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I know there are already a lot of posts out there, but I couldn't find this exact combination in any of them. Comparing two samples (Prices associated with men and with women), but I have neither the same sample size ($n = 790$ vs $n=795$) nor equal variance or normality.

My hypothesis is whether the prices for Women are greater than prices for Men. The Wilcoxon test is significant on a 4 % level.

Can I actually say anything helpful since so many assumptions are violated? Would another test be better?

EDIT Some additional infos: 

Prices Women: Median 28.00, Mean 28.47,  Std. Dev. 17.17,  Skewness 0.91, Kurtosis 5.41
Prices Men:   Median 26.00, Mean, 29.08, Std. Dev. 22.43,  Skewness 2.39, Kurtosis 12.74
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    $\begingroup$ I found some other resources which state the opposite: data.library.virginia.edu/the-wilcoxon-rank-sum-test "This is where the Wilcoxon Rank Sum Test comes in. It only makes the first two assumptions of independence and equal variance." Or: kasuya.ecology1.org/stats/utest01e.html $\endgroup$
    – Heike
    Commented Dec 28, 2018 at 11:09
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    $\begingroup$ The assumptions of the Wilcoxon rank-sum test are well described at en.wikipedia.org/wiki/…. They do not include any of the characteristics you mention. $\endgroup$
    – whuber ♦
    Commented Dec 28, 2018 at 16:02
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    $\begingroup$ @whuber: the wiki page you linked seems to disagree with you! Specifically 3. Under the null hypothesis H0, the distributions of both populations are equal implies equal variance (if exists). $\endgroup$
    – kjetil b halvorsen ♦
    Commented Dec 29, 2018 at 22:34
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    $\begingroup$ @Kjetil I apologize for being vague. I agree with you that the Wikipedia article stipulates "equal distributions" as the null hypothesis. I was focusing on the alternative hypothesis, which does not require equal variances. In some formulations of the Wilcoxon test, the alternative is more restricted and does imply equal variances. There are implications for how one interprets the test. $\endgroup$
    – whuber ♦
    Commented Dec 29, 2018 at 22:53
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    $\begingroup$ This test is in many ways better thought of as estimating -- in your example -- the probability that the price for males is greater than that for females, or its complement -- with a popular significance procedure on the side. For that probability to be estimated only which way the inequality runs is needed (modulo small print about ties). $\endgroup$
    – Nick Cox
    Commented Dec 30, 2018 at 10:10

2 Answers 2

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tl;dr if you want to interpret the rejection of the null hypothesis as evidence that prices for women are greater than those for men, then you do need the assumption of equal variance (in fact, equal distributions) between the two populations. If you are satisfied with showing that the distribution of prices for women differs in some way from that of men, then you don't need the extra assumption.

You don't need to worry about unequal sample size (this will affect the power of the test, but not its validity) or Normality.

For what it's worth, testing whether one group's values are larger on average than another group's when their variances also differ is a surprisingly deep question, even for Normally distributed data (where it's known as the Behrens-Fisher problem).

Referring to the Wikipedia page: the "very general formulation" says:

  1. Under the null hypothesis H0, the distributions of both populations are equal.[3]
  2. The alternative hypothesis H1 is that the distributions are not equal.

The next paragraph says:

Under more strict assumptions than the general formulation above, e.g., if the responses are assumed to be continuous and the alternative is restricted to a shift in location, i.e., $F_1(x) = F_2(x + δ)$, we can interpret a significant Mann–Whitney U test as showing a difference in medians ...

(emphasis added)

Note to technical readers: I think this is a reasonable summary, but if anyone wants to be more rigorous, feel free to comment or edit or post an alternative answer ...

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Ben Bolker's answer is great. I just wanted to add an answer to the "Would another test be better?" part.

If you want to be able to conclude that prices for women are greater than those for men without the assumptions of equal distributions under the null, Brunner-Munzel's test can be recommended. For a full technical description of this test, see Chapter 3 in https://link.springer.com/content/pdf/10.1007/978-3-030-02914-2.pdf. For a non-technical introduction and comparison to the Wilcoxon test, see https://journals.sagepub.com/doi/full/10.1177/2515245921999602 (Disclaimer: I am the author of this one).

A disclaimer: this test is not optimal in any strict sense. However, it seems to be the most reasonable test for the nonparametric Behrens-Fisher problem available at the moment, just as Welch's t test is not optimal in any sense but most reasonable if you want to test equal means. Indeed, Brunner-Munzel's test is almost Welch's t test on ranked data, while the Wilcoxon test is close to Student's t test on ranked data.

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