Fitting mixed Poisson regression models using quasi-likelihood methods

This paper is to investigate the use of the quasi-likelihood, extended quasi-likelihood, and pseudo-likelihood approach to estimating and testing the mean parameters with respect to two variance models, M1: psi = mu(theta)(1 + mu phi) and M2: psi = mu(theta)(1 + tau). Simulation was conducted to compare the bias and standard deviation, and type I error of the Wald tests, based on the model-based and robust variance estimates, using the three semi-parametric approaches under four mixed Poisson models, two variance structures, and two sample sizes. All methods perform reasonably well in terms of bias. Type I error of the Wald test, based on either the model-based or robust estimate, tends to be larger than the nominal level when over-dispersion is moderate. The extended quasi-likelihood method with the variance model M1 performs more consistently in terms of the efficiency and controlling the type I error than with the model M2, and better than the pseudo-likelihood approach with either the M1 or M2 model. The model-based estimate seems to perform better than the robust estimate when the sample size is small.