The Univariate Distribution of Hierarchical Composite Endpoints and the Condorcet Non-transitivity Paradox

Hierarchical Composite Endpoints (HCEs), as analyzed with Generalized Pairwise Comparisons (GPC) statistics, are general methods of constructing endpoints in clinical trials across various therapeutic areas to establish the efficacy of novel treatments. Although GPC statistics do not require distributional assumptions for estimation, ignoring the underlying distributions can hinder the interpretation of treatment effects. We provide a formal definition of HCEs in a special case based on the "most-important outcome" principle over a fixed timeframe of evaluation. This can be a limitation, but it offers important advantages. In this case, although HCEs involve multiple outcomes, they result in univariate distributions, allowing for the application of the Brunner-Konietschke formula for the estimation of Mann-Whitney effect (MWE) variance. The Condorcet paradox in clinical trials can cause treatment effects to be nonaccumulative. We discuss classes of the stochastically ordered component distributions of the univariate HCE as a sufficient condition to avoid this paradox, emphasizing that HCEs should preferably be defined with outcomes where a consistent treatment benefit is expected. Maraca plots can be used to diagnose violations of conditions that could lead to a Condorcet paradox. We also provide a rank-based method for estimating the MWE when a threshold is used for pairwise comparison between patients in two groups and discuss the implications of using thresholds on the statistical power for the success odds test. Our formal definition of HCEs under stochastically ordered distributions provides a strong theoretical foundation for the use of HCEs in clinical trials, including when a threshold is applied to pairwise comparisons.