# A conditional Bayesian approach for testing independence in two-way contingency tables

## Keywords:

Bayes Factor, bayesian P-value, hierarchical Bayes, important sampling, noncentral hypergeometric distribution## Abstract

Bayesian methods for exact small-sample analysis with categorical data in contingency tables are considered. Point null hypotheses versus two-sided hypothesis are tested concerning log odds ratios in these tables with fixed row margins. The conditional distribution of the sufficient statistics for interesting parameters conditional on sufficient statistics of other nuisance parameters in the model is obtained and used to eliminate the effect of nuisance parameters. This distribution is Fisher's multivariate noncentral hypergeometric distribution. Three Bayesian approaches, hierarchical Bayes, empirical Bayes, and noninformative Bayes are considered and compared by simulation studies.

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