MODELLING BINARY DATA Second Edition CHAPMAN & HALL/CRC Texts in Statistical Science Series Series Editors C. Chatfield, University of Bath, UK Jim Lindsey, University of Liège, Belgium Martin Tanner, Northwestern University, USA J. Zidek, University of British Columbia, Canada Analysis of Failure and Survival Data Epidemiology — Study Design and Peter J. Smith Data Analysis The Analysis and Interpretation of M.
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Rao Second Edition Statistical Analysis of Reliability Data R. Kimber, The Theory of Linear Models T. Jørgensen MODELLING BINARY DATA Second Edition David Collett School of Applied Statistics The University of Reading, UK CHAPMAN & HALL/CRC A CRC Press Company Boca Raton London New York Washington, D. CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2003 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.
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CCC is a not-for-profit organization that pro- vides licenses and registration for a variety of users. For organizations that have been granted a pho- tocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.com and the CRC Press Web site at http://www.com To Janet Contents 1 Introduction 1 1.2 The scope of this book 14 1.3 Use of statistical software 15 1.4 Further reading 16 2 Statistical inference for binary data 19 2.1 The binomial distribution 19 2.2 Inference about the success probability 23 2.3 Comparison of two proportions 31 2.4 Comparison of two or more proportions 38 2.5 Further reading 42 3 Models for binary and binomial data 45 3.3 Methods of estimation 50 3.4 Fitting linear models to binomial data 53 3.5 Models for binomial response data 56 3.6 The linear logistic model 58 3.7 Fitting the linear logistic model to binomial data 59 3.8 Goodness of fit of a linear logistic model 65 3.9 Comparing linear logistic models 71 3.10 Linear trend in proportions 78 3.11 Comparing stimulus-response relationships 81 3.12 Non-convergence and overfitting 85 3.13 Some other goodness of fit statistics 87 3.14 Strategy for model selection 91 3.15 Predicting a binary response probability 98 3.16 Further reading 101 4 Bioassay and some other applications 103 4.1 The tolerance distribution 103 4.2 Estimating an effective dose 106 4.5 Non-linear logistic regression models 118 CONTENTS 4.6 Applications of the complementary log-log model 122 4.7 Further reading 128 5 Model checking 129 5.1 Definition of residuals 130 5.2 Checking the form of the linear predictor 135 5.3 Checking the adequacy of the link function 146 5.4 Identification of outlying observations 150 5.5 Identification of influential observations 154 5.6 Checking the assumption of a binomial distribution 168 5.7 Model checking for binary data 169 5.8 Summary and recommendations 185 5.9 Further reading 193 6 Overdispersion 195 6.1 Potential causes of overdispersion 195 6.2 Modelling variability in response probabilities 199 6.3 Modelling correlation between binary responses 201 6.4 Modelling overdispersed data 202 6.5 A model with a constant scale parameter 206 6.6 The beta-binomial model 211 6.8 Further reading 213 7 Modelling data from epidemiological studies 215 7.1 Basic designs for aetiological studies 216 7.2 Measures of association between disease and exposure 219 7.3 Confounding and interaction 223 7.4 The linear logistic model for data from cohort studies 226 7.5 Interpreting the parameters in a linear logistic model 230 7.6 The linear logistic model for data from case-control studies 242 7.7 Matched case-control studies 250 7.8 Further reading 264 8 Mixed models for binary data 269 8.1 Fixed and random effects 269 8.2 Mixed models for binary data 270 8.4 Mixed models for longitudinal data analysis 284 8.5 Mixed models in meta-analysis 291 8.6 Modelling overdispersion using mixed models 293 8.7 Further reading 300 9 Exact Methods 303 9.1 Comparison of two proportions using an exact test 303 9.2 Exact logistic regression for a single parameter 307 CONTENTS 9.3 Exact hypothesis tests 312 9.4 Exact confidence limits for βk 317 9.5 Exact logistic regression for a set of parameters 318 9.8 Further Reading 323 10 Some additional topics 325 10.1 Ordered categorical data 325 10.2 Analysis of proportions and percentages 329 10.3 Analysis of rates 330 10.4 Analysis of binary time series 331 10.5 Modelling errors in the measurement of explanatory variables 331 10.6 Multivariate binary data 332 10.7 