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I am wondering if there exist any advantage of using the Mann-Whitney U test over a permutation test in the following setting:

Suppose I have a group of animals with eggs in their brood chamber, from this group I assign some to a control vessel and some to a treatment vessel. Then I wait till the newborns hatch in each vessel and I measure the size of the newborns.

I want to test the hypothesis that the newborns came from the same population, i.e, there is no effect of the treatment with respect to control in the size of newborns. Under this hypothesis, I can construct a permutation test to assess if the mean difference between the size of the animals in each vessel is just random or there is evidence that suggest that the difference could not have happened by chance.

Is there any reason to use Mann–Whitney U test over just this simple permutation test?

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There are important reasons to use the Wilcoxon-Mann-Whitney two-sample rank-sum test in this context. The most important reason is that the test is transformation-invariant, i.e., you get the same inference whether you log-transform the weights or not. Secondly, this rank test is more robust to outliers. Third, it extends to more complex situations such as needing covariate adjustment (the proportional odds model is the generalization of the Wilcoxon test).

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  • $\begingroup$ The null hypothesis [for the Wilcoxon test is] that, for randomly selected values X and Y from two populations, the probability of X being greater than Y is equal to the probability of Y being greater than X. [Wikipedia Mann–Whitney U test]. I am assuming the proportional odd model means ordinal logistic regression. Thus I think the generalization in parentheses refers to 1. adding explanatory regressor variables, and 2. being able to compare more than two dependent variables. Is this a correct interpretation? $\endgroup$
    – Single Malt
    Commented Sep 25, 2021 at 18:10
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    $\begingroup$ The proportional odds model is for one Y. But yes to the other. Also it handles ties better than Wilcoxon $\endgroup$
    – Frank Harrell
    Commented Sep 26, 2021 at 2:05
  • $\begingroup$ Incorporating your comment, the generalization is in two “dimensions”: 1. The dependent variable has the number of ordinal variables (whereas Wilcoxon-Mann-Whitney has only two variables, neither of which, by the nature of the test needs to be termed dependent ). 2. Optional addition of explanatory variables. Is this correct? Handling ties better is more of a bonus, but could be viewed as a slight generalization as in a sense uses more of the data. $\endgroup$
    – Single Malt
    Commented Sep 26, 2021 at 18:06
  • $\begingroup$ In the Wilcoxon test the dependent variable is the ordinal or continuous measure you are comparing, and the independent variable is the group indicator. $\endgroup$
    – Frank Harrell
    Commented Sep 26, 2021 at 22:09
  • $\begingroup$ This may have solved my confusion. This link treats the X and Y as peers en.wikipedia.org/wiki/Mann–Whitney_U_test, but as per your comment it can also be viewed as an ordinal or continuous variable against an independent group indicator. $\endgroup$
    – Single Malt
    Commented Sep 27, 2021 at 13:25

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