The analysis of stratified 2 x 2 contingency tables.

We consider the problem of testing for independence against the consistent superiority of one treatment over another when the response variable is binary and is compared across two treatments in each of several strata. Specifically, we consider the randomized clinical trial setting. A number of issues arise in this context. First, should tables be combined if there are small or zero margins? Second, should one assume a common odds ratio across strata? Third, if the odds ratios differ across strata, then how does the standard test (based on a common odds ratio) perform? Fourth, are there other analyzes that are more appropriate for handling a situation in which the odds ratios may differ across strata? In addressing these issues we find that the frequently used Cochran-Mantel-Haenszel test may have a poor power profile, despite being optimal when the odds ratios are common. We develop novel tests that are analogous to the Smirnov, modified Smirnov, convex hull, and adaptive tests that have been proposed for ordered categorical data.