3 - Generalized Linear Models

Summary

In this chapter we embed the logistic regression model as well as the classical regression model into the framework of generalized linear models. Generalized linear models (GLMs), which have been proposed by Nelder and Wedderburn (1972), may be seen as a framework for handling several response distributions, some categorical and some continuous, in a unified way. Many of the binary response models considered in later chapters can be seen as generalized linear models, and the same holds for part of the count data models in Chapter 7.

The chapter may be read as a general introduction to generalized linear models; continuous response models are treated as well as categorical response models. Therefore, parts of the chapter can be skipped if the reader is interested in categorical data only. Basic concepts like the deviance are introduced in a general form, but specific forms that are needed in categorical data analysis will also be given in the chapters where the models are considered. Nevertheless, the GLM is useful as a background model for categorical data modeling, and since McCullagh and Nelder's (1983) book everybody working with regression models should be familiar with the basic concept.

Basic Structure

A generalized linear model is composed from several components. The random component specifies the distribution of the conditional response y_i given x_i, whereas the systematic component specifies the link between the expected response and the covariates.