Compound Poisson Regression Models

Abstract

Count data are easily modelled in GLIM using the Poisson distribution. However, in modelling such data the counts are often aggregated over one or more factors, or important explanatory variables are unavailable and as a result the fit obtained is often poor. This paper examines a method of allowing for this unexplained variation by introducing an independent random variable into the linear model for the Poisson mean, giving a compound Poisson model for the observed data. By assuming a known form for the distribution of this random variable, in particular the normal distribution, and using a combination of numerical integration, the EM algorithm and iteratively reweighted least squares, maximum likelihood estimates can be obtained for the parameters. Macros for implementing this technique are presented and its use is illustrated with several examples.