Analyzing dependent proportions in cluster randomized trials: Modeling inter-cluster correlation via copula function

When two interventions are randomized to multiple sub-clusters within a whole cluster, accounting for the within sub-cluster (intra-cluster) and between sub-clusters (inter-cluster) correlations is needed to produce valid analyses of the effect of interventions. With the growing interest in copulas and their applications in statistical research, we demonstrate, through applications, how copula functions may be used to account for the correlation among responses across sub-clusters. Copulas having asymmetric dependence property may prove useful for modeling the relationship between random functions especially in clinical, health and environmental sciences where response data are in general skewed. These functions can in general be used to study scale-free measures of dependence, and they can be used as a starting point for constructing families of bivariate distributions, with a view to simulations. The core contribution of this paper is to provide an alternative approach for estimating the inter-cluster correlation using copula to accurately estimate the treatment effect when the outcome variable is measured on the dichotomous scale. Two data sets are used to illustrate the proposed methodology.