000 13 10 1 0.818 0.1163 0.6192 1.000 18 8 1 0.716 0.1397 0.4884 1.000 23 7 1 0.614 0.1526 0.3769 0.999 31 5 1 0.491 0.1642 0.2549 0.946 34 4 1 0.368 0.1627 0.1549 0.875 48 2 1 0.184 0.1535 0.0359 0.944 group=Nonmaintained lower upper time n.risk n.event survival std.err 95% CI 95% CI 5 12 2 0.8333 0.1076 0.6470 1.000 8 10 2 0.6667 0.1361 0.4468 0.995 12 8 1 0.5833 0.1423 0.3616 0.941 23 6 1 0.4861 0.1481 0.2675 0.883 27 5 1 0.3889 0.1470 0.1854 0.816 30 4 1 0.2917 0.1387 0.1148 0.741 33 3 1 0.1944 0.1219 0.0569 0.664 43 2 1 0.0972 0.0919 0.0153 0.620 45 1 1 0.0000 NA NA NA The summary function applied to this class of data provides a radically different output than those from the previous regression models. Here, you get the estimated survival probability at many time points, along with

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standard deviations and confidence intervals for each of the two treatment groups. The estimated survival for the nonmaintained group at 30 weeks is 0.29, whereas it is 0.49 at 31 weeks for the maintained group. Printing the fit of the survival analysis directly yields certain summary statistics for each of the two groups, such as the estimated mean and median survival times. > leuk.fit Call: survfit(formula = Surv(time, status) ∼ group, data = leukemia) 0.95 0.95 n events mean se(mean) median LCL UCL group= 11 7 52.6 19.83 31 18 NA Maintained group= 12 11 22.7 4.18 23 8 NA Nonmaintained With this output, we see that the median survival time for the maintained group is 31 weeks but is only 23 weeks for the nonmaintained group. The next obvious question is whether or not the difference in survival between these two groups is statistically significant. Using the survdiff function, we can conduct a statistical test for the difference between two survival distributions: > survdiff(Surv(time, status)∼group, leukemia) Group N Observed Expected (O-E)^2/E (O-E)^2/V Maintained 11 7 10.69 1.27 3.4 Nonmaintained 12 11 7.31 1.86 3.4 Chisq= 3.4

on 1 degrees of freedom, p= 0.0653

This is only the beginning of S-Plus’s survival analysis capabilities. Use the Help system to find out more options to be used with these models.

8.4.5 Endnote The aim of this book is not to teach statistics to the reader, although much is explained in the text. The focus is primarily to show how to understand and use S-Plus, independent of one’s knowledge of statistics or computing. If your interest in using S-Plus to perform statistics has been sparked and you want to learn more, we refer you to other sources. In particular, you may want to look at Venables and Ripley (1997) or Chambers and Hastie (1992).

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8.5 Exercises Exercise 8.1 In the regression model presented in this chapter, we assumed that education was a continuous variable. Often however, we encounter variables that have discrete levels and must perform a regression model with them. Create four education groupings of roughly equal size, and an indicator variable for each grouping. The first indicator variable should be equal to 1 for each observation in the first grouping and 0 for all other observations. The other indicator variables are similarly defined. Perform a multiple regression with these new indicator variables in place of the continuous version of education. What is the mean education level per grouping? Does the mean fertility differ between the groupings? What additional conclusions can you make from this analysis over the one presented in the chapter? Hint: Use the quantile, cut, and cbind functions.

Exercise 8.2 This exercise is designed to give you some practice with logistic regression, model building, and stepwise regression. Using the Kyphosis data set as in the section on logistic regression in this chapter, perform the following. • Fit a model to the kyphosis data containing only the variable Start as a predictor. • Use update to add a term for Age. • Use anova to compare the fits of the two models. Which is better? • If the model with both terms is better, use update to add the term Number, if not, use update to add Number directly to the model with Start. Compare these models. What do you conclude? You have now constructed a model using one approach. Compare this model with that obtained using stepwise regression. Are the two models the same? Which model do you think is the “best?” (Hint: fit a model with all terms because the stepwise regression procedure (step) removes terms.) Plot the fit of the best fitting model. What can be said from the diagnostic plots, does the model fit well? Try using the plot.gam function and then answer the same question.

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Exercise 8.3 In this exercise, we will lead you through the steps necessary to analyze the drug.mult data set. You will first need to familiarize yourself with the data set by reading its description in the Help system. You will note that it has two factors: gender and time. The time factor is “hidden” in the layout of the data, in that there is a response vector at each time point. An ANOVA model is the typical procedure to use for a continuous outcome variable and factor variables as the predictor variables. We have listed the tasks you will need to perform: • Rearrange the data into a new data frame such that there are variables for the response, sex, time, and subject. (Define sex and subject as class factor and time as class ordered.) • Perform an ANOVA model with response as the outcome and sex as the predictor. What do the results tell you about the effect of sex? • Perform an ANOVA model with response as the outcome and time as the predictor. What do the results tell you about the effect of time? • Perform an ANOVA model with response as the outcome and both sex and time as predictors. What do the results tell you about the effects of sex and time when both are in the model (simultaneous adjustment)? • Perform an ANOVA model with response as the outcome and predictors of sex, time, and the interaction of sex and time. What do the results tell you about the interaction of sex and time? The analysis to this point is fine as a first step, but it not technically correct because it has assumed that the response is independent from one time point to the next. As the same subjects are used at each of the time points, their responses are surely correlated. Continue the analysis with some more advanced techniques. • Confirm that the observations are correlated across the time points. (Hint: The cor and var functions work on matrices.) • Perform a “split-plot” ANOVA. This type of model uses the aov function in conjunction with the Error function. What do the results tell you about the effects of sex and time? • Perform a multivariate analysis of variance (MANOVA) on the data. What do the results tell you about the effect of sex? (Hint: Use the manova function on the original data set.) Summarize the results using the univariate=T option. Does this shed any further light on the results, or confuse things further? What do all the results actually tell us about the effects of sex and time? Do all the results agree? What have we forgotten to do in this analysis? We

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have here a lot of results from fancy statistical techniques and, worst of all, they do not all agree with each other. What we have forgotten is the simple data summaries and graphics to show us what information is really in the data. Most statistical models rely on assumptions that may or may not be met. Simple summary statistics and graphical descriptions of the data cannot be overemphasized. Furthermore, they should be the first step of a data analysis, not something done at the end when the results from models are confusing. Nonetheless, the analysis for this exercise will conclude with a few graphical methods that are of tremendous help in trying to interpret effects in ANOVA models. • Perform summary statistics on the variables (not shown in solution). • Explore the main effects of the sex, time, and subject factors by using the graphical function plot.design. • Explore the interaction of sex and time by using the graphical function interaction.plot. Can you now make more informed conclusions about the effects of time and sex? Can you see why some people claim that statistics is an art rather than a science? For statisticians reading the book, the analysis is still not complete (although the solution is). The MANOVA test is probably too conservative to use for the final conclusion. The split-plot results rely on the assumption that the variance–covariance matrix is spherical. Looking at the variance– covariance matrix (and/or correlation matrix), it seems likely that it is not spherical. You could make a Huynh–Feldt adjustment to the degrees of freedom in the split-plot test and recompute the p-value, but this is only an approximation to departures from nonsphericity. The best option is to perform a longitudinal analysis of the data. With this modeling approach, it is possible to choose a model that is between the generality of the MANOVA model and the simplicity of the simple ANOVA (or split-plot) model. Longitudinal models can be fit with the lme function, standing for linear mixed effects. The statistics involved with longitudinal techniques are far beyond the scope of this book, but for those with the desire, read the documention for the lme function and carry this analysis one step further.

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8.6 Solutions Solution to Exercise 8.1 To make our lives easier, we first extract the vectors of interest so that we don’t have to deal with matrices. > education fertility quants.educ quants.educ 0% 25% 50% 75% 100% 1 6 8 12 53 We will use the cut function to use the cutpoints defined by the quantiles to create a variable that contains a 1 for the first group, a 2 for the second group, and so on. Note that we have also used the include.lowest option to include the minimum value. Groups are normally defined starting with any value greater than the lower cutpoint and up to and including the upper cutpoint. This method’s flaw is that the minimum value gets left out. > cuts.educ tapply(education, cuts.educ, length) 1+ thru 6 6+ thru 8 8+ thru 12 12+ thru 53 14 11 11 11 > tapply(education, cuts.educ, mean) 1+ thru 6 6+ thru 8 8+ thru 12 12+ thru 53 3.785714 7.363636 10.72727 24 To make our indicator variables, we first make four vectors and fill them with zeros. We then put ones into the vectors in the appropriate elements.

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> + > > > >

educ1 fit.c summary(fit.c) Call: glm(formula = Kyphosis ∼ Start + Number, family = binomial, data = kyphosis) Deviance Residuals: Min 1Q Median 3Q Max -1.903483 -0.5938547 -0.3885682 -0.2490264 2.214024 Coefficients: Value Std. Error t value (Intercept) -1.0288475 1.22345046 -0.8409392 Start -0.1849421 0.06397197 -2.8909871 Number 0.3574336 0.19963697 1.7904177

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(Dispersion Parameter for Binomial family taken to be 1 ) Null Deviance: 83.23447 on 80 degrees of freedom Residual Deviance: 64.53647 on 78 degrees of freedom Number of Fisher Scoring Iterations: 4 Correlation of Coefficients: (Intercept) Start Start -0.6285140 Number -0.8487089 0.2077516 We see that the term Number is not statistically significant, but note that the t-value of Start has decreased compared to its previous values indicating that the two predictors are not completely independent of one another. This is all the more reason to perform the chi-squared test from the anova function. > anova(fit.c, test="Chi") Analysis of Deviance Table Binomial model Response: Kyphosis Terms added sequentially (first to last) Df Deviance Resid. Df Resid. Dev Pr(Chi) NULL 80 83.23447 Start 1 15.16229 79 68.07218 0.00009865 Number 1 3.53571 78 64.53647 0.06006055 We see here that the residual deviance has been reduced by 3.5 with the addition of the term Number which is nearly statistically significant (p=0.06). Since the p-value is so close to 0.05, we choose to keep the term Number in the final model although this is perhaps not fully justified. The default for the stepwise regression function in S-Plus (step) is to start with a full model and drop terms that are deemed to be unnecessary. Therefore, we simply need to feed the full model to step and see what it does. > fit.all fit.all Start: AIC= 69.3799 Kyphosis ∼ Age + Number + Start

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Single term deletions Model: Kyphosis ∼ Age + Number + Start scale:

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Df Sum of Sq RSS Cp <none> 70.26871 78.26871 Age 1 2.877010 73.14572 79.14572 Number 1 3.336562 73.60527 79.60527 Start 1 9.308866 79.57757 85.57757 Call: glm(formula = Kyphosis ∼ Age + Number + Start, family = binomial, data = kyphosis) Coefficients: (Intercept) Age Number Start -2.036932 0.01093048 0.410601 -0.20651 Degrees of Freedom: 81 Total; 77 Residual Residual Deviance: 61.37993 The stepwise deletion procedure hasn’t removed any of the terms. However, we just saw that a stepwise addition procedure (done manually) doesn’t arrive at the full model. This occurs frequently because of the criteria set for the automatic stepwise procedure. If they had been fine-tuned, we might have reached the same final model. In any case, we feel more confident in the model that we built manually (fit.c). We should now look at some diagnostic plots to see if the overall model fits the data well. A plot of fit.c is not very helpful in answering our question (see Figure 8.2). Because the response variable kyphosis is dichotomous and takes only the values 0 and 1, the standard diagnostic plots, that work so well with Gaussian distributed data, don’t convey much information. In this setting, it is much more useful to use one of the special plotting functions, plot.glm and plot.gam, which have been specially designed to plot more appropriate diagnostic graphs for these type of models. We have chosen to use plot.gam using the following code. > par(mfrow=c(1, 2)) > plot.gam(fit.c, resid=T, scale=6) The option resid specifies that partial residuals are to be plotted, and the scale option specifies the number of units that are to be plotted on the y-axis (important when trying to keep the same axis for all graphs). The resulting graph is shown in Figure 8.3. The partial residuals show that the

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linear relationship used with both predictor variables included in the model is approximately correct. Another question that could be asked at this point is how well the model could be used to predict outcomes. With this model, would we have predicted any false positives? False negatives? These question, and lots of related ones, are left to the reader to explore.

Solution to Exercise 8.3 The drug.mult data set is in matrix format with one column for each time point and one row for each subject. There is also a column indicating the gender of each patient. With this type of a setup, the response is in several columns, one for each level of one of the factors of interest. For ANOVA models, S-Plus requires that all the responses be together in one column, so we need to transform the data to a more suitable arrangement. We create the factor variables using the rep function and then concatenate the responses using the c function. > subj sex > + >

time nd.aov.both summary(nd.aov.both) Df Sum of Sq Mean Sq F Value Pr(F) sex 1 60.80167 60.80167 19.75149 0.000278263 time 3 49.10833 16.36944 5.31763 0.007861779 Residuals 19 58.48833 3.07833 Notice that the variable time is highly significant in the joint model. This tells us that the mean response differs between males and females, and also between the time points. The next question to consider is the following. Is there a significant interaction between sex and time? One example of an interaction would be to have a large mean for males and a small mean for females at the early time points, but to have the reverse at the late time points. In the model below, we add the interaction term. > nd.aov.inter summary(nd.aov.inter) Df Sum of Sq Mean Sq F Value Pr(F) sex 1 60.80167 60.80167 22.26827 0.0002317 time 3 49.10833 16.36944 5.99522 0.0061389 sex:time 3 14.80167 4.93389 1.80701 0.1864698 Residuals 16 43.68667 2.73042 We are now reasonably satisfied that the interaction between time and sex is not significant and stick to the conclusion that there are main effects of both variables. There is an obvious problem with the analyses we have performed up to this point that all boils down to correlation. Take any particular subject. Is his/her response at time 1 independent from his/her response at time 2? Although it is theoretically possible that these responses are independent, it

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is probable that they are correlated. The ANOVA model, however, assumes that all responses are independent from one another. The design of the study is somewhat typical and can be run in the ANOVA setting under certain other conditions. If we introduce a random effect for the subjects, we can make an adjustment for the correlation that exists. The type of model we run in this case is referred to as a split-plot ANOVA model and is fit in S-Plus with the simple addition of one extra term: > nd.split.plot summary(nd.split.plot) Error: subj Df Sum of Sq Mean Sq F Value Pr(F) sex 1 60.80167 60.80167 19.32256 0.01173 Residuals 4 12.58667 3.14667 Error: Within Df Sum of Sq Mean Sq F Value Pr(F) time 3 49.10833 16.36944 6.316184 0.0081378 sex:time 3 14.80167 4.93389 1.903751 0.1828514 Residuals 12 31.10000 2.59167 The output is slightly different in that it is split into two parts. In the “Error: subj” section, the “Residuals” term refers to the subjects nested within sex. This term has 4 degrees of freedom, n−1 for the females (=2) and n−1 for the males (=2). In the “Error: Within” section, the “Residuals” term stands for the interaction between sex, time, and subject nested within sex. These are perhaps statistical details and can be ignored. The main point is that we have new tests for sex, time, and their interaction. Notice that the interaction is still nonsignificant, and that sex and time are still both significant. Notice that the p-value for time has hardly changed at all, whereas the p-value for sex, although still significant, has increased due to the correction for correlation. The split-plot ANOVA model assumes that the variance–covariance matrix across the time points is spherical. In practice, the definition of sphericity is met with the compound symmetry variance structure that assumes a common variance along the diagonal and a different but constant covariance in the off-diagonals. This seems to be a fairly restrictive assumption, but one that fits the data quite well in many practical situations. Without going into a test to see whether or not the sphericity assumption of the split-model holds, it is also possible to run a multivariate analysis of variance (MANOVA) on the data. This type of model is designed for responses that are correlated, be it different response variables (pulse and blood pressure) or, as in our case, responses at different time points.

