Simultaneous Testing of McNemar’s Problem for Several Populations

McNemar’s problem concerns the hypothesis of equal probabilities for the unlike pairs of correlated binary variables. We consider four different extensions to this problem, each for testing simultaneous equality of proportions of unlike pairs in c independent populations of correlated binary variables, but each under different assumptions and/or additional hypotheses. For each extension both the likelihood ratio test and the goodness-of-fit chi-square test are given. When c = 1, all cases reduce to McNemar’s problem. For c ≥ 2, however, the tests are quite different, depending on exactly how the hypothesis and alternatives of McNemar are extended. An example illustrates how widely the results may differ, depending on which extended framework is appropriate.