Generalized Linear Models

Preface ix

Notation and Acronyms xi

Chapter 1 Exponential Families 1

1.1. Definition 1

1.2. Mean, variance, and variance function 3

1.3. Examples of exponential families 4

1.4. Maximum likelihood estimation 9

1.5. Technical appendix 18

1.5.1. Some useful results from probability 18

1.5.2. Negative binomial distribution and Poisson-gamma mixtures 19

1.6. Exercises 20

Chapter 2. From Linear Models to GLMs 25

2.1. Reminders on the linear model 27

2.1.1. Matrix form of the linear model 28

2.1.2. Some examples of linear models 29

2.1.3. Least-squares approximation 31

2.1.4. Asymptotics of the LSE 34

2.1.5. Linear Gaussian model 37

2.2. Three components of a generalized linear model 43

2.2.1. The random component 43

2.2.2. The linear predictor 44

2.2.3. The link function 45

2.3. Estimation in generalized linear models 46

2.3.1. Maximum likelihood 46

2.3.2. Asymptotic properties and inference 48

2.3.3. Estimating the dispersion parameter 50

2.4. Some examples 52

2.4.1. Logistic regression model 52

2.4.2. Poisson regression model 59

2.4.3. Gamma regression model 60

2.5. Generalized linear models in R: Poisson regression example 61

2.5.1. Confidence intervals and hypothesis tests 65

2.5.2. AIC and BIC, variable selection 68

2.5.3. Prediction, confidence intervals for a prediction 69

2.6. Technical appendix 71

2.6.1. Some probability distributions 71

2.6.2. Cochran's theorem 72

2.7. Exercises 72

Chapter 3. Censored and Missing Data in GLMs 83

3.1. Censored data 83

3.1.1. Introduction 83

3.1.2. Poisson regression with a right-censored response 85

3.1.3. Gamma regression with a right-censored response 93

3.2. Missing data problems 101

3.2.1. Introduction 101

3.2.2. Missing data typology 102

3.2.3. Methods for treating missing data 103

3.2.4. A missing data problem in the Poisson model 113

3.2.5. A missing data problem in the gamma regression model 127

3.3. Technical appendix 135

3.3.1. Two lemmas 135

3.3.2. Proof of theorem 3.3 140

3.3.3. Proof of theorem 3.4 143

3.3.4. Proof of theorem 3.5 145

3.3.5. Proof of theorem 3.6 149

3.3.6. Elements of empirical processes 150

3.4. Exercises 154

Chapter 4. Zero-Inflated Models 159

4.1. Introduction 159

4.1.1. Overdispersion 159

4.1.2. Excess of zeros 163

4.2. Zero-inflated Poisson models and extensions 166

4.2.1. The zero-inflated Poisson model 166

4.2.2. Semi-parametric ZIP models 170

4.2.3. Zero-inflated generalized Poisson model 174

4.2.4. A zero-inflation test 177

4.3. Zero-inflated negative binomial model 183

4.3.1. Negative binomial model 183

4.3.2. ZINB model 184

4.3.3. The ZIP model versus the ZINB model 186

4.4. Zero-inflated binomial model 187

4.5. Censored and missing data: examples of problems 190

4.5.1. Censored ZIP model 190

4.5.2. Missing covariables in the ZIB model 192

4.5.3. Missing covariables in the ZIP model 195

4.6. Marginal zero-inflated models 200

4.6.1. Introduction 200

4.6.2. MZIP and MZINB models 203

4.7. Exercises 204

References 209

Index 217