  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 nonparametric tests of group
difference (Wilcoxon rank-sum test, Fligner-Plicello test,
Cochran-Mantel-Haenszel test, test based on the regression on ranks and the
rank ANCOVA test). We show that the win ratio gives an interpretable treatment
effect measure with corresponding test to detect treatment effect difference
under minimal assumptions.
