  The win ratio (WR) is a novel statistic used in randomized controlled trials
that can account for hierarchies within event outcomes. In this paper we report
and study the long-run non-transitive behavior of the win ratio and the closely
related Area Under the Receiver Operating Characteristics Curve (AUC) and argue
that their transitivity cannot be taken for granted. Crucially, traditional
within-group statistics (i.e., comparison of means) are always transitive,
while the WR can detect non-transitivity. Non-transitivity provides valuable
information on the stochastic relationship between two treatment groups, which
should be tested and reported. We specify the necessary conditions for
transitivity, the sufficient conditions for non-transitivity, and demonstrate
non-transitivity in a real-life large randomized controlled trial for the WR of
time-to-death. Our results can be used to rule out or evaluate the possibility
of non-transitivity and show the importance of studying the strength of
stochastic relationships.
