The Wilcoxon Mann-Whitney test is much misunderstood. It calculates a very important and useful parameter (so it isn't nonparametric!). This parameter is the probability that an observation from one group will be higher than an observation from the other group.

In treatment trials, for example, this amounts to the probability that an individual will fare better on the treatment than in the control group.

While it is less powerful with normal data, it can be more powerful when data are not normal (and normal data are pretty rare!).

I wrote a paper on it at one stage, in the Stata Journal. Conroy RM. What hypotheses do “nonparametric” two-group tests actually test? The Stata Journal. 2012;12(2):1–9.

David,

you are basically right, the Mann-Whitney-U-Test can be used when the assumptions for a parametric test are not met. In my understanding (b) should be that the dependent variable is not normally distributed within each group. Maybe that was the case?

Furthermore, the sample for calculating a t-test should be big enough, the smaller the sample the weaker the parametric test. In accordance with the central limit theorem, slight deviations from the assumptions will result in weak or even wrong results for a small sample. Even though there are no real reference values, some think a sample size of N=30 should be big enough, some say N=50 is required for running a t-test when assumptions are slightly violated.

Maybe the sample size was just small?

Other than that, I don't see the need for a Mann-Whitney-U-Test either...

Thank you, Sebastian, for the very informative response. You are correct about (b), that the assumption of normality is required within each group. Thanks for the correction. Regarding your other idea, the sample size is sufficient, even for a threshold of N=50, so I will inquire if non-normality was the reason for the Mann-Whitney U test.

Cheers!

The Wilcoxon Mann-Whitney test is much misunderstood. It calculates a very important and useful parameter (so it isn't nonparametric!). This parameter is the probability that an observation from one group will be higher than an observation from the other group.

In treatment trials, for example, this amounts to the probability that an individual will fare better on the treatment than in the control group.

While it is less powerful with normal data, it can be more powerful when data are not normal (and normal data are pretty rare!).

I wrote a paper on it at one stage, in the Stata Journal. Conroy RM. What hypotheses do “nonparametric” two-group tests actually test? The Stata Journal. 2012;12(2):1–9.

Jochen, thank you for the detailed response, but I think you have misunderstood (b). You began with "This statement is wrong..." but then went on to basically restate what Sebastian and I had already agreed upon, based on his clarification.

As a further clarification, while you state that rank-based tests are acceptable to use with non-normally distributed data (which, again, Sebastian and I had already agreed upon) it seems that even in this case, the Welch's modified t-test may be preferable to a nonparametric test such as the Mann-Whitney U. See the following reference:

Skovlund E and Fenstad GU (2001). Should we always choose a nonparametric test when comparing two apparently nonnormal distributions? Journal of Clinical Epidemiology 54(1):86-92.

Regarding (c), thank you for the correction and the information about the Welch's t-test when variances are unequal. It seems that I was not alone in my misunderstanding, as a literature search led to some papers suggesting that the Mann-Whitney U is frequently used based on the (erroneous) belief that it is a safe alternative to the Student's t-test when variances are unequal. Several authors commented on this misuse of the Mann-Whitney U, and suggested (as you did) to instead use Welch's t-test for unequal variances.

For the benefit of others, here are some of the references for papers comparing the Student's t-test, Mann-Whitney U, and Welch's t-test:

(1) Buxton GE (2006). The unequal variance t-test is an underused alternative to Student's t-test and the Mann-Whitney U test. Behavioral Ecology 688-690

(2) Fagerland MW and Sandvik L (2009). Performance of five two-sample location tests for skewed distributions with unequal variances. Contemporary Clinical Trials 30: 490-496.

David, thanks for your clarifications. I overlooked that you already agreed on (b).

I hope I will not stir a controversy if I say openly that I doubt the authors of (at least some of) those papers understand what they are doing. For instance, Fagerland and Sandvik recommend using the Welch t-test, after trimming the samples, to compare the medians. They write "it is important to accurately define the population differences of interest, and to interpret test results in strict adherence to the tests' null hypotheses", yet they recommend comparing the medians using a test whose null hypothesis does not involve said medians at all. It seems that it is "important" but just not important enough for them to actually do it.

Note further that a comparison of the medians is logically unrelated to the variances, whereas one of the "selling points" of the paper is they make recommendations for populations with unequal variances, because that is intrinsically problematic for comparison.

Wiser advice would be: if you want to compare the medians, just use a test where the null hypothesis is a median comparison (e.g. the sign test). Do not write a paper to tell us, among five tests none of which performs a median comparison, which one is "recommended" for comparing medians depending on extraneous factors such as the sample variances.