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In statistics, the Mann–Whitney ''U'' test (also called the Mann–Whitney–Wilcoxon (MWW), Wilcoxon rank-sum test (WRS), or Wilcoxon–Mann–Whitney test) is a nonparametric test of the null hypothesis that two samples come from the same population against an alternative hypothesis, especially that a particular population tends to have larger values than the other. It can be applied on unknown distributions contrary to ''t''-test which has to be applied only on normal distributions, and it is nearly as efficient as the ''t''-test on normal distributions. The Wilcoxon rank-sum test is not the same as the Wilcoxon signed-rank test, although both are nonparametric and involve summation of ranks. The Wilcoxon rank-sum test is applied to independent samples. The Wilcoxon signed-rank test is applied to matched or dependent samples. ==Assumptions and formal statement of hypotheses== Although Mann and Whitney〔 developed the MWW test under the assumption of continuous responses with the alternative hypothesis being that one distribution is stochastically greater than the other, there are many other ways to formulate the null and alternative hypotheses such that the MWW test will give a valid test. A very general formulation is to assume that: # All the observations from both groups are independent of each other, # The responses are ordinal (i.e. one can at least say, of any two observations, which is the greater), # Under the null hypothesis H0, the probability of an observation from the population ''X'' exceeding an observation from the second population ''Y'' equals the probability of an observation from ''Y'' exceeding an observation from ''X'' : P(''X'' > ''Y'') = P(''Y'' > ''X'') or P(''X'' > ''Y'') + 0.5·P(''X'' = ''Y'') = 0.5. A stronger null hypothesis commonly used is "The distributions of both populations are equal" which implies the previous hypothesis. # The alternative hypothesis H1 is "the probability of an observation from the population ''X'' exceeding an observation from the second population ''Y'' is different from the probability of an observation from ''Y'' exceeding an observation from ''X'' : P(''X'' > ''Y'') ≠ P(''Y'' > ''X'')." The alternative may also be stated in terms of a one-sided test, for example: P(''X'' > ''Y'') > P(''Y'' > ''X'') Under more strict assumptions than those 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 MWW test as showing a difference in medians. Under this location shift assumption, we can also interpret the MWW 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Mann–Whitney U test」の詳細全文を読む スポンサード リンク
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