Generalized Linear Models and Extensions

From the likelihood point of view, generalized linear models (GLMs) can simply be regarded as a flexible class of models, all of which share a convenient form of likelihood function. This chapter presents the general form of the likelihood equations that underlie the maximum likelihood modelling of exponential family data. It presents methods for model evaluation and comparison. The model that is used in the first case study in the chapter is a standard logistic regression on proportion data, and no obvious problems with the model fit are found. Often proportion data and count data will not be well modelled by a binomial distribution and a Poisson distribution. The chapter presents practical strategies for this situation, including the use of natural generalizations of the binomial and Poisson distributions, and the use of quasi-likelihood. It ends with a second case study that employs these strategies in an analysis of count data. Controlled Vocabulary Terms generalised additive model; generalized binomial distribution; likelihood ratio test; logistic regression; Poisson distribution; quasi-likelihood