Clustered overdispersed count data are commonly analyzed by the generalized linear mixed model (GLMM) including gamma random effects. Routinely, the GLMM is fitted by maximizing the marginal likelihood. However, the derivation of the marginal joint distribution is complicated and the marginalization is done numerically. Therefore, it can be too time-consuming or intractable with very large data. Therefore, we propose a less intensive estimator.
It is rooted in the split-sample method, but with a single cluster by stratum, leading to the cluster-by-cluster(CbC) estimator. It requires two stages. (1) a generalized linear model is fitted to each cluster separately. (2) global estimates for the fixed effects and overdispersion parameter are computed using weights. For the variance of the random effects, a method-of-moments estimator is proposed.
Based on simulations, it is fast and shows good statistical properties. For the fixed effects, it is unbiased and highly efficient. For the variance components, it is asymptotically efficient as the size of the clusters increases. Moreover, this method is illustrated using a large database belonging to a network of Belgian general practices.