Reproducibility probability estimation and testing for the Wilcoxon rank-sum test

The reproducibility probability (RP) of a statistically significant outcome is the true power of a statistical test and its estimate is a useful indicator of the stability of the test result. RP-testing consists in testing statistical hypotheses using an RP-estimator as test statistic. In the parametric framework, the RP-based test and the classical one are equivalent, while in the nonparametric one to perform RP-testing is possible only approximately. In this work, we evaluate through a wide simulation study the performances of several semi-parametric and nonparametric RP-estimators (RPEs) for the Wilcoxon rank-sum (WRS) test. RPEs have two tasks: to perform RP-testing and to estimate the RP. To compare RPEs performances we adopt risk indexes (e.g. mean square error (MSE)) and an index of agreement between the outcomes of the WRS test and the RP-based test. Results indicate that the rate of disagreement tends to zero as the sample size increases; the overall rate of disagreement provided by semi-parametric RPEs with finite samples (size 20-200 per group) is 0.15%, and that of nonparametric ones is 0.58%. Concerning risk measures, there is not an RPE dominating the others; for high power values, nonparametric RPEs present the lowest MSE; on average, the semi-parametric RPE based on the upper bound of the variance of the test statistic performs best; nevertheless, the relative gains between the best and the worst are quite small (5-10%). To conclude, well-approximated RP-testing for the WRS test can be performed by adopting a semi-parametric RPE. Since nonparametric plug-in based RPEs perform well in presence of high reproducibility, their adoption is suggested for evaluating the stability of test results and, for example, those of clinical trials.