A simulation study comparing likelihood and non-likelihood approaches in   analyzing overdispersed count data
  Overdispersed count data are modelled with likelihood and non-likelihood approaches. Likelihood approaches include the Poisson mixtures with three distributions, the gamma, the lognormal, and the inverse Gaussian distributions. Non-likelihood approaches include the robust sandwich estimator and quasilikelihood. In this simulation study, overdispersed count data were simulated under the Poisson mixtures with the gamma, the lognormal and the inverse Gaussian distributions, then analyzed with the five likelihood and non-likelihood approaches. Our results indicated that 1) when the count data are mildly overdispersed, there are virtually no differences in type I error rate, standard error of the main effect, and empirical power among the five methods; 2) when the count data are very overdispersed, none of these five approaches is robust to model misspecification as evaluated by type I error rate, standard error of the main effect, and empirical power. This simulation study raises caution on using non-likelihood method for analyzing very overdispered count data because of likely higher type I error and inappropriate power levels. Unlike non-likelihood approaches, likelihood approaches allow for statistical tests based on likelihood ratios and for checking model fit and provide basis for power and sample size calculations. When likelihood approaches are used, we suggest comparing likelihood values to select the appropriate parametric method for analyzing very overdispersed count data. 
Stanley Xu, Gary Grunwald, Richard Jones