Bayesian inference on contingency tables with uncertainty about independence for small areas

A scientist might have vague information about independence/dependence in a two-way table, and a statistician might proceed with estimation conditional on this piece of information. However, one needs to take into account the uncertainty in this information which can increase variability. We develop a Bayesian method to solve this problem when estimation is needed for the cells of a $ r \times c $ r×c contingency table and there is uncertainty about independence or dependence. In our problem, there are several small areas and a $ r \times c $ r×c table is constructed for each area. We use the hierarchical Dirichlet-multinomial model to analyze the counts from these small areas. The key idea in our method is that the cell probabilities of each area is expressed as a convex combination of the cell probabilities under independence and the cell probabilities under dependence, where each area has its own unknown weight. We show how to fit the model using the Gibbs sampler even though many of the conditional posterior densities are nonstandard. As a by product of our method, we have actually produced a test of independence which is competitive to the chi-square test for a single table. To illustrate our method, we have used an example on body mass index and bone mineral density data obtained from NHANES III. We have shown some important differences among the three scenarios (independence, dependence and the convex combination of these two) when Bayesian predictive inference is done for the finite population means corresponding to each cell of the $ r \times c $ r×c table.