Foundations of Linear and Generalized Linear Models

CHAPTER 4 Generalized Linear Models: Model Fitting and Inference

We now extend our scope from the linear model to the generalized linear model (GLM). This extension encompasses (1) non-normal response distributions and (2) link functions of the mean equated to the linear predictor. Section 1.1.5 introduced examples of GLMs: Loglinear models using the log-link function for a Poisson (count) response and logistic models using the logit-link function for a binomial (binary) response.

Section 4.1 provides more details about exponential family distributions for the random component of a GLM. In Section 4.2 we derive likelihood equations for the maximum likelihood (ML) estimators of model parameters and show their large-sample normal distribution. Section 4.3 summarizes the likelihood ratio, score, and Wald inference methods for the model parameters. Then in Section 4.4 we introduce the deviance, a generalization of the residual sum of squares used in inference, such as to compare nested GLMs. That section also presents residuals for GLMs and ways of checking the model. Section 4.5 presents two standard methods, Newton–Raphson and Fisher scoring, for solving the likelihood equations to fit GLMs. Section 4.6 discusses the selection of explanatory variables for a model, followed by an example. A chapter appendix shows that fundamental results for linear models about orthogonality of fitted values and residuals do not hold exactly for GLMs, but analogs hold for an adjusted, weighted ...