Comparison of subject-specific and population averaged models for count data from cluster-unit intervention trials

Maximum likelihood estimation techniques for subject-specific (SS) generalized linear mixed models and generalized estimating equations for marginal or population-averaged (PA) models are often used for the analysis of cluster-unit intervention trials. Although both classes of procedures account for the presence of within-cluster correlations, the interpretations of fixed effects including intervention effect parameters differ in SS and PA models. Furthermore, closed-form mathematical expressions relating SS and PA parameters from the two respective approaches are generally lacking. This paper investigates the special case of correlated Poisson responses where, for a log-linear model with normal random effects, exact relationships are available. Equivalent PA model representations of two SS models commonly used in the analysis of nested cross-sectional cluster trials with count data are derived. The mathematical results are illustrated with count data from a large non-randomized cluster trial to reduce underage drinking. Knowledge of relationships among parameters in the respective mean and covariance models is essential to understanding empirical comparisons of the two approaches.