Chapter 3 - Introduction to the Generalized Linear Model: The Simplest Model for Count Data

Abstract

Statistical models must account for the signal and the noise part of an observed response. Generalized linear models (GLMs) represent the archetypal statistical model. They use a probability distribution to describe the noise and a linear model for the signal. The linear model can be described by the design matrix and the parameter vector. We give examples of this model description because it is important for a thorough understanding of linear models and their implementation in WinBUGS. GLMs extend the concept of a linear model, such as in ANOVA or regression, to probability distributions other than the normal, for example, the Poisson or the binomial distribution. A link function transforms the mean response, and the linear model is applied to this transformation of the mean response. GLMs are easy to fit in WinBUGS, and their description in the BUGS language greatly clarifies the concept of a GLM. We illustrate Poisson and binomial GLMs to model temporal variation in counts. We show how a classical analysis using maximum likelihood in R yields estimates that numerically closely match those from a Bayesian analysis in WinBUGS with vague priors. For every example, we illustrate convergence assessment of Markov chains by visual and formal means.