An alternative specification of generalized linear mixed models

Consider stratified data in which Y_{i 1}, ..., Y_{i ni} denote real-valued response variables corresponding to the observations from stratum i, i = 1, ..., m and suppose that Y_{i j} follows an exponential family distribution with canonical parameter of the form θ_{i j} = x_{i j} β + γ_i. In analyzing data of this type, the stratum-specific parameters are often modeled as random effects; a commonly-used approach is to assume that γ₁, ..., γ_m are independent, identically distributed random variables. The purpose of this paper is to consider an alternative approach to defining the random effects, in which the stratum means of the response variable are assumed to be independent and identically distributed, with a distribution not depending on β. It will be shown that inferences about β based on this formulation of the generalized linear mixed model have many desirable properties. For instance, inferences regarding β are less sensitive to the choice of random effects distribution, are less subject to bias from omitted stratum-level covariates and are less affected by separate between- and within-cluster covariate effects.