How reliable are the multiple comparison methods for odds ratio?

The homogeneity tests of odds ratios are used in clinical trials and epidemiological investigations as a preliminary step of meta-analysis. In recent studies, the severity or mortality of COVID-19 in relation to demographic characteristics, comorbidities, and other conditions has been popularly discussed by interpreting odds ratios and using meta-analysis. According to the homogeneity test results, a common odds ratio summarizes all of the odds ratios in a series of studies. If the aim is not to find a common odds ratio, but to find which of the sub-characteristics/groups is different from the others or is under risk, then the implementation of a multiple comparison procedure is required. In this article, the focus is placed on the accuracy and reliability of the homogeneity of odds ratio tests for multiple comparisons when the odds ratios are heterogeneous at the omnibus level. Three recently proposed multiple comparison tests and four homogeneity of odds ratios tests with six adjustment methods to control the type-I error rate are considered. The reliability and accuracy of the methods are discussed in relation to COVID-19 severity data associated with diabetes on a country-by-country basis, and a simulation study to assess the powers and type-I error rates of the tests is conducted.