Generalized Linear Models (GLM) is a general class of statistical models that includes many commonly used models as special cases. For example, the class of GLMs that includes linear regression, analysis of variance and analysis of covariance, is a special case of GLIMs. GLIMs also include log-linear models for analysis of contingency tables, prohib/logit regression, Poisson regression and much more. This book gives an overview of GLMs and presents practical examples of their use. Although the approach is applied, the basic theory of GLMs is presented in a compact way. The exponential family of distributions is discussed as well as the Maxium Likelihood estimation and ways of assessing the fit of the model. Response variables as continuous variables, as binary/binomial variables, as counts and as ordinal response variables are discussed. There are many practical examples using the Genmod software of the SAS package. Theory and applications of a more complex nature, like quasi-likelihood procedures, repeated measures models, mixed models and analysis of survival data is also covered.
Author(s): Ulf Olsson
Publisher: Studentlitteratur AB
Year: 2002
Language: English
Pages: 243
Tags: Математика;Теория вероятностей и математическая статистика;Математическая статистика;
Title Page......Page 2
Copyright Page......Page 3
Contents......Page 4
Preface......Page 10
1.1 The role of models......Page 12
1.2 General Linear Models......Page 13
1.3 Estimation......Page 14
1.4.2 Sums of squares decomposition......Page 15
1.5 Inference on single parameters......Page 17
1.7 Different types of tests......Page 18
1.8.1 Simple linear regression......Page 19
1.8.2 Multiple regression......Page 21
1.8.3 t tests and dummy variables......Page 23
1.8.4 One-way ANOVA......Page 24
1.8.5 ANOVA: Factorial experiments......Page 29
1.8.6 Analysis of covariance......Page 32
1.9 Estimability......Page 34
1.11.1 Computer software for GLM:s......Page 35
1.11.2 Model building strategy......Page 36
1.11.3 A few SAS examples......Page 37
1.12 Exercises......Page 38
2.1.1 Types of response variables......Page 42
2.1.3 Response as a binary variable......Page 43
2.1.4 Response as a proportion......Page 44
2.1.5 Response as a count......Page 45
2.1.7 Ordinal response......Page 46
2.2 Generalized linear models......Page 47
2.3.2 The binomial distribution......Page 48
2.3.4 The function......Page 49
2.4 The link function......Page 51
2.6 Maximum likelihood estimation......Page 53
2.7 Numerical procedures......Page 55
2.8.1 The deviance......Page 56
2.8.3 Akaike’s information criterion......Page 57
2.9.1 Wald tests......Page 58
2.9.3 Score tests......Page 59
2.10 Descriptive measures of fit......Page 60
2.11 An application......Page 61
2.12 Exercises......Page 64
3.2 The Hat matrix......Page 66
3.3.1 Pearson residuals......Page 67
3.3.2 Deviance residuals......Page 68
3.3.6 The choice of residuals......Page 69
3.4.1 Leverage......Page 70
3.5 Partial leverage......Page 71
3.6 Overdispersion......Page 72
3.6.1 Models for overdispersion......Page 73
3.7 Non-convergence......Page 74
3.8.1 Residual plots......Page 75
3.8.2 Variance function diagnostics......Page 77
3.8.4 Transformation of covariates......Page 78
3.9 Exercises......Page 79
4.1.1 Simple linear regression......Page 80
4.1.2 Simple ANOVA......Page 82
4.2 The choice of distribution......Page 83
4.3.1 The Chi-square distribution......Page 84
4.3.3 An application with a gamma distribution......Page 86
4.4 The inverse Gaussian distribution......Page 88
4.5.1 Plot of residuals against predicted values......Page 89
4.5.3 Plots of residuals against covariates......Page 90
4.5.4 Influence diagnostics......Page 92
4.6 Exercises......Page 94
5.1.1 The probit link......Page 96
5.1.3 The complementary log-log link......Page 97
5.2.1 The Bernoulli distribution......Page 98
5.2.2 The Binomial distribution......Page 99
5.3 Probit analysis......Page 100
5.4 Logit (logistic) regression......Page 102
5.5.1 Model building......Page 103
5.5.2 Model building tools......Page 107
5.5.3 Model diagnostics......Page 108
5.6 Odds ratios......Page 109
5.7 Overdispersion in binary/binomial models......Page 111
5.7.2 Modeling as a beta-binomial distribution......Page 112
5.7.3 An example of over-dispersed data......Page 113
5.8 Exercises......Page 115
6.1 Log-linear models: introductory example......Page 122
6.1.2 When independence does not hold......Page 123
6.2.1 The multinomial distribution......Page 124
6.2.4 Relation to contingency tables......Page 125
6.3 Analysis of the example data......Page 126
6.4 Testing independence in an r×c crosstable......Page 128
6.5.1 A three-way table......Page 129
6.5.3 Genmod analysis of the drug use data......Page 130
6.5.4 Interpretation through Odds ratios......Page 131
6.6.1 Binary response......Page 132
6.7 Capture-recapture data......Page 133
6.8 Poisson regression models......Page 137
6.9 A designed experiment with a Poisson distribution......Page 140
6.10 Rate data......Page 142
6.11.1 Modeling the scale parameter......Page 144
6.11.2 Modeling as a Negative binomial distribution......Page 145
6.12 Diagnostics......Page 146
6.13 Exercises......Page 148
7.1 Arbitrary scoring......Page 156
7.3 Proportional odds......Page 159
7.4 Latent variables......Page 161
7.5 A Genmod example......Page 164
7.6 Exercises......Page 166
8.1 Variance heterogeneity......Page 168
8.2 Survival models......Page 169
8.2.1 An example......Page 170
8.3 Quasi-likelihood......Page 173
8.4 Quasi-likelihood for modeling overdispersion......Page 174
8.5 Repeated measures: the GEE approach......Page 176
8.6 Mixed Generalized Linear Models......Page 179
8.7 Exercises......Page 183
Some basic definitions......Page 190
Some special types of matrices......Page 191
Calculations on matrices......Page 192
Multiplication by a scalar......Page 193
The inverse of a matrix......Page 194
The rank of a matrix......Page 195
Eigenvalues and eigenvectors......Page 196
Further reading......Page 197
The likelihood function......Page 198
Properties of Maximum Likelihood estimators......Page 199
The Newton-Raphson method......Page 200
Fisher’s scoring......Page 201
Bibliography......Page 202
Solutions to the exercises......Page 208
Index......Page 240
