Editing Mann–Whitney U test (section)

==Relation to other tests==

===Comparison to Student's ''t''-test===
The Mann–Whitney ''U'' test tests a null hypothesis that the [[probability distribution]] of a randomly drawn observation from one group is the same as the probability distribution of a randomly drawn observation from the other group against an alternative that those distributions are not equal (see [[Mann–Whitney U test#Assumptions and formal statement of hypotheses]]). In contrast, a [[t-test]] tests a null hypothesis of equal means in two groups against an alternative of unequal means. Hence, except in special cases, the Mann–Whitney ''U'' test and the t-test do not test the same hypotheses and should be compared with this in mind.  
;Ordinal data: The Mann–Whitney ''U'' test is preferable to the ''t''-test when the data are [[Level of measurement#Ordinal scale|ordinal]] but not interval scaled, in which case the spacing between adjacent values of the scale cannot be assumed to be constant.
;Robustness:As it compares the sums of ranks,<ref name="Motulsky 2007">Motulsky, Harvey&nbsp;J.; ''Statistics Guide'', San Diego, CA: GraphPad Software, 2007, p. 123</ref> the Mann–Whitney ''U'' test is less likely than the ''t''-test to spuriously indicate significance because of the presence of [[outlier]]s. However, the Mann–Whitney ''U'' test may have worse [[Type I and type II errors|type I error]] control when data are both heteroscedastic and non-normal.<ref>{{Cite journal|last=Zimmerman|first=Donald W.|date=1998-01-01|title=Invalidation of Parametric and Nonparametric Statistical Tests by Concurrent Violation of Two Assumptions|journal=The Journal of Experimental Education|volume=67|issue=1|pages=55–68|doi=10.1080/00220979809598344|issn=0022-0973}}</ref>
;Efficiency:When normality holds, the Mann–Whitney ''U'' test has an (asymptotic) [[Efficiency (statistics)|efficiency]] of 3/{{pi}} or about&nbsp;0.95 when compared to the ''t''-test.<ref name="Lehmann 1999">Lehamnn, Erich&nbsp;L.; ''Elements of Large Sample Theory'', Springer, 1999, p. 176</ref> For distributions sufficiently far from normal and for sufficiently large sample sizes, the Mann–Whitney ''U'' test is considerably more efficient than the ''t''.<ref name="Conover 1980">Conover, William&nbsp;J.; [https://onlinepubs.trb.org/onlinepubs/nchrp/cd-22/manual/v2chapter6.pdf ''Practical Nonparametric Statistics''], John Wiley & Sons, 1980 (2nd Edition), pp. 225–226</ref> This comparison in efficiency, however, should be interpreted with caution, as Mann–Whitney and the t-test do not test the same quantities. If, for example, a difference of group means is of primary interest, Mann–Whitney is not an appropriate test.<ref>{{Cite journal|last1=Lumley|first1=Thomas|last2=Diehr|first2=Paula|author2-link=Paula Diehr|last3=Emerson|first3=Scott|last4=Chen|first4=Lu|date=May 2002|title=The Importance of the Normality Assumption in Large Public Health Data Sets|journal=Annual Review of Public Health|volume=23|issue=1|pages=151–169|doi=10.1146/annurev.publhealth.23.100901.140546|pmid=11910059| doi-access=free|issn=0163-7525}}</ref>

The Mann–Whitney ''U'' test will give very similar results to performing an ordinary parametric two-sample [[t test|''t''-test]] on the rankings of the data.<ref>{{cite journal |doi=10.2307/2683975 |title=Rank Transformations as a Bridge Between Parametric and Nonparametric Statistics |first1=William&nbsp;J. |last1=Conover |first2=Ronald&nbsp;L. |last2=Iman |author-link2=Ronald L. Iman |journal=[[The American Statistician]] |volume=35 |issue=3 |year=1981 |pages=124–129 |jstor=2683975 }}</ref>
{| class="wikitable float-right"
|+Relative efficiencies of the Mann–Whitney test versus the two-sample ''t''-test if ''f'' = ''g'' equals a number of distributions<ref>{{Cite book |last=Vaart |first=A. W. van der |url=http://dx.doi.org/10.1017/cbo9780511802256 |title=Asymptotic Statistics |date=1998-10-13 |publisher=Cambridge University Press |doi=10.1017/cbo9780511802256 |isbn=978-0-511-80225-6}}</ref>
!Distribution
!Efficiency
|-
|Logistic
|<math>\pi^2/9</math>
|-
|Normal
|<math>3/\pi
</math>
|-
|Laplace
|3/2
|-
|Uniform
|1
|}

===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]].