A hierarchical generalized linear model is developed in which the data are allowed to determine the distributional assumptions of the dispersion sub-models while considering fixed and random effects. In addition the model allows for covariates to be included in both the mean and dispersion sub-models.

In recent times the pseudo-likelihood and double extended quasi-likelihood models were developed for modeling data from a hierarchical generalized linear structure. The pseudo-likelihood model is appropriate for any response distribution from the exponential family, but its systematic component is approximated by restricting to a normal random variable. In addition, the pseudo-likelihood model only allows for a constant dispersion correction, and cannot include dispersion covariates. The double extended quasi-likelihood model is appropriate for any response and random effect distributions from the exponential family, and does allow dispersion sub-models with dispersion covariates. However, the double extended quasi-likelihood model imposes a specific mean-variance relationship for both dispersion sub-models.

In hierarchical generalized linear models it is natural to account for components of overdispersion separately at each level of clustering. The dispersion sub-models developed can be used to model components of overdispersion. In this research the double generalized extended quasi-likelihood model consists of power functions which are defined by the mean-variance relationships in the dispersion sub-models as determined by the data, and thus are expected to more accurately model the dispersion in the data.

The models developed are compared to the pseudo-likelihood and double extended quasi-likelihood models through a simulation study. The performances of these models are assessed through data simulated with various distributional assumptions, and also in data simulated with overdispersion present at different levels. The double generalized extended quasi-likelihood is shown to be reliable and performs consistently better than existing models. These models are used to analyze a commonly known respiratory data set.