We extend the results of Luo et al. (2015, Biometrics 71, 139-145) regarding the alternative hypotheses for the win ratio from hazard orders to the upper quadrant stochastic order on the plane. This extension substantially widens the range of alternatives against which the win ratio is known to be consistent. It incorporates alternatives induced by simple and popular copula models that are left out by the characterization of Luo et al. (2015). We also discuss how our results may be generalized to win ratios in multivariate and stratified settings.
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