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One drawback is that we will lose the ability to test for an effect of time. The MANOVA assumes a general covariance structure such that the assumptions are always met (still must have Gaussian distributed data). This generality comes with a price called reduced power. Since it can test against any alternative hypothesis, it does not necessarily have power to test the particular hypothesis we have. This means that the MANOVA test might say that the effect of gender is not significant when it really is. If, however, this test says that gender is significant related to the response, then we can be sure of our result. > nd.manova summary(nd.manova) Df Pillai approx. F num den P-value Trace df df gender 1 0.928614 3.25209 4 1 0.391236 Residuals 4 > summary(nd.manova, univariate=T) Response: Y.1 Df Sum of Sq Mean Sq F Value Pr(F) gender 1 7.260000 7.260000 2.93531 0.161819 Residuals 4 9.893333 2.473333 Response: Y.2 Df Sum of Sq Mean Sq F Value Pr(F) gender 1 13.20167 13.20167 3.345017 0.1414016 Residuals 4 15.78667 3.94667 Response: Y.3 Df Sum of Sq Mean Sq F Value Pr(F) gender 1 4.681667 4.681667 35.55696 0.003971604 Residuals 4 0.526667 0.131667 Response: Y.4 Df Sum of Sq Mean Sq F Value Pr(F) gender 1 50.46 50.46 11.54691 0.02732576 Residuals 4 17.48 4.37 The p-value for gender shows that it is not significantly related to the response (p=0.39). The univariate ANOVAs show that the gender effect is significant at the last two time points. What do we do now? Which of the analyses do we believe? This is how many statisticians proceed with data analysis. The point is that much of the information contained in the data can be seen through the use of summary statistics and simple graphs. Furthermore, these are

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precisely the strengths of S-Plus and should have been the starting point of our analysis. So, we will now start at the beginning. You have been asked to perform the summary statistics on your own. There are two types of graphs that are of particular interest to ANOVA type models, namely plot.design and interaction.plot. We put the syntax below and show the resulting graphs in Figure 8.4. mean of response 78 79 80 81 82

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> par(mfrow=c(2, 1)) > plot.design(newdrug) > interaction.plot(time, sex, response, data=newdrug) By default, these two functions plot the mean of the response at the different levels of the factors. As an option, you can also pass a different function to be used such as the median. From the graphs, we quickly see that the mean response is higher for males than for females and that the mean response is higher at the later time points that at the earlier ones. In the interaction plot, the mean response is plotted for each gender across the time points. Note that, with the possible exception of the last time point, the curves are parallel, implying that there is no interaction between sex and time.

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One quick check of the sphericity assumption of the split-plot model is to look at the correlation and variance matrices of the data. > attach(drug.mult) > cor(cbind(Y.1, Y.2, Y.3, Y.4)) Y.1 Y.2 Y.3 Y.4 Y.1 1.0000000 0.7684940 0.7084921 0.3772931 Y.2 0.7684940 1.0000000 0.7282511 0.5883456 Y.3 0.7084921 0.7282511 1.0000000 0.8165430 Y.4 0.3772931 0.5883456 0.8165430 1.0000000 > var(cbind(Y.1, Y.2, Y.3, Y.4)) Y.1 Y.2 Y.3 Y.4 Y.1 3.430667 3.427333 1.339333 2.576 Y.2 3.427333 5.797667 1.789667 5.222 Y.3 1.339333 1.789667 1.041667 3.072 Y.4 2.576000 5.222000 3.072000 13.588 > detach() The variance changes from 3.4 to 5.8 to 1.0 to 13.6. This alone is evidence that a spherical variance structure is probably not sufficient to model our data. That leaves us with the alternative of the MANOVA model and the conclusion that the mean response does not differ between males and females. That is a conservative conclusion and many more analyses could be done to “complete” this analysis. In particular, a longitudinal model could be fit using the lme function that fits a variance structure that is between the spherical assumption of the split-plot model and the general structure of the MANOVA model. There are also ways of arranging the data to get a MANOVA type of test of time (take differences from baseline for each patient and then test if the mean vector containing only three responses differs from the zero vector). We started this exercise convinced that there is an effect of gender on the response (i.e., that the mean response is higher for males than for females). After some investigation involving summary statistics, graphical methods, statistical methods, and checking of assumptions, it is rather likely that there is no real difference in the mean responses between the sexes. (The authors will gladly look at any further analyses of this data set if sent by E-mail.)

9 Programming

”Writing program code is a good way of debugging your thinking.” Bill Venables on S-News February 1997

Now that you have learned the elementary commands in S-Plus and many ways of applying them, it is time to discover its advanced programming functionalities. This chapter introduces list structures, loops, and writing functions, covers debugging matters, and gives some more details on how to use S-Plus as a programming language.

9.1 Lists A list is typically a very flexible structure in which to store data. Its main usage is in situations where some data belong together by their contents, but the structure is very heterogeneous. Imagine a database of somebody’s floppy disks, each disk with a number, a short text description of the contents, and a list of files it contains. It would not be a good idea to store this kind of data in a matrix or data frame, because the number of files is different for each disk. A single list can contain different element types; for example, a vector, a matrix, and another list, all at the same time. A list is constructed by calling the function list and enumeration of the objects to go into it. Here is an example for creating a list with three elements, each of them being a vector. > L > > >

x y z L

length(L) 3 You can add a new element by assigning the new element to an unused index: > L[[4]] L[[length(L)+1]] L[[2]] > > >

