Improved Exact Inference about Conditional Association in Three-Way Contingency Tables

We propose modified exact inferential methods for contingency tables. Ordinary “exact” inference is conservative, because of the discreteness. For estimating a common odds ratio in several 2 × 2 tables, two modifications of the ordinary “exact” confidence interval maintain at least a fixed confidence level but tend to be much narrower. One approach inverts results of a test with a modified P value utilizing the test statistic and table probabilities. The second approach inverts one two-sided test rather than two one-sided tests. This approach is much less conservative when the true odds ratio is relatively small or large. We also generalize results of Cohen and Sackrowitz and relate modified P values to construction of exact, unbiased, and admissible tests for an ordinal alternative to conditional independence.