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.
Keywords: Cochran–Mantel–Haenszel test; Fligner–Policello test; Hodges–Lehmann estimator; Kansas City cardiomyopathy questionnaire; Wilcoxon test; Win ratio; adjustment; clinical trial; dapagliflozin in patients with heart failure and reduced ejection fraction; estimand; heart failure; intercurrent event; location test; missing data; number needed to treat; patient reported outcome; rank analysis; rank analysis of covariance; stratification; symptom score; van Elteren test; win probability.