An Introduction to Generalized Linear Models Third Edition

Annette J. Dobson

University of Queensland Herston, Australia

Adrian G. Barnett

Queensland University of Technology  Kelvin Grove, Australia

Contents

Preface

1 Introduction 1

1.1 Background 1

1.2 Scope 1

1.3 Notation 5

1.4 Distributions related to the Normal distribution 7

1.5 Quadratic forms 11

1.6 Estimation 12

1.7 Exercises 15

2 Model Fitting 19

2.1 Introduction 19

2.2 Examples 19

2.3 Some principles of statistical modelling 32

2.4 Notation and coding for explanatory variables 37

2.5 Exercises 40

3 Exponential Family and Generalized Linear Models 45

3.1 Introduction 45

3.2 Exponential family of distributions 46

3.3 Properties of distributions in the exponential family 48

3.4 Generalized linear models 51

3.5 Examples 52

3.6 Exercises 55

4 Estimation 59

4.1 Introduction 59

4.2 Example: Failure times for pressure vessels 59

4.3 Maximum likelihood estimation 64

4.4 Poisson regression example 66

4.5 Exercises 69

5 Inference 73

5.1 Introduction 73

5.2 Sampling distribution for score statistics 74

5.3 Taylor series approximations 76

5.4 Sampling distribution for MLEs 77

5.5 Log-likelihood ratio statistic 79

5.6 Sampling distribution for the deviance 80

5.7 Hypothesis testing 85

5.8 Exercises 87

6 Normal Linear Models 89

6.1 Introduction 89

6.2 Basic results 89

6.3 Multiple linear regression 95

6.4 Analysis of variance 102

6.5 Analysis of covariance 114

6.6 General linear models 117

6.7 Exercises 118

7 Binary Variables and Logistic Regression 123

7.1 Probability distributions 123

7.2 Generalized linear models 124

7.3 Dose response models 124

7.4 General logistic regression model 131

7.5 Goodness of fit statistics 135

7.6 Residuals 138

7.7 Other diagnostics 139

7.8 Example: Senility and WAIS 140

7.9 Exercises 143

8 Nominal and Ordinal Logistic Regression 149

8.1 Introduction 149

8.2 Multinomial distribution 149

8.3 Nominal logistic regression 151

8.4 Ordinal logistic regression 157

8.5 General comments 162

8.6 Exercises 163

9 Poisson Regression and Log-Linear Models 165

9.1 Introduction 165

9.2 Poisson regression 166

9.3 Examples of contingency tables 171

9.4 Probability models for contingency tables 175

9.5 Log-linear models 177

9.6 Inference for log-linear models 178

9.7 Numerical examples 179

9.8 Remarks 183

9.9 Exercises 183

10 Survival Analysis 187

10.1 Introduction 187

10.2 Survivor functions and hazard functions 189

10.3 Empirical survivor function 193

10.4 Estimation 195

10.5 Inference 198

10.6 Model checking 199

10.7 Example: Remission times 201

10.8 Exercises 202

11 Clustered and Longitudinal Data 207

11.1 Introduction 207

11.2 Example: Recovery from stroke 209

11.3 Repeated measures models for Normal data 213

11.4 Repeated measures models for non-Normal data 218

11.5 Multilevel models 219

11.6 Stroke example continued 222

11.7 Comments 224

11.8 Exercises 225

12 Bayesian Analysis 229

12.1 Frequentist and Bayesian paradigms 229

12.2 Priors 233

12.3 Distributions and hierarchies in Bayesian analysis 238

12.4 WinBUGS software for Bayesian analysis 238

12.5 Exercises 241

13 Markov Chain Monte Carlo Methods 243

13.1 Why standard inference fails 243

13.2 Monte Carlo integration 243

13.3 Markov chains 245

13.4 Bayesian inference 255

13.5 Diagnostics of chain convergence 256

13.6 Bayesian model fit: the DIC 260

13.7 Exercises 262

14 Example Bayesian Analyses 267

14.1 Introduction 267

14.2 Binary variables and logistic regression 267

14.3 Nominal logistic regression 271

14.4 Latent variable model 272

14.5 Survival analysis 275

14.6 Random effects 277

14.7 Longitudinal data analysis 279

14.8 Some practical tips for WinBUGS 286

14.9 Exercises 288

Appendix 291

Software 293

References 295

Index 303

Preface

The original purpose of the book was to present a unified theoretical and

conceptual framework for statistical modelling in a way that was accessible to

undergraduate students and researchers in other fields.

The second edition was expanded to include nominal and ordinal logistic

regression, survival analysis and analysis of longitudinal and clustered data.

It relied more on numerical methods, visualizing numerical optimization and

graphical methods for exploratory data analysis and checking model fit. These

features have been extended further in this new edition.

The third edition contains three new chapters on Bayesian analysis. The

fundamentals of Bayesian theory were written long before the development of

classical theory but practical Bayesian analysis has only recently become avail-

able. This availability is mostly thanks to Markov chain Monte Carlo methods

which are introduced in Chapter 13. The increased availability of Bayesian

analysis means that more people with a classical knowledge of statistics are

trying Bayesianmethods for generalized linear models. Bayesian analysis offers

significant advantages over classical methods because of the ability formally

to incorporate prior information, greater flexibility and an ability to solve

complex problems.

This edition has also been updated with Stata and R code, which should

help the practical application of generalized linear models. The chapters on

Bayesian analyses contain R and WinBUGS code.

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