Precise and accurate power of the rank-sum test for a continuous outcome

ABSTRACT

Accurate power calculations are essential in small studies containing expensive experimental units or high-stakes exposures. Herein, power of the Wilcoxon Mann–Whitney rank-sum test of a continuous outcome is formulated using a Monte Carlo approach and defining $P(X<Y)\equiv p$ as a measure of effect size, where $X$ and $Y$ denote random observations from two distributions hypothesized to be equal under the null. Effect size $p$ fosters productive communications because researchers understand $p=0.5$ is analogous to a fair coin toss, and $p$ near 0 or 1 represents a large effect. This approach is feasible even without background data. Simulations were conducted comparing the empirical power approach to existing approaches by Rosner & Glynn, Shieh and colleagues, Noether, and O’Brien-Castelloe. Approximations by Noether and O’Brien-Castelloe are shown to be inaccurate for small sample sizes. The Rosner & Glynn and Shieh, Jan & Randles approaches performed well in many small sample scenarios, though both are restricted to location-shift alternatives and neither approach is theoretically justified for small samples. The empirical method is recommended and available in the R package wmwpow.