In the health and social sciences, researchers often encounter categorical data for which complexities come from a nested hierarchy and/or cross-classification for the sampling structure. A common feature of these studies is a non-standard data structure with repeated measurements which may have some degree of clustering. In this paper, methodology is presented for the joint estimation of quantities of interest in the context of a stratified two-stage sample with bivariate dichotomous data. These quantities are the mean value pi of an observed dichotomous response for a certain condition or time-point and a set of correlation coefficients for intra-cluster association for each condition or time period and for inter-condition correlation within and among clusters. The methodology uses the cluster means and pairwise joint probability parameters from each cluster. They together provide appropriate information across clusters for the estimation of the correlation coefficients.213731752