Post-hoc comparison tests for odds ratios

The null hypothesis of homogeneity of odds ratio across the strata for a 2x2 tables is tested by several tests. When the null hypothesis is rejected, it means that at least one of the odds ratios significantly differs from others. Post-hoc tests are used after the null hypothesis of equality of groups is rejected Those tests aim to reveal the true differences between groups. In this paper, we propose three post-hoc tests that control family-wise type-I error. The first and second approaches are based on the difference between the two odds ratios. The third approach uses the modification of the Breslow-Day test. These tests provide the homogeneous groups by their odds ratios and enables to calculate the common odds ratio. The suggested tests are applied to several COVID-19 data sets and the results are discussed. The proposed methods can also be used to compare different risk factors on a certain event or outcome.