On approximate likelihood inference in a poisson mixed model

A two-step estimation approach is proposed for the fixed-effect parameters, random effects and their variance σ² of a Poisson mixed model. In the first step, it is proposed to construct a small σ²-based approximate likelihood function of the data and utilize this function to estimate the fixed-effect parameters and σ². In the second step, the random effects are estimated by minimizing their posterior mean squared error. Methods of Waclawiw and Liang (1993) based on so-called Stein-type estimating functions and of Breslow and Clayton (1993) based on penalized quasilikelihood are compared with the proposed likelihood method. The results of a simulation study on the performance of all three approaches are reported.