On the intransitivity of the win ratio

The win-ratio analysis of controlled clinical trials uses pairwise comparisons between patients in the treatment and control group based on a primary outcome, say time to death, with indeterminacies resolved where possible by a secondary outcome, say time to hospitalization. The resulting preferences may not be transitive. Intransitivity occurs when potential follow-up times vary between patients and rankings from the primary events differ from those from secondary events. We characterize the structure of closed loops, derive some general properties of win-ratio preferences and provide simple numerical illustrations. Under realistic assumptions, unless all potential follow-up times are equal, intransitivities are certain to occur in sufficiently large samples, but their overall frequency is low.