**StATS: ****Sample size calculation for a nonparametric test (March 8, 2005)**

I got an email inquiry about how to calculate power for a Wilcoxon signed ranks test. I don't have a perfect reference for this, but I do have a brief discussion on sample size calculations for the Mann Whitney U test as part of my pages on selecting an appropriate sample size. The same considerations would apply for the Wilcoxon test. In response, the email author sent me a link to

**Computing sample size for data to be analyzed with nonparametric tests.**. GraphPad Software. Accessed on 2005-03-08. www.graphpad.com/library/BiostatsSpecial/article_154.htm

which offers the following advice:

If you plan to use a nonparametric test, compute the sample size required for a t test and add 15%.

This assumes a reasonably high number of subjects (at least a few dozen) and a distribution which is not really unusual. I had not heard this rule; the author cites pages 76-81 of Lehmann, Nonparametrics: Statistical Methods Based on Ranks [BookFinder4U link]. I don't have this book, so I can only guess as to the basis for this formula.

This rule could be based, I suppose, on the lower bound for the Asymptotic Relative Efficiency (ARE) of the Mann Whitney U test versus the t-test, which is 0.864. This says that no matter what the distribution, the ARE of the Mann Whitney U test can never be worse than 0.864 for a reasonably broad class of probability distributions. Inverting that gives you an increase in the sample size by a factor of 1.157. The same statement would also apply for the Wilcoxon Signed Ranks test, which can never have an ARE less than 0.864 compared to the paired t-test.

This page was written by Steve Simon while working at Children's Mercy Hospital. Although I do not hold the copyright for this material, I am reproducing it here as a service, as it is no longer available on the Children's Mercy Hospital website. Need more information? I have a page with general help resources. You can also browse for pages similar to this one at Category: Sample size justification.