Editing Mann–Whitney U test (section)

==Assumptions and formal statement of hypotheses==
Although [[Henry Mann]] and Donald Ransom Whitney<ref name="mannwhitney1947" /> developed the Mann–Whitney ''U'' test under the assumption of [[Continuous probability distribution|continuous]] responses with the [[alternative hypothesis]] being that one distribution is [[stochastic ordering|stochastically greater]] than the other, there are many other ways to formulate the [[null hypothesis|null]] and alternative hypotheses such that the Mann–Whitney ''U'' test will give a valid test.<ref name="FayProschan2010">{{cite journal |last1=Fay |first1=Michael&nbsp;P. |last2=Proschan |first2=Michael&nbsp;A. |journal=[[Statistics Surveys]] |year=2010 |pages=1–39 |volume=4 |doi=10.1214/09-SS051 |title=Wilcoxon–Mann–Whitney or ''t''-test? On assumptions for hypothesis tests and multiple interpretations of decision rules |pmc=2857732 |mr=2595125 |pmid=20414472 }}</ref>

A very general formulation is to assume that:

# All the observations from both groups are [[statistical independence|independent]] of each other,
# The responses are at least [[ordinal measurement|ordinal]] (i.e., one can at least say, of any two observations, which is the greater),
# Under the null hypothesis ''H''<sub>0</sub>, the distributions of both populations are identical.<ref>[https://www.jstor.org/stable/2283092], See Table 2.1 of Pratt (1964) "Robustness of Some Procedures for the Two-Sample Location Problem." ''Journal of the American Statistical Association.'' 59 (307): 655–680. If the two distributions are normal with the same mean but different variances, then Pr[''X''&nbsp;>&nbsp;''Y'']&nbsp;=&nbsp;Pr[''Y''&nbsp;<&nbsp;''X''] but the size of the Mann–Whitney test can be larger than the nominal level. So we cannot define the null hypothesis as Pr[''X''&nbsp;>&nbsp;''Y'']&nbsp;=&nbsp;Pr[''Y''&nbsp;<&nbsp;''X''] and get a valid test.</ref>
# The alternative hypothesis ''H''<sub>1</sub> is that the distributions are not identical.

Under the general formulation, the test is only [[Consistency (statistics)#Tests|consistent]] when the following occurs under ''H''<sub>1</sub>:

# The probability of an observation from population ''X'' exceeding an observation from population ''Y'' is different (larger, or smaller) than the probability of an observation from ''Y'' exceeding an observation from ''X''; i.e.,  {{math|1=P(''X'' > ''Y'') ≠ P(''Y'' > ''X'')}} or {{math|1=P(''X'' > ''Y'') + 0.5 · P(''X'' = ''Y'') ≠ 0.5}}.

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., {{math|1=''F''<sub>1</sub>(''x'') = ''F''<sub>2</sub>(''x'' + ''δ'')}}, we can interpret a significant Mann–Whitney ''U'' test as showing a difference in medians. Under this location shift assumption, we can also interpret the Mann–Whitney ''U'' test as assessing whether the [[Hodges–Lehmann estimate]] of the difference in central tendency between the two populations differs from zero. The [[Hodges–Lehmann estimate]] for this two-sample problem is the [[median]] of all possible differences between an observation in the first sample and an observation in the second sample.

Otherwise, if both the dispersions and shapes of the distribution of both samples differ, the Mann–Whitney ''U'' test fails a test of medians. It is possible to show examples where medians are numerically equal while the test rejects the null hypothesis with a small p-value.<ref>{{cite journal |last1=Divine |first1=George W. |last2=Norton |first2=H. James |last3=Barón |first3=Anna E. |last4=Juarez-Colunga |first4=Elizabeth |title=The Wilcoxon–Mann–Whitney Procedure Fails as a Test of Medians |journal=The American Statistician |date=2018 |volume=72 |issue=3 |pages=278–286 |doi=10.1080/00031305.2017.1305291 |doi-access=free }}</ref><ref>{{cite journal |last1=Conroy |first1=Ronán  |title=What Hypotheses do "Nonparametric" Two-Group Tests Actually Test? |journal=Stata Journal |date=2012 |volume=12 |issue=2 |pages=182–190 |doi=10.1177/1536867X1201200202 |s2cid=118445807 |url=https://www.researchgate.net/publication/279580873 |access-date=24 May 2021|doi-access=free }}</ref><ref>{{cite journal |last1=Hart |first1=Anna |title=Mann–Whitney test is not just a test of medians: differences in spread can be important |journal=BMJ |date=2001 |volume=323 |issue=7309 |pages=391–393 |doi=10.1136/bmj.323.7309.391 |doi-access=free |pmid=11509435 |pmc=1120984 }}</ref> 

The Mann–Whitney ''U'' test / Wilcoxon rank-sum test is not the same as the [[Wilcoxon signed-rank test|Wilcoxon ''signed''-rank test]], although both are nonparametric and involve summation of [[Ranking (statistics)|ranks]]. The Mann–Whitney ''U'' test is applied to independent samples. The Wilcoxon signed-rank test is applied to matched or dependent samples.