A Continuous-Weighted Win Ratio for Hierarchical Composite Endpoints

Hierarchical composite endpoints are commonly used in clinical trials when component out- comes differ in clinical importance. The win ratio compares patients across treatment groups according to a pre-specified order of clinical priority and has an intuitive interpretation. A strict hierarchical rule, however, uses a lower-priority endpoint only when all higher-priority endpoints are tied or non-informative. This may reduce power when the treatment effect is mainly ex- pressed through lower-priority outcomes. Recent threshold-based extensions relax this hierar- chy by allowing lower-priority outcomes to contribute when higher-priority outcomes are within pre-specified margins. We propose a continuous-weighted win ratio for hierarchical composite endpoints. The method replaces hard transitions between endpoint levels with smooth weights, so that a lower-priority endpoint can contribute gradually when the higher-priority endpoint is close to tied. The resulting estimator is a two-sample U-statistic. We derive its large-sample distribution using the Hoeffding projection and provide a consistent plug-in variance estimator. We also describe the role of the tuning parameter through local alternatives and local efficiency. The proposed class includes the strict win ratio and hard-threshold rules as special or limit- ing cases. Simulation studies show that relaxing the strict hierarchy can improve power when treatment effects are concentrated on lower-priority endpoints, but may reduce power when the highest-priority endpoint carries the main treatment signal.