A robust adjustment to McNemar test when the data are clustered

Eliasziw and Donner (1991) proposed an adjusted McNemar test that can be used to compare two proportions estimated from paired clustered binary data. The adjustment factor they proposed is formulated in terms of an intra-cluster correlation. They considered two estimators for the intra-cluster correlation. However, both estimators presented estimation issues; the first estimator cannot be calculated when the number of discordant pairs is small while the second estimator is not consistent when the Dirichlet-multinomial distribution assumption is violated. To overcome the two estimation issues, we propose an alternative adjustment factor that is formulated in terms of a different intra-cluster correlation, which can be estimated consistently using both discordant and accordant pairs. The consistency of our estimator of the new intra-cluster correlation is guaranteed even when the Dirichlet-multinomial distribution assumption is violated. In addition, we show that the proposed adjusted McNemar test performs as well as Eliasziw and Donner’s adjusted McNemar test in terms of size and power.