Description:
The common nonparametric rank test for comparing the distributions of two groups has many names, including the Mann-Whitney U test, the Wilcoxon Rank-Sum test, the Mann-Whitney-Wilcoxon test, and the Wilcoxon-Mann-Whitney test. Often, this test is used as the nonparametric alternative to the traditional two-sample t-test.
Suppose that researchers wish to compare two types of local anesthesia to determine whether there is a difference in time to loss of pain. Subjects will be randomized to treatment, the treatment will be administered, and the time to loss of pain measured. The researchers wish to find the sample size needed to obtain 90% power, if the true difference is 3, 4, or 5 minutes.
Past experiments of this type have had standard deviations in the range of 1 to 5 minutes. It is anticipated that the standard deviation of the two groups will be equal.
It is unknown which treatment has lower time to loss of pain, so a two-sided test will be used.
The researchers will be performing a Mann-Whitney U test instead of the t-test because it is anticipated that the distribution of the two populations is not Normal. The researchers have found that the distribution shape that most closely resembles the two populations is Tukey’s Lambda distribution with Skewness value 0.37 and an Elongation parameter of 0.011. (Tukey’s Lambda distribution reshapes the Normal distribution with skewness and kurtosis modifications.)
The numeric results section shows the needed sample sizes for the various parameter combinations.
The sample size chart shows the effect of the presumed difference and the assumed standard deviation.
Selecting a specific scenario, the researchers would like to know how the power is affected if a Normal distribution is assumed instead.
The power drops down below 90% by a moderate amount.
Further, they wish to know the power and alpha that would be achieved if a Tukey’s Lambda distribution is used, but a regular t-test is used instead of the Mann-Whitney U test.
These comparisons can be used to show the advantages of using the Mann-Whitney U test when the data is non-Normal, and the standard t-test when the data is Normal.
