Sample size calculation for clustered binary data with sign tests using different weighting schemes

We propose a sample size calculation approach for testing a proportion using the weighted sign test when binary observations are dependent within a cluster. Sample size formulas are derived with nonparametric methods using three weighting schemes: equal weights to observations, equal weights to clusters, and optimal weights that minimize the variance of the estimator. Sample size formulas are derived incorporating intracluster correlation and the variability in cluster sizes. Simulation studies are conducted to evaluate a finite sample performance of the proposed sample size formulas. Empirical powers are generally close to nominal levels. The number of clusters required increases as the imbalance in cluster size increases and the intracluster correlation increases. The estimator using optimal weights yields the smallest sample size estimate among three estimators. For small values of intracluster correlation the sample size estimates derived from the optimal weight estimator are close to that derived from the estimator assigning equal weights to observations. For large values of intracluster correlation, the optimal weight sample size estimate is close to the sample size estimate assigning equal weights to clusters.