Cochran–Mantel–Haeszel stratified-adjusted test for multiple odds ratios

The Cochran–Mantel–Haeszel test (thereby CMH test) is commonly applied to test whether exposure to a risk factor has significant impact on the clinical outcome. However, studies in medical research often require the identification of several significant risk factors simultaneously. In the literature, one of the ubiquitous approaches that strongly control the familywise error rate for multiple tests is the Holm’s step-down procedure. It elegantly applies the first-degree Bonferroni inequality at each step to make inference decisions. On the other hand, when the number of to-be-tested hypotheses is large, Holm’s procedure unavoidably inherits the conservativeness of the Bonferroni inequality, which makes it almost useless when dealing with large number of populations. In this paper, we propose a sequentially rejective procedure for simultaneous inference on odds ratios. We prove that, when testing multiple odds ratios, the new procedure is uniformly more powerful than the Holm’s procedure. Simulation studies show that the improvement is substantial in some scenarios.