Sample size formula for general win ratio analysis

Originally proposed for the analysis of prioritized composite endpoints, the win ratio has now expanded into a broad class of methodology based on general pairwise comparisons. Complicated by the non-i.i.d. structure of the test statistic, however, sample size estimation for the win ratio has lagged behind. In this article, we develop general and easy-to-use formulas to calculate sample size for win ratio analysis of different outcome types. In a nonparametric setting, the null variance of the test statistic is derived using U-statistic theory in terms of a dispersion parameter called the standard rank deviation, an intrinsic characteristic of the null outcome distribution and the user-defined rule of comparison. The effect size can be hypothesized either on the original scale of the population win ratio, or on the scale of a "usual" effect size suited to the outcome type. The latter approach allows one to measure the effect size by, for example, odds/continuation ratio for totally/partially ordered outcomes and hazard ratios for composite time-to-event outcomes. Simulation studies show that the derived formulas provide accurate estimates for the required sample size across different settings. As illustration, real data from two clinical studies of hepatic and cardiovascular diseases are used as pilot data to calculate sample sizes for future trials.