Adjusted win ratio with stratification: Calculation methods and interpretation.

The win ratio is a general method of comparing locations of distributions of two independent, ordinal random variables, and it can be estimated without distributional assumptions. In this paper we provide a unified theory of win ratio estimation in the presence of stratification and adjustment by a numeric variable. Building step by step on the estimate of the crude win ratio we compare corresponding tests with well known non-parametric tests of group difference (Wilcoxon rank-sum test, Fligner-Policello test, van Elteren test, test based on the regression on ranks, and the rank analysis of covariance test). We show that the win ratio gives an interpretable treatment effect measure with corresponding test to detect treatment effect difference under minimal assumptions.