Editing Mann–Whitney U test (section)

===Different distributions===
The Mann–Whitney ''U'' test is not valid for testing the null hypothesis <math>P(Y>X)+0.5P(Y=X)= 0.5</math> against  the alternative hypothesis <math>P(Y>X)+0.5P(Y=X)\neq 0.5</math>), without assuming that the distributions are the same under the null hypothesis (i.e., assuming <math>F_1=F_2</math>).<ref name="FayProschan2010" /> To test between those hypotheses, better tests are available. Among those are the [[Brunner_Munzel_Test|Brunner-Munzel]] and the Fligner–Policello test.<ref>{{Cite book| publisher = Springer International Publishing| last1 = Brunner| first1 = Edgar| last2 = Bathke| first2 = Arne C.| last3 = Konietschke| first3 = Frank| title = Rank and pseudo-rank procedures for independent observations in factorial designs: Using R and SAS| location = Cham| series = Springer Series in Statistics| date = 2018| doi = 10.1007/978-3-030-02914-2| url = http://link.springer.com/10.1007/978-3-030-02914-2| isbn = 978-3-030-02912-8
}}</ref> Specifically, under the more general null hypothesis <math>P(Y>X)+0.5P(Y=X)= 0.5</math>, the Mann–Whitney ''U'' test can have inflated type I error rates even in large samples (especially if the variances of two populations are unequal and the sample sizes are different), a problem the better alternatives solve.<ref name="karch">{{Cite journal| doi = 10.1177/2515245921999602| issn = 2515-2459| volume = 4| issue = 2| last = Karch| first = Julian D.| title = Psychologists Should Use Brunner–Munzel's Instead of Mann–Whitney's ''U'' Test as the Default Nonparametric Procedure| journal = Advances in Methods and Practices in Psychological Science| date = 2021| doi-access = free| hdl = 1887/3209569| hdl-access = free}}</ref> As a result, it has been suggested to use one of the alternatives (specifically the Brunner–Munzel test) if it cannot be assumed that the distributions are equal under the null hypothesis.<ref name="karch" />

====Alternatives====
If one desires a simple shift interpretation, the Mann–Whitney ''U'' test should ''not'' be used when the distributions of the two samples are very different, as it can give erroneous interpretation of significant results.<ref>{{cite journal |doi=10.1006/anbe.2001.1691 |title=Mann–Whitney ''U'' test when variances are unequal | volume=61 |issue=6 | year=2001 |journal=Animal Behaviour |pages=1247–1249 | last1 = Kasuya | first1 = Eiiti|s2cid=140209347 }}</ref> In that situation, the [[Welch's t-test|unequal variances]] version of the ''t''-test may give more reliable results.

Similarly, some authors (e.g., Conover{{full citation needed|date=November 2012}}) suggest transforming the data to ranks (if they are not already ranks) and then performing the ''t''-test on the transformed data, the version of the ''t''-test used depending on whether or not the population variances are suspected to be different. Rank transformations do not preserve variances, but variances are recomputed from samples after rank transformations.

The [[Brown–Forsythe test]] has been suggested as an appropriate non-parametric equivalent to the [[F-test|''F''-test]] for equal variances.{{citation needed|date=February 2012}}

A more powerful test is the [[Brunner_Munzel_Test|Brunner-Munzel test]], outperforming the Mann–Whitney ''U'' test in case of violated assumption of exchangeability.<ref>{{cite journal |doi=10.1177/2515245921999602 |title=Psychologists Should Use Brunner–Munzel's Instead of Mann–Whitney's ''U'' Test as the Default Nonparametric Procedure | volume=4 |issue=2 | year=2021 |journal= Advances in Methods and Practices in Psychological Science| last1 = Karch | first1 = Julian | url = https://journals.sagepub.com/doi/full/10.1177/2515245921999602| hdl=1887/3209569 |s2cid=235521799 | hdl-access=free }}</ref>

The Mann–Whitney ''U'' test is a special case of the [[proportional odds model]], allowing for covariate-adjustment.<ref>{{cite journal |title=Violation of Proportional Odds is Not Fatal | last1 = Harrell | first1 = Frank| date = 20 September 2020 | url = https://www.fharrell.com/post/po/}}</ref>

See also [[Kolmogorov–Smirnov test]].