Relative Efficiency of Unequal Versus Equal Cluster Sizes for the Nonparametric Weighted Sign Test Estimators in Clustered Binary Data

We performed an analysis of clustered binary data from multiple observations for each participant in which any 2 observations from a participant are assumed to have a common correlation coefficient. In the weighted sign test on proportion in clustered binary data, 3 weighting schemes were considered: equal weights to observations, equal weights to clusters, and optimal weights that minimize the variance of the estimator. Because the distribution of cluster sizes may not be exactly specified before the trial starts, the sample size is usually determined using an average cluster size without taking into account any potential imbalance in cluster size, even though cluster size usually varies among clusters. In this article, we investigate the relative efficiency (RE) of unequal versus equal cluster sizes for clustered binary data using the weighted sign test estimators. The REs are computed as a function of correlation among observations for each participant and the various cluster size distributions. The required sample size for unequal cluster sizes will not exceed the sample size for an equal cluster size multiplied by the maximum RE. It is concluded that the maximum RE for various cluster size distributions considered here does not exceed 1.50, 1.61, and 1.12 for equal weights to observations, equal weights to clusters, and optimal weights, respectively. It suggests sampling 50%, 61%, and 12% more clusters, respectively, depending on the weighting schemes than the number of clusters computed using an average cluster size.