Rank-based estimators of global treatment effects for cluster randomized trials with multiple endpoints on different scales

Cluster randomized trials commonly employ multiple endpoints. When a single summary of treatment effects across endpoints is of primary interest, global methods represent a common analysis strategy. However, specification of the required joint distribution is non-trivial, particularly when the endpoints have different scales. We develop rank-based interval estimators for a global treatment effect referred to here as the "global win probability, or the mean of multiple Wilcoxon Mann-Whitney probabilities, and interpreted as the probability that a treatment individual responds better than a control individual on average. Using endpoint-specific ranks among the combined sample and within each arm, each individual-level observation is converted to a "win fraction" which quantifies the proportion of wins experienced over every observation in the comparison arm. An individual's multiple observations are then replaced with a single "global win fraction" by averaging win fractions across endpoints. A linear mixed model is applied directly to the global win fractions to obtain point, variance, and interval estimates adjusted for clustering. Simulation demonstrates our approach performs well concerning confidence interval coverage and type I error, and methods are easily implemented using standard software. A case study using public data is provided with corresponding R and SAS code.