Bayesian Test of Different Association Structures in Two-Way Contingency Tables

Bayesian methods for exact small-sample analysis with categorical data in $Itimes J$ contingency tables are considered. Different structures of association are defined and tested concerning log odds ratios in these tables with fixed row margins. The conditional distribution of sufficient statistics for interesting parameters conditional on the sufficient statistics of other nuisance parameters in the model is obtained and used to eliminate the effect of nuisance parameters. The resulting distribution for the table is Fisher's multivariate noncentral hypergeometric distribution. For Bayesian approach, although computation under this distribution is complicated, a common Bayesian model is considered. Bayes factor is used as a measure of evidence for Bayesian testing of different association structures. The performance of our testing Bayesian approach is compared with that of the classical corrected likelihood ratio test by some simulation studies. Also the Bayesian test of “homogenous association” is applied on a real data set.