Selection of random effects distributions in mixed counts models: A quasi-likelihood approac

Generalized linear models with responses comprised of counts data are clustered into homoscedastic groups by random effects that are considered to follow either a normal or a gamma distribution. There are data sets for which it is shown that the standard errors of the random effects estimates are only slightly adjusted from those of the normal and gamma distributions when the random effects use a power function, mean-variance relationship, or quasi distribution. The adjusted standard errors is demonstrated by subjecting the random effects of selected data sets to the power function quasi distribution, after first using these data to estimate a value for the power function exponent, a method not used in the literature. The efficacy of the quasi distribution is measured by comparing it with normal and gamma distributions' random effects standard errors, model overdispersion, and the standardized random effects deviance residuals diagnostic plots. Comparison results show that the power function quasi distribution model of random effects benefits is data set-dependent.