Poisson regression is the standard or base count response regression model. We have seen in previous discussion that other count models deal with data that violate the assumptions carried by the Poisson model. Since the model does play such a central role in count response modeling, we begin with an examination of its derivation and structure, as well as how it can be parametermized to model rates. The concept of overdispersion is introduced in this chapter, together with two tests that have been used to assess its existence and strength.

Derivation of the Poisson model

A primary assumption is that of equidispersion, or the equality of the mean and variance functions. When the value of the variance exceeds that of the mean, we have what is termed overdispersion. Negative binomial regression is a standard way to deal with certain types of Poisson overdispersion; we shall find that there are a variety of negative binomial based models, each of which address the manner in which overdispersion has arisen in the data. However, to fully appreciate the negative binomial model and its variations, it is important to have a basic understanding of the derivation of the Poisson as well as an understanding of the logic of its interpretation.

Maximum likelihood models, as well as the canonical form members of generalized linear models, are ultimately based on an estimating equation derived from a probability distribution.