modelling binary data

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BINARY. DATA. Second Edition. David Collett. School of Applied Statistics. The University ... 11.3 Using packages to perform some non-standard analyses. 341.
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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.C.

Contents

Introduction 1.1 Some examples 1.2 The scope of this book 1.3 Use of statistical software 1.4 Further reading

1 1 14 15 16

Statistical inference for binary data 2.1 The binomial distribution 2.2 Inference about the success probability 2.3 Comparison of two proportions 2.4 Comparison of two or more proportions 2.5 Further reading

19 19 23 31 38 42

Models for binary and binomial data 3.1 Statistical modelling 3.2 Linear models 3.3 Methods of estimation 3.4 Fitting linear models to binomial data 3.5 Models for binomial response data 3.6 The linear logistic model 3.7 Fitting the linear logistic model to binomial data 3.8 Goodness of fit of a linear logistic model 3.9 Comparing linear logistic models 3.10 Linear trend in proportions 3.11 Comparing stimulus-response relationships 3.12 Non-convergence and overfitting 3.13 Some other goodness of fit statistics 3.14 Strategy for model selection 3.15 Predicting a binary response probability 3.16 Further reading

45 45 47 50 53 56 58 59 65 71 78 81 85 87 91 98 101

Bioassay and some other applications 4.1 The tolerance distribution 4.2 Estimating an effective dose 4.3 Relative potency 4.4 Natural response • 4.5 Non-linear logistic regression models

103 103 106 111 114 118

CONTENTS

4.6 Applications of the complementary log-log model 4.7 Further reading

122 128

5 Model checking 5.1 Definition of residuals 5.2 Checking the form of the linear predictor 5.3 Checking the adequacy of the link function 5.4 Identification of outlying observations 5.5 Identification of influential observations 5.6 Checking the assumption of a binomial distribution 5.7 Model checking for binary data 5.8 Summary and recommendations 5.9 Further reading

129 130 135 146 150 154 168 169 185 193

6 Overdispersion 6.1 Potential causes of overdispersion 6.2 Modelling variability in response probabilities 6.3 Modelling correlation between binary responses 6.4 Modelling overdispersed data 6.5 A model with a constant scale parameter 6.6 The beta-binomial model 6.7 Discussion 6.8 Further reading

195 195 199 201 202 206 211 212 213

7 Modelling data from epidemiological studies 7.1 Basic designs for aetiological studies 7.2 Measures of association between disease and exposure 7.3 Confounding and interaction 7.4 The linear logistic model for data from cohort studies 7.5 Interpreting the parameters in a linear logistic model 7.6 The linear logistic model for data from case-control studies 7.7 Matched case-control studies 7.8 Further reading

215 216 219 223 226 230 242 250 264

8 Mixed models for binary data 8.1 Fixed and random effects 8.2 Mixed models for binary data 8.3 Multilevel modelling 8.4 Mixed models for longitudinal data analysis 8.5 Mixed models in meta-analysis 8.6 Modelling overdispersion using mixed models 8.7 Further reading

269 269 270 277 284 291 293 300

9 Exact Methods 9.1 Comparison of two proportions using an exact test 9.2 Exact logistic regression for a single parameter

303 303 307

CONTENTS

9.3 9.4 9.5 9.6 9.7 9.8

Exact hypothesis tests Exact confidence limits for (3k Exact logistic regression for a set of parameters Some examples Discussion Further Reading

312 317 318 319 322 323

10 Some additional topics 10.1 Ordered categorical data 10.2 Analysis of proportions and percentages 10.3 Analysis of rates 10.4 Analysis of binary time series 10.5 Modelling errors in the measurement of explanatory variables 10.6 Multivariate binary data 10.7 Analysis of binary data from cross-over trials 10.8 Experimental design

325 325 329 330 331 331 332 333 333

11 Computer software for modelling binary data 11.1 Statistical packages for modelling binary data 11.2 Interpretation of computer output 11.3 Using packages to perform some non-standard analyses 11.4 Further reading

335 335 339 341 349

Appendix A Values of logit(p) and probit(p)

351

Appendix B Some derivations B.I An algorithm for fitting a GLM to binomial data B.2 The likelihood function for a matched case-control study

353 353 357

Appendix C Additional data sets C.I Tonsil size C.2 Toxicity of rotenone C.3 Food poisoning C.4 Analgesic potency of four compounds C.5 Vasoconstriction of the C.6 Treatment of neuralgia C.7 HIV infection C.8 Aplastic anaemia * C.9 Cancer of the cervix CIO Endometrial cancer

361 361 361 362 362 363 363 365 365 367 367

fingers

References

369

Index of examples

379

Index

381