Negative multinomial regression models for clustered event counts

In this paper, I consider a negative multinomial regression model for clustered count data. One example of such data is quarterly counts of a surgical procedure performed from 1988 through 1991 in a national sample of 175 U.S. hospitals. In this data set, each hospital contributes a number of quarterly counts. The quarterly counts contributed by the same hospital form a cluster. The potential problem for clustered count data irs that the multiple counts in the same cluster may not be independent. When they are not independent, Poisson and mixed Poisson models for overdispersed count data including negative binomial models are inappropriate. In contrast, the negative multinomial regression model makes explicit allowance for correlated observations by subjecting the multiple counts in the same cluster to a cluster-specific random effect. A gamma-distributed cluster-specific effect in this formulation leads to the negative multinomial regression model. I describe maximum likelihood estimation and illustrate the model through examples.