  We propose in this paper a random intercept Poisson model in which the random
effect distribution is assumed to follow a generalized log-gamma (GLG)
distribution. We derive the first two moments for the marginal distribution as
well as the intraclass correlation. Even though numerical integration methods
are in general required for deriving the marginal models, we obtain the
multivariate negative binomial model for a particular parameter setting of the
hierarchical model. An iterative process is derived for obtaining the maximum
likelihood estimates for the parameters in the multivariate negative binomial
model. Residual analysis are proposed and two applications with real data are
given for illustration.
