Modelling Binary Data

381 Pages·1991·10.961 MB·English

by  D. Collett (auth.)

Modelling Binary
Data
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Modelling Binary
Data
D. Collett
Department of Applied Statistics,
University of Reading, UK
lal11
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
First edition 1991
©D. Collett 1991
Originally published by Chapman & Hall in 1991
Softcover reprint of the hardcover 1st edition 1991
Typeset in 9J/lli Times by Interprint Limited, Malta
ISBN 978-0-412-38790-6 ISBN 978-1-4899-4475-7 (eBook)
DOI 10.1007/978-1-4899-4475-7
Apart from any fair dealing for the purposes of research or private study, or criticism or review,
as permitted under the UK Copyright Designs and Patents Act, 1988, this publication may not be
reproduced, stored, or transmitted, in any form or by any means, without the prior permission
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sent to the publishers at the London address printed on this page.
The publisher makes no representation, express or implied, with regard to the accuracy of
the information contained in this book and cannot accept any legal responsibility or liability for
any errors or omissions that may be made.
The GLIM macros contained in Appendix C may be reproduced and used, with
no implied warranty from the author or publisher.
A catalogue record for this book is available from the British Library
Library of Congress Cataloging-in-Publication Data
Collett, D., 1952-
Modelling binary data / D. Collett.
p. cm.
Includes bibliographical references and index.
ISBN 978-0-412-38790-6
1. Analysis of variance. 2. Distribution (Probability theory)
3. Linear models (Statistics) I. Title.
QA279.C64 1991
519.5'38 - dc20 91-23845
CIP
To Janet
Contents
Preface xi
1 Introduction 1
1.1 Some examples 1
1.2 The scope of this book 14
1.3 Use of statistical software 15
Further reading 16
2 Statistical inference for binary data 17
2.1 The binomial distribution 17
2.2 Inference about the success probability 21
2.3 Comparison of two proportions 29
2.4 Comparison of two or more proportions 37
Further reading 42
3 Models for binary and binomial data 43
3.1 Statistical modelling 43
3.2 Linear models 45
3.3 Methods of estimation 48
3.4 Fitting linear models to binomial data 51
3.5 Models for binomial response data 53
3.6 The linear logistic model 56
3.7 Fitting the linear logistic model to binomial data 56
3.8 Goodness of fit of a linear logistic model 62
3.9 Comparing linear logistic models 67
3.10 Linear trends in proportions 74
3.11 Comparing stimulus-response relationships 77
3.12 Non-convergence and overfitting 81
3.13 A further example on model selection 83
3.14 Predicting a binary response probability 88
Further reading 91
4 Bioassay and some other applications 92
4.1 The tolerance distribution 92
viii Contents
4.2 Estimating an effective dose 96
4.3 Relative potency 101
4.4 Natural response 105
4.5 Non-linear logistic regression models 108
4.6 Applications of the complementary log-log model 112
Further reading 118
5 Model checking 120
5.1 Definition of residuals 121
5.2 Checking the form of the linear predictor 126
5.3 Checking the adequacy of the link function 138
5.4 Identification of outlying observations 141
5.5 Identification of influential observations 146
5.6 Checking the assumption of a binomial distribution 160
5.7 Model checking for binary data 162
5.8 Summary and recommendations 177
5.9 A further example on the use of diagnostics 179
Further reading 186
6 Overdispersion 188
6.1 Potential causes of overdispersion 188
6.2 Modelling variability in response probabilities 192
6.3 Modelling correlation between binary responses 194
6.4 Modelling overdispersed data 195
6.5 The special case of equal n, 199
6.6 The beta-binomial model 204
6.7 Random effects in a linear logistic model 205
6.8 Comparison of methods 215
6.9 A further example on modelling overdispersion 216
Further reading 221
7 ModeUing data from epidemiological studies 223
7.1 Basic designs for aetiological studies 224
7.2 Measures of association between disease and exposure 227
7.3 Confounding and interaction 231
7.4 The linear logistic model for data from cohort studies 234
7.5 Interpreting the parameters in a linear logistic model 238
7.6 The linear logistic model for data from case-control studies 251
7.7 Matched case-control studies 260
7.8 A matched case-control study on sudden infant death syndrome 268
Further reading 275
8 Some additional topics 277
8.1 Analysis of proportions and percentages 277
Contents ix
8.2 Analysis of rates 278
8.3 Analysis of binary data from cross-over trials 278
8.4 Random effects modelling 279
8.5 Modelling errors in the measurement of explanatory variables 280
8.6 Analysis of binary time series 281
8.7 Multivariate binary data 281
8.8 Experimental design 282
9 Computer software for modelling binary data 284
9.1 Statistical packages for modelling binary data 285
9.2 Computer-based analyses of example data sets 290
9.3 Using packages to perform some non-standard analyses 326
9.4 Summary of the relative merits of packages for modelling binary data 336
Further reading 338
Appendix A 340
Appendix B 342
B.1 An algorithm for fitting a generalized linear model to
binomial data 342
B.2 The likelihood function for a matched case--control study 346
Appendix C 349
C.1 GUM macro for computing Anscombe residuals 349
C.2 GUM macro for constructing a half-normal plot of the
standardised deviance residuals with simulated envelopes 350
C.3 GUM macro for constructing a smoothed line 350
C.4 GUM macro for computing the LIp-statistic 351
C.5 GUM macro for computing the LID-statistic 352
C.6 GUM macro for computing the C-statistic 353
C.7 GUM macro for implementing the Williams procedure
for modelling overdispersion 353
References 355
Index of Examples 362
Index 364
Preface
Data are said to be binary when each observation falls into one of two categories,
such as alive or dead, positive or negative, defective or non-defective, and success
or failure. In this book, it is shown how data of this type can be analysed using a
modelling approach. There are several books that are devoted to the analysis of
binary data, and a larger number that include material on this topic. However, there
does appear to be a need for a textbook of an intermediate level which dwells on
the practical aspects of modelling binary data, which incorporates recent work on
checking the adequacy of fitted models, and which shows how modern computa
tional facilities can be fully exploited. This book is designed to meet that need.
The book begins with a description of a number of studies in which binary data
have been collected. These data sets, and others besides, are then used to illustrate
the techniques that are presented in the subsequent chapters. The majority of the
examples are drawn from the agricultural, biological and medical sciences, mainly
because these are the areas of application in which binary data are most frequently
encountered. Naturally, the methods described in this book can be applied to
binary data from other disciplines.
Underlying most analyses of binary data is the assumption that the observations
are from a binomial distribution, and in Chapter 2, a number of standard statistical
procedures based on this distribution are described. The modelling approach is
then introduced in Chapter 3, with particular emphasis being placed on the use of
the linear logistic model. The analysis of binary data from biological assays, and
other applications of models for binary data, are covered in Chapter 4.
A distinguishing feature of this book is the amount of material that has been
included on methods for assessing the adequacy of a fitted model, and the
phenomenon of overdispersion. Chapter 5 provides a comprehensive account of
model checking diagnostics, while models for overdispersed data, that is, data
which are more variable than the binomial distribution can accommodate, are
reviewed in Chapter 6. Both of these chapters include a summary of the methods
that are most likely to be useful on a routine basis.
A major area of application of the linear logistic model is to the analysis of data
from epidemiological studies. In Chapter 7, we see how this model provides the
basis of a systematic approach for analysing data from cohort studies and both
unmatched and matched case-control studies. Chapter 8 contains a brief discussion
of some additional topics in the analysis of binary data.