Power calculation of adjusted McNemar's test based on clustered data of varying cluster size

McNemars test is often used to compare two proportions estimated from paired observations. When the observations are sampled in clusters, adjustment is needed to ensure that the size of McNemars test does not exceed the nominal level. Eliasziw and Donner (1991) developed an adjustment to McNemars test that involves first estimating the correlation between discordant pairs within a cluster, then using the estimate of the correlation to adjust the usual McNemar's test statistic. Gönen (2004) derived two approximations for calculating the power and sample size for the adjusted McNemar's test. He reported that the accuracy of the two approximations is compromised for large value of intracluster correlation and small value of proportion of discordant pairs; the error of the approximation can be higher than 10 per cent. In this paper, we extend his power formula, developed under fixed cluster size assumption, to accommodate the case where the cluster sizes are not constant. We show via simulations that the theoretical powers calculated from our proposed power formula are close to their empirical counterparts under a variety of settings. More significantly, in the case of fixed cluster size, our reduced power formula provides a more accurate power approximation than Gönen's power formula regardless of the values of intracluster correlation and the proportion of discordant pairs.