It is well known that the generalized linear mixed model is useful
for analyzing the overdispersion and correlation structure for multivariate discrete
data. In this paper, we derive an approximation of the density function for the
generalized linear mixed model. This approximation is found to satisfy the properties
of probability density function under some conditions. Therefore, this approximation
can be regarded as a class of multivariate distributions. Estimation of the parameters
in this class can be carried out by the maximum likelihood method. We give the
likelihood ratio criteria for testing several covariance structures. Several simulation
studies were also conducted for the Poisson log-normal model when the proposed
density function is regarded as an approximate likelihood of the generalized linear mixed
model.