Analysis of binary data from cross-over trials 333 10.8 Experimental design 333 11 Computer software for modelling binary data 335 11.1 Statistical packages for modelling binary data 335 11.2 Interpretation of computer output 339 11.3 Using packages to perform some non-standard analyses 341 11.4 Further reading 349 Appendix A Values of logit(p) and probit(p) 351 Appendix B Some derivations 353 B.1 An algorithm for fitting a GLM to binomial data 353 B.2 The likelihood function for a matched case-control study 357 Appendix C Additional data sets 361 C.2 Toxicity of rotenone 361 C.4 Analgesic potency of four compounds 362 C.5 Vasoconstriction of the fingers 363 C.6 Treatment of neuralgia 363 C.9 Cancer of the cervix 367 C.10 Endometrial cancer 367 References 369 Index of examples 379 Index 381 Preface to the second edition The aim of the first edition of this book was to describe the modelling approach to binary data analysis, with an emphasis on practical applications.
That edi- tion was prepared in 1989–90, but in the intervening period there have been a number of important methodological and computational advances. These include the development of techniques for analysing data with more than one level of variation through the use of mixed models, and procedures that lead to an exact version of logistic regression. These methodological advances have been accompanied by developments in statistical computing, with the result that modern computer-based methods for modelling binary data can now be implemented using a wide range of computer packages. This new edition has been prepared so that the text may continue to realise the aims of the first edition, by providing a comprehensive practical guide to statistical methods for use in modelling binary data.
I hope that this book will also continue to meet the needs of statisticians in the pharmaceutical industry; those engaged in agricultural, biological, epidemiological, industrial and medical research; numerate scientists in universities and research institutes, and students fol- lowing undergraduate or postgraduate programmes that feature statistical modelling. The first seven chapters correspond to those in the first edition. Specifically, Chapter 1 introduces a number of example data sets, and statistical proce- dures based on the binomial distribution are described in Chapter 2. Chap- ter 3 introduces the modelling approach, with emphasis being given to the linear logistic model, and Chapter 4 covers bioassay, non-linear logistic mod- els and some other applications.
Model checking diagnostics are described and illustrated in Chapter 5, and the phenomenon of overdispersion is discussed in Chapter 6. Chapter 7 describes the use of linear logistic models in the analysis of data from epidemiological studies, and shows how the estimated parameters in such models can be interpreted in terms of odds ratios. The opportunity has been taken to revise and update the material in each of these chapters. In particular, the emphasis of the first edition on the GLIM software has been eliminated, so that the illustrative examples are now independent of any specific package.
Two chapters have been added. Chapter 8 presents an introduction to mixed models for binary data analysis. The use of these models in multilevel mod- elling, longitudinal data analysis and meta-analysis is considered in detail. Material on the use of mixed models for overdispersion, originally in Chap- ter 6, is also included in this chapter.
Exact methods for modelling binary PREFACE TO THE SECOND EDITION data, which include Fisher’s exact test as a special case, are introduced in Chapter 9. This chapter shows how exact estimates of parameters in a logistic regression model can be found, and how exact hypothesis tests about such parameters can be conducted. By their very nature, the topics considered in Chapters 8 and 9 are a little more sophisticated, and so the mathematical level of these chapters is slightly higher than that of the earlier chapters. Additional topics are covered in Chapter 10, which includes a substantial sec- tion devoted to modelling ordered categorical data.
Chapter 11, on computer software, has been re-written to reflect developments in this area over the last ten years. There are now so many different packages that can be used in modelling binary data that it is no longer practicable to give a compre- hensive guide to the output from them.