Negative binomial regression

In this and subsequent chapters we shall discuss the nature and utility of some 25 varieties of negative binomial regression that are useful for modeling count response data. In addition, we also examine certain models that are related to the negative binomial family of models. This chapter will primarily be devoted to an examination of the derivation of the negative binomial model and to the two foremost methods of its estimation. We also consider how the probabilities generated from a negative binomial model differ from the Poisson, as well as how they differ among various negative binomial models based on both mean and ancillary parameters. Varieties of negative binomial I mentioned that the basic negative binomial model can be enhanced to allow for the modeling of a wide range of count response situations. The Poisson can likewise be enhanced to adjust for data that violate, for instance, its essential distributional assumptions. In fact, many of the same distributional problems face both Poisson and negative binomial models. We therefore find similar approaches to the handling of such data for both the Poisson and negative binomial. These include models such as zero-inflated Poisson (ZIP), which is directly related to the zero-inflated negative binomial (ZINB). Other models without a specific negative binomial correlate are also discussed, e.g. generalized Poisson, which has a heterogeneity parameter like the NB-2 model, but which can also be used to model underdispersion.