A Poisson Mixed Model with Nonnormal Random Effect Distribution

TL;DR

This paper introduces a novel Poisson mixed model with a generalized log-gamma random effect, deriving moments and correlation, and providing estimation methods with real data applications.

Contribution

It develops a Poisson mixed model with a nonnormal random effect distribution and derives explicit moments and estimation procedures.

Findings

Derived moments and intraclass correlation for the model

Obtained multivariate negative binomial as a special case

Applied the model to real datasets successfully

Abstract

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.