Introduction to General and Generalized Linear Models

by Madsen; Henrik

• ISBN13: 9781420091557
• ISBN10: 1420091557
• Edition: 1st
• Format: Hardcover
• Copyright: 2010-11-09
• Publisher: CRC Press

Summary

Since the mathematics behind generalized linear models is often difficult to follow while the mathematics behind general linear models is well understood, this text describes the methodology behind both models in a parallel setup. After introducing a likelihood framework that is sufficient to cover both approaches, the authors present general linear models, including analysis of covariance, before moving on to more complicated generalized linear models using the same likelihood-based approach. Numerous simulated and real-world examples, implemented using R and SAS, illustrate the methods discussed. The text also provides exercises to further develop understanding.

Table of Contents

Preface	p. xi
Notation	p. xiii
Introduction	p. 1
Examples of types of data	p. 2
Motivating examples	p. 3
A first view on the models	p. 5
The likelihood principle	p. 9
Introduction	p. 9
Point estimation theory	p. 10
The likelihood function	p. 14
The score function	p. 17
The information matrix	p. 18
Alternative parameterizations of the likelihood	p. 20
The maximum likelihood estimate (MLE)	p. 21
Distribution of the ML estimator	p. 22
Generalized loss-function and deviance	p. 23
Quadratic approximation of the log-likelihood	p. 23
Likelihood ratio tests	p. 25
Successive testing in hypothesis chains	p. 27
Dealing with nuisance parameters	p. 33
Problems	p. 38
General linear models	p. 41
Introduction	p. 41
The multivariate normal distribution	p. 42
General linear models	p. 44
Estimation of parameters	p. 48
Likelihood ratio tests	p. 53
Tests for model reduction	p. 58
Collinearity	p. 64
Inference on parameters in parameterized models	p. 70
Model diagnostics: residuals and influence	p. 73
Analysis of residuals	p. 77
Representation of linear models	p. 78
General linear models in R	p. 81
Problems	p. 83
Generalized linear models	p. 87
Types of response variables	p. 89
Exponential families of distributions	p. 90
Generalized linear models	p. 99
Maximum likelihood estimation	p. 102
Likelihood ratio tests	p. 111
Test for model reduction	p. 115
Inference on individual parameters	p. 116
Examples	p. 117
Generalized linear models in R	p. 152
Problems	p. 153
Mixed effects models	p. 157
Gaussian mixed effects model	p. 159
One-way random effects model	p. 160
More examples of hierarchical variation	p. 174
General linear mixed effects models	p. 179
Bayesian interpretations	p. 185
Posterior distributions	p. 191
Random effects for multivariate measurements	p. 192
Hierarchical models in metrology	p. 197
General mixed effects models	p. 199
Laplace approximation	p. 201
Mixed effects models in R	p. 218
Problems	p. 219
Hierarchical models	p. 225
Introduction, approaches to modeling of overdispersion	p. 225
Hierarchical Poisson Gamma model	p. 226
Conjugate prior distributions	p. 233
Examples of one-way random effects models	p. 237
Hierarchical generalized linear models	p. 242
Problems	p. 243
Real life inspired problems	p. 245
Dioxin emission	p. 246
Depreciation of used cars	p. 249
Young fish in the North Sea	p. 250
Traffic accidents	p. 251
Mortality of snails	p. 252
Supplement on the law of error propagation	p. 255
Function of one random variable	p. 255
Function of several random variables	p. 255
Some probability distributions	p. 257
The binomial distribution model	p. 259
The Poisson distribution model	p. 262
The negative binomial distribution model	p. 264
The exponential distribution model	p. 266
The gamma distribution model	p. 268
The inverse Gaussian distribution model	p. 275
Distributions derived from the normal distribution	p. 280
The Gamma-function	p. 284
List of symbols	p. 285
Bibliography	p. 287
Index	p. 293
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