Chapter 6: Generalized Linear Models: Estimation

Abstract

The previous chapter defined glms and studied the components of a glm. This chapter discusses the estimation of the unknown parameters in the glm: the regression parameters and possibly the dispersion parameter ϕ. Because glms assume a specific probability distribution for the responses from the edm family, maximum likelihood estimation procedures are used for parameter estimation, and general formulae are developed for the glm context. We first derive the score equations and information for the glm context, which are used to form algorithms for estimating the regression parameters for glms. The residual deviance is then defined as a measure of the residual variability across n observations after fitting the model. The standard errors of the regression parameters are developed and matrix formulations are used to estimate the regression parameters. We then explore the important connection between the algorithms for fitting linear regression models and glms. Techniques are then developed for estimating ϕ We conclude with a discussion of using r to fit glms.

The challenge for the model builder is to get the most out of the modelling process by choosing a model of the right form and complexity so as to describe those aspects of the system which are perceived as important.

Chatfield [ 1 , p. 27]