Means: One and Two Groups
Sample size planning for a two-group Mann-Whitney U (Wilcoxon rank-sum) test using the Noether approximation for continuous outcomes without ties. Specify the competing probability, power, significance level, and allocation ratio; the calculator finds the smallest integer group sizes that meet the target power.
The calculator uses the Noether (1987) approximation for the two-sample Mann-Whitney U / Wilcoxon rank-sum test. This method uses this procedure for continuous data without ties and defines the effect size as the competing probability P1 = P(Y1 ≥ Y2). Under the null hypothesis, P1 = 0.5.
Let N = N1 + N2 be the total evaluable sample size and let c = N1/N. For a two-sided test, the Noether approximation is:
For a one-sided test, z1-α/2 is replaced by z1-α. Because actual group sizes must be integers, the calculator evaluates candidate integer group sizes near the continuous solution and reports the smallest design that achieves at least the requested power under the same approximation.
The allocation ratio is entered as R = N2/N1. Equal allocation corresponds to R = 1. For an unequal allocation, the calculator searches integer N1 and N2 values while keeping N2/N1 as close as possible to the requested ratio and retaining the target-power requirement.
If a dropout rate is specified, the required evaluable group sizes are inflated using N1′ = N1/(1 - DR) and N2′ = N2/(1 - DR), with each enrollment size rounded up, consistent with the relevant methodological literature.
This calculator is intended for two independent groups with continuous responses and no ties, matching the assumptions of the this method Noether procedure. The Mann-Whitney effect size P1 is a probability-based measure of separation between the two distributions; it is not automatically the same thing as a difference in means or medians unless additional location-shift assumptions are appropriate.
Noether, G. E. (1987). Sample size determination for some common nonparametric tests. Journal of the American Statistical Association, 82(398), 645–647.
Lehmann, E. L. (1975). Nonparametrics: Statistical Methods Based on Ranks. Holden-Day.
Zhao, Y. D., Rahardja, D., & Qu, Y. (2008). Sample size calculation for the Wilcoxon-Mann-Whitney test adjusting for ties. Statistics in Medicine, 27, 462–468.
the software. this method Sample Size Software: Mann-Whitney U or Wilcoxon Rank-Sum Tests (Noether), Chapter 257.
See the analysis this calculator plans for applied to published trial results, with the estimates, confidence intervals and interpretation explained.