LOG‐LINEAR MODELS FOR MEAN AND DISPERSION IN MIXED POISSON REGRESSION MODELS

Abstract

Summary This paper is concerned with the analysis of repeated measures count data overdispersed relative to a Poisson distribution, with the overdispersion possibly heterogeneous. To accommodate the overdispersion, the Poisson random variable is compounded with a gamma random variable, and both the mean of the Poisson and the variance of the gamma are modelled using log linear models. Maximum likelihood estimates (MLE) are then obtained. The paper also gives extended quasi‐likelihood estimates for a more general class of compounding distributions which are shown to be approximations to the MLEs obtained for the gamma case. The theory is illustrated by modelling the determination of asbestos fibre intensity on membrane filters mounted on microscope slides.