Generalized linear mixed models are flexible tools for modeling non-normal data and are usefulfor accommodating overdispersion in Poisson regression models with random effects. Theirmain difficulty resides in the parameter estimation because there is no analytic solution for themaximization of the marginal likelihood. Many methods have been proposed for this purpose andmany of them are implemented in software packages. The purpose of this study is to comparethe performance of three different statistical principles -marginal likelihood, extended likelihood,Bayesian analysis - via simulation studies. Real data on contact wrestling are used for illustration.