Bayesian test of independence and conditional independence of two ordinal variables

For analysis of contingency tables with large sample size, classical approaches using approximate methods have high power. However, when the sample size is small or some cells have frequencies less than 5, classical approaches are so conservative. Also asymptotic behavior may be poor when the table contains small counts. So, Bayesian test of independence for two-way contingency tables with ordinal variables is considered. The conditional independence of two ordinal variables given values of a controlling variable is also considered. To do these tests, gamma and partial gamma are used as association parameters for ordinal variables. Since gamma has a complex posterior form, it is intractable to compute directly the necessary inferential measures. So, a Dirichlet distribution is used as a prior distribution for the vector of cell probabilities, then the use of computational methods such as the Monte Carlo algorithm is introduced to generate samples from posterior distribution of gamma. Also, the Bayesian P-value and Bayes factor are obtained. In a simulation study, the choice of appropriate prior distribution for gamma is discussed and also the performance of gamma is compared to that of kappa. It is shown that, in contingency tables with ordinal variables, it is better to apply gamma as a measure of association. Some sensitivity analysis to the choice of prior are also performed on real applications.