Modification of the generalized quasi-likelihood model in the analysis of the Add Health study

The relationship between the mean and variance is an implicit assumption of parametric modeling. While many distributions in the exponential family have a theoretical mean-variance relationship, it is often the case that the data under investigation are correlated, thus varying from the relation. We present a generalized method of moments estimation technique for modeling certain correlated data by adjusting the mean-variance relationship parameters based on a canonical parameterization. The proposed mean-variance form describes overdispersion using two parameters and implements an adjusted canonical parameter which makes this approach feasible for all distributions in the exponential family. Test statistics and confidence intervals are used to measure the deviations from the mean-variance relation parameters. We use the modified relation as a means of fitting generalized quasi-likelihood models to correlated data. The performance of the proposed modified generalized quasi-likelihood model is demonstrated through a simulation study and the importance of accounting for overdispersion is highlighted through the evaluation of adolescent obesity data collected from a U.S. longitudinal study.