L

L L$comment names(L) # the names of the elements "number" "bool" "message" "comment" > names(L)[2] names(L) # the new names "number" "true and false" "message" "comment" To access the list element with the name “true and false,” put the name in double quotes (because of the spaces in its name). > L$"true and false" Note Using named list elements is a good idea. In this way, the order and position of the list elements are completely irrelevant and can always be changed. ⊳ Note Named list elements can be addressed in several ways. The following commands all do the same thing: extracting the element “number” from the list L. > > > >

L$number L$"number" L[["number"]] L[[names(L)=="number"]]

Single brackets can be used as well, but the returned result differs slightly. The above expressions return a vector, whereas the expressions below return a list of length 1. > L["number"] > L[names(L)=="number"] It is not advisable to use single brackets ("[" and "]") to reference list elements, as these are also used for array-like structures. Using double brackets or the $ operator makes programs easier to read. For S-Plus for UNIX 3.x and S-Plus for Windows 4.5/2000, single brackets are useful for accessing more than one element at a time. The statement L[[names(L)=="number"]] generates an error, as a vector of (TRUE/FALSE) indices is used. ⊳

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9.1.3 Applying the Same Function to List Elements It is often very convenient to keep objects of the same type in a list, for example, the country data we have used earlier. S-Plus provides some useful functions that enable the application of a function to all elements of the list, making the task easy and efficient.

lapply The tool lapply is designed for working on all elements of a list using the same function. We use the country data again and calculate some statistics of the list elements: > + + >

countries lapply(countries, sum) $GDP: 4440 $Pop: 155 $Area: 1027 Remember that all standard arithmetic is also a function within S-Plus. So we can also subtract 1 from each element of the list by entering > lapply(countries, "-", 1) You see that all additional parameters passed to lapply are passed unmodified to the function "-". The minus operator "-" is a function like any other. You could enter 5-2 or alternatively "-"(5, 2). In order to see its declaration, use get("-"). The index selection functions ¨[¨ and ¨[[¨ can be used in the same way. We extract the first element of each list component by supplying the parameter 1 to the function ¨[¨.

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> lapply(countries, "[", 1) $GDP: 227 $Pop: 8 $Area: 84 You might have noticed as well that we have lost the names of the countries as labels. We can add them to each list element as follows: > + > +

country.names sapply(countries, sum) GDP Pop Area 4440 155 1027 whereas using sapply to calculate summary statistics of all list elements gives

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> sapply(countries, summary) GDP Pop Area 227.0 7.00 41.00 292.2 7.75 73.25 924.0 33.00 221.00 1110.0 38.75 256.70 1742.0 64.00 404.50 2365.0 82.00 544.00

rapply If our data structure is a more complicated list, there is another function that might help: rapply descends down through all levels of a list if we have a list of lists, and it lets us define a function to apply only to objects of a certain class.1 One needs to specify the object to process and the function to apply. Optionally, one can set a restriction to use only objects of certain classes by specifying classes="vector of class names" and how to treat the result (the default is unlisting the list). Here is a simple example. Let us define a list of four components, the fourth being a list again. > x rapply(x, rev, how="replace") $four: $four$B: "G" "F" "E" $four$A: 3 2 1 $three: 123 122 121 120 119 118 117 116 115 114 113 112 111 $two: "d" "c" "b" "a" $one: 5 4 3 2 1 1 The

function rapply was introduced in S-Plus Version 5.0 and is not part of S-Plus for UNIX 3.x or S-Plus for Windows 4.5/2000.

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You can see that the elements were reversed on all levels. The list element $four is now in position 1, and all its components are reversed as well. Let us calculate the mean of all numeric elements in the list: > rapply(x, mean, "numeric", how="replace") Problem in rapply(x, mean, "numeric", how="replace..: must supply a function of one argument Use traceback() to see the call stack S-Plus informs us that the function to apply must have exactly one argument. The function mean has one argument only (plus an option), such that we have to use a little trick: > f rapply(x, f, "numeric", how="replace") $one: 3 $two: "a" "b" "c" "d" $three: 117 $four: $four$A: 2 $four$B: "E" "F" "G"

9.1.4 Unlisting a List Sometimes, it is necessary to extract the elements of a list to compose them again into a single structure. To do this, the function unlist takes all elements of a list and combines them in the most global data structure. For example, a list with vectors and matrices with numeric contents becomes one long vector containing all the numbers; a list with numeric and character contents becomes a character mode variable.

9.1.5 Generating a List by Using split The split function splits up one variable according to another. The result is a list with as many elements as there are distinct values in the second variable. For example,

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> split(barley$yield, barley$site) generates a list of length 6, as there are six sites in the Barley data. Each list component contains the yield for one of the sites. The names of the list elements correspond to the levels of the site variable. The split data are now ready to apply a function to the yield data of each site separately.

9.2 Writing Functions Just like a macro in other languages, a function is simply a sequence of commands put together into a single new command, the function’s name. In contrast to macros, S-Plus functions offer the same flexibility and capability as other modern programming languages. Functions have arguments passed to them, and results are returned to the calling routine. All other variables defined in the function’s body are kept internally and disappear after the function is executed. In S-Plus, a function and its usage are very similar to their usage in mathematics. In mathematics, you write y = f (x), as you do in S-Plus: > y cube cube(3) 27 but you can also supply a vector, > x cube(x/2) 0.125 1.000 3.375 8.000 15.625 call it recursively, > cube(cube(x)) 1 512 19683 262144 1953125

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or pass a structure like a matrix. > x x 1 3 2 4 > cube(x) 1 27 8 64 A function can have more than one argument (or none at all). We will imitate the system’s division function. Yes, the operand "/" in S-Plus is itself a function, which can be accessed and modified exactly like any other function. Enter get("/") to see the declaration. The following function simply divides the first argument supplied by the second: > divide divide(7, 4) 1.75 # two numbers as arguments > a b divide(b, a) # two vectors as arguments 4 3.5 3.14286 2.875 2.6666 2.5 > divide(1:10, 3) # a vector and a number 0.3333 0.6666 1.0000 1.3333 1.6666 2.0000 2.3333 2.6666 3.0000 3.3333 The more general syntax of a function declaration is as follows: function-name f ?f would give a description of our function and its arguments, together with the little help text we entered. As it is very easy to document functions in this fashion, you should make it a custom to add a few lines of comments to each function declaration.

9.2.2 Scope of Variables We will now cover the scope of variables declared within functions in more detail. To begin, note that all variables declared inside the body of a function are local and vanish after the function is executed. Their scope (the environment where the variables are known) does not extend beyond the function’s limits. To demonstrate this effect, we use the following example. Pay attention to the value of z before and after execution of the function f. > > > > >

z f x y z

f() Moreover, the parameters can use other parameters in the default settings. See the next example and note that y is a function of x in the declaration of y, and that error is a function of both x and y. Example 9.2. A function with cross-referencing default parameters The parameters of the function declared use other parameters to set default values. f f2(x=3) to obtain the same result (that x gets assigned the value 3 in the function). Sometimes, like in the boxplot function, it is useful to have the ... in the beginning, followed by optional arguments for special purposes. On the other hand, it is not useful to put required arguments behind the ... parameter. The following example shows how ... arguments are treated. Example 9.3. Usage of ... as a function argument The following function shows how to treat an unspecified number of parameters using the ... expression. We compare the distributions of the data passed to the function. To do this, we calculate the minimum and maximum over all data and set the x-axis to the same boundaries for all histograms: f f(sin(pi)) # pass an expression "sin(pi)" In this way, you could add a title to a graph containing what was passed as x:

> title(deparse(substitute(x))) If a function has more than just one parameter, the x in the expression deparse(substitute(x)) can, of course, be replaced by any other parameter name. Another useful tool is the functon sys.call(). It returns the complete call to the function in form of a list. The first element is the name of the function, the second element is the first argument, etc.

9.3 Iteration An iteration is, in principle, a loop or repeatedly executed instruction cycle, with only a few changes in each cycle. In programming languages that are not matrix or array oriented, like C or Fortran, even simple matrix multiplication needs three nested loops (over rows, columns, and the indices). Since S-Plus is matrix oriented, these operations are much more efficient and easy to formulate in mathematical terms. This means they are faster than loops and the code is much easier to read and write. Pay attention to the following guideline, which is elementary but needs to be kept in mind when writing S-Plus code. Note Whenever possible, try to avoid loops. Ninety-nine percent of the time, an operation on matrices or lists is much more elegant as well as much faster. Try to use vectorized statements and functions like apply, lapply, rapply, sapply, tapply, by, split, etc. ⊳ Here is an easy example. If we define a vector x with 100,000 elements and assign the square of x to y, without using a loop, the following two lines take about half a second to execute:

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> x y x for(i in 1:10) { print(i) } If there is only a single statement after the for command, the brackets { } can be omitted, but for ease of reading the code and to make sure the intended statements are included in the loop, we recommend using them at all times. The object ivalues can be of any type; a vector, a variable, a matrix, or anything else. S-Plus walks through all the elements of the argument specified, replacing the loop variable (here i) by the elements in ivalues successively. The looping variable can be of any mode, like numeric, character, boolean, or a mixture of these. Here are a few examples. Example 9.5. A numeric looping variable > for(i in c(3, 2, 9, 6)) { print(i^2) } 9 4 81 36 Example 9.6. A character looping variable > transport.media for(vehicle in transport.media) { print(vehicle) } "car" "bus" "train" "bike"

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Note There is no explicit return argument in loops. Thus, you must always use the print function to print values within a loop. Using a statement like > for(i in 1:4) { i }

to print out the current value of i does not result in printouts of the values i takes. ⊳

9.3.2 The while Loop In contrast to the for loop, the while statement is used if the total number of iteration cycles is not known before the loop starts. For example, if you want to iterate until a certain criterion is fulfilled, the while loop is the appropriate statement to use. Example 9.7. A simple while loop Calculate the sum over 1,2,3,..., until the sum is larger than 1000. n 1000][1]

9.3.5 Large Loops There are situations where a loop cannot be avoided. If these loops have many cycles or if the block of commands is long, S-Plus might need a lot of memory and maybe even stop execution of the command due to lack of memory. This is often the situation where one starts to write the program in C. It can still be used from within S-Plus, as we will see later. However, let us point out a function that might turn out to be useful. The problem with large loops is that S-Plus does not free memory until completion of an expression. The function For circumvents this problem by basically splitting up a for loop into a series of top-level expressions. The syntax is slightly different than the for function, but, most important, all variables used within a For loop must be global variables, not local ones defined within functions.

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Furthermore, all variables defined within the For loop will be global ones in the local directory, even after the loop completes. Here is an example of a function f with a simple loop: f x f(x) 6 7 8 NULL The function was run now, and the last expression of the For statement was printed at the end of each cycle. Let us check the value of x: > x 8 The global variable x is modified! This illustrates our statement that all assignments are done within the global environment. We have made another (possible) mistake. The argument to the function, x, is completely ignored by For. The global variable x is used, no matter what gets passed to f. This is demonstrated by calling f again: > f(99) 9 10 11 NULL > x 11 Obviously, one needs to experiment a little with the function and its side effects. If the for statement comprises a set of commands, enclose them in braces ({ and }) and separate the statements by semicolons (;). A larger loop is mostly run as a batch job, so you might want to save your commands to a file and run a batch job (described in Section 13.4, p. 350).

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9.4 Debugging: Searching for Errors Now it is time to write your own functions. If you haven’t tried this yet, don’t worry. You’ll have plenty of opportunities in the exercise section. Of course, when we write functions, they don’t always do what we want. This is the trouble with computers in general. They do exactly what we tell them to do, but this is very often not what we want them to do. This section is about searching for sources of errors. We will deal with four types of errors: • Syntax errors While the system is reading the definition of a function, it encounters a problem and stops reading the definition. An example is to close more brackets than we have opened. We are given an error message, typically beginning with Syntax Error: ... • Invalid arguments The function received arguments that do not allow it to correctly execute the function. Calling a routine to do a calculation on numbers and passing character variables to it causes such errors. • Execution errors While executing the function, S-Plus encounters an error like an undefined operation. One such example is plotting x against y with x and y having different lengths. S-Plus stops the execution and we need to track where the error source originates. Most of the time, S-Plus tells us Error in ... If the execution was aborted in a statement that is not the origin of the error, as when we try to plot only missing values, we need to find the source with tools referenced in the following item. • Logical errors The S-Plus system completes the operation without detecting an error. However, the result returned to us is wrong; for example, the mean value of a data set being smaller than the smallest value. We need to track the whole function for a logical error that caused a programming error. The above list of problems and errors is, in general, in order of difficulty of finding the source of the problem. We will now tackle them systematically, introducing the tools S-Plus has to handle them.

9.4.1 Syntax Errors Syntax errors are detected most easily. We might have forgotten a closing bracket or confused the name of a function with something else, and the

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system tells us while reading in the declaration, that it disagrees with what it has just read: Syntax error: Unbalanced parentheses, expected "}", before ")" at this point: Such a message is typically followed by the line of code in question, and having another look at the code should clarify what the problem was. Note If S-Plus gives a syntax error when reading something in from a file, and you cannot see directly what the problem was - maybe because you cannot find the line of code - mark the code using the mouse, and copy and paste it into the command window. S-Plus will react on the input directly and you can see from where the problem is coming. ⊳ When executing a longer sequence of commands (e.g., if the source command is used to read in commands from a file), it helps to set > options(echo=T) To change S-Plus’s sensitivity to warnings, you can change the option warn to the values from −1 to 3. Setting > options(warn=3) lets S-Plus terminate the current execution whenever a warning occurs. By default (which is options(warn=0)), warnings are collected and returned at the end of a statement’s execution.

9.4.2 Invalid Arguments When writing a function, some errors can be checked for automatically. For example, if you write a function that gets an input parameter x, and x is not allowed to be negative, you can check for it and exit the function in a controlled manner by issuing an error message: f traceback() 5: eval(action, sys.parent()) 4: doErrorAction("Problem in x^2: Non-numeric first operand", 1000) 3: x^2 2: f("aha") 1: Message: Problem in x^2: Non-numeric first operand We see the list of calls, and the problem becomes clear. The function f was called as f("aha"). Within f, an expression x^2 was executed, and an error action was started with the message “Problem in xˆ2: Non-numeric first operand.” Now we know where the error occurred. If we see the problem directly, fine. If not, we might want to examine the variable’s contents during the function’s execution, or get some more information on the other commands being processed. This is covered in the following section.

9.4.4 Logical Errors If you are familiar with other programming languages like C, you might know that these languages typically come with a debugger. The purpose of a debugger is to trace the state of a program while it is being executed. To do this, the program is not executed all at once, but interrupted in certain places or executed line by line. S-Plus has these facilities, too. If we know where a problem occurs but are unsure what the problem is, we might want to interrupt the program in that specific place and look at the contents of the variables. There are several ways of doing it.

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Note The functionality covered in this chapter describes the implementation in S-Plus Version 5.0 and higher. Most of the tools exist in previous versions like S-Plus for UNIX 3.4 and S-Plus for Windows 4.5 and 2000. The syntax and options differ slightly though. ⊳

browser The browser function called within another function interrupts the execution and lets the user query contents of variables and inspect levels of calls. We can also overwrite any variables defined in the function by assigning something to them. The call can be inserted manually into a function, as in f trace(f, browser) to call browser on entering the function f,

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> trace(f, exit=browser) to enter browser on exiting the function f, and > trace(f, browser, exit=browser) to enter browser on entering and exiting f. Replace f by any function name. The function untrace ends the tracing process. Call untrace(f) to stop tracing the function f or untrace() to stop all tracing.

debugger If you want to run a debugger after your function crashed, tell S-Plus to dump all current frames on exiting the function. S-Plus will write out all levels of calls and their variables on exiting. You can then run the debugger. As dumping out all frames might take some time, depending on the complexity of the function, the system default when an error occurs is to dump only the levels of function calls. By default, the settings are > options(error=expression(dump.calls())) If we want to go through the levels of function calls and examine the variables’ contents, we need to change the setting to > options(error=expression(dump.frames())) and run the function again. The dump will then be done (if the function crashes again), and we can inspect the state of the system by using the debugger: > debugger() If your function works fine, you might want to reset the error action to just dumping calls.

recover S-Plus can automatically enter a debugging tool if an error occurs. The function recover has basically the same functionality as browser. We use it to examine the state of the system in case of an error. The function recover does not exist under S-Plus for Windows 4 and 2000. Have a look at the function inspect instead (which does not exist under S-Plus Version 5 for UNIX). S-Plus is told to call recover each time an error occurs by setting an option as follows: > options(error=expression(if(interactive()) recover() + else dump.calls())) If S-Plus is not run interactively but in batch mode, recover should not be called, as it expects interactive input via the keyboard. That is why we check if S-Plus is in the interactive mode by using interactive.

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If an error occurs, you are prompted with a message and asked if you want to debug the problem: Problem in x^2: Non-numeric first operand Debug ? ( y|n ): Enter a “y” to start the recover function. Table 9.1 lists the basic commands that are common to browser, recover, and debugger. Table 9.1. Some commands in browser, debugger, and recover

Command ? where up (down) q object any S-Plus expression

Effect Help. Show the names of all variables in the current evaluation frame1 Display current frame list Go up (down) in the list of evaluation frames Quit Show the contents of object Execute the expression and keep results or assignments in the current frame

For S-Plus for UNIX 3.x and S-Plus for Windows 4.5/2000, there is no frame navigation. Use ? for help and 0 to quit. 1 Global variables defined in the interactive environment are also known within any frame. Pick the debugging tool most appropriate for your current problem. Most likely you will have to experiment a little with the different tools and options. Finally, note that you can also debug S-Plus internal routines if necessary. Chapter 10 on object-oriented programming has some more details and an extended example of a debugging session in the exercises.

9.5 Output Using the cat Function In addition to the simple print function, which prints a variable’s value, there is a much more powerful tool: the cat function. If you know the cat function from the C language, you will find many similarities. Here are some examples: > x cat(x) 1 2 3 > cat("Hello.") Hello. > x cat("x has a value of", x, ".") x has a value of 7 . S-Plus and C have similar control characters (see Table 9.2). These control characters can be used in any string, and the S-Plus functions will deal with them accordingly. For example, a graph title going over two lines can be created by inserting a new line control character somewhere in the title string and entering title("First\nand second line"). Note that there is no space around the \n, as this would add an extra space to the title. Table 9.2. Control sequences in S-Plus

Control Sequence

Effect

\n \t \\ \" \b \r \octalcode

New line Tabulator Backslash (\) " Backspace Carriage return Number coding a character in octal format (not ASCII, see a PostScript manual)

9.6 The paste Function The paste function is one of S-Plus’s most universal functions. It takes any number of arguments and concatenates them into strings. Some examples are given as follows: • Title strings for graphics

> s1 s2 title(s1, s2)

• Names for matrix and array rows and columns: Assume that a matrix x with some medical data exists. > row.names col.names dimnames(x) paste(..., sep=" ", collapse=NULL) sep is the field separator, which is set to " " by default. A field separator is put in the space between two concatenated objects. The argument collapse is the object separator, empty by default, such that the objects to be pasted together do not get collapsed. If collapse is modified to "", the result is a long string instead of a vector of strings, if the arguments were vectors. Here are some more examples of how paste can be used: > str paste("String", 1:length(str), "(", str, ") has", + nchar(str), "characters.") "String 1 ( A ) has 1 characters." "String 2 ( vector ) has 6 characters." "String 3 ( of ) has 2 characters." "String 4 ( strings ) has 7 characters." We can solve the problem of writing “character” and “characters,” respectively, very elegantly: > paste("String ", 1:length(str), " (", str, ") has ", + nchar(str), " character", + ifelse(nchar(str)==1, "", "s"), ".", sep="") "String 1 (A) has 1 character." "String 2 (vector) has 6 characters." "String 3 (of) has 2 characters." "String 4 (strings) has 7 characters." The next example uses different settings of sep and collapse. > x names(x) paste(names(x), "=", 1:3, ".") "X = 1 ." "Y = 2 ." "Z = 3 ." > paste(names(x), "=", 1:3, ".", sep="") "X=1." "Y=2." "Z=3." > paste(names(x), "=", 1:3, ".", collapse="") "X = 1 .Y = 2 .Z = 3 ." > paste(names(x), "=", 1:3, ".", sep="", collapse="") "X=1.Y=2.Z=3." > paste(names(x), "=", 1:3, ".", sep="", collapse=" ") "X=1. Y=2. Z=3."

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9.7 Exercises Exercise 9.1 Take a list L generated as follows. > L L sapply(L, mean) If we wanted to, we could generate the matrix containing mean, variance, and number of observations of each list element with a single command: > x dimnames(x) trace(show, browser) calls browser each time the function show is entered. > trace(show, browser, exit=browser) calls browser on entering and exiting the function show. > trace(show, exit=browser) calls browser on exiting the function show. To stop tracing of all functions, the function untrace can be used: > untrace() If there are too many calls to the function we want to trace, we can only trace it for calls with objects of certain classes like country. The function traceMethod is called as follows to trace only calls to show with objects of class country: > traceMethod("show", "country", browser) The arguments are the same as for trace.

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10.4 Help Besides the standard online help, > help("helptopic") there are some ways to query object-oriented help pages. To query the definition of a class, for example, matrix, enter > class ? matrix If we are interested in the methods provided, we can ask for the generic function and its associated library of class-specific functions. > method ? show displays the list of class-specific show functions. We can also ask directly for the show function for objects of class country by entering > method ? show("country")

10.5 Summary and Overview Recall again what we have done in this chapter. We have created a new class country. We have defined a function to generate objects of the new class, and we have added class-specific methods to show and plot objects of class country. We have also added a new method "

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