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| In [[statistics]], the '''Mann–Whitney ''U''''' test (also called the '''Mann–Whitney–Wilcoxon''' ('''MWW'''), '''Wilcoxon rank-sum test''', or '''Wilcoxon–Mann–Whitney test''') is a [[non-parametric statistics|non-parametric]] [[statistical hypothesis test|test]] of the [[null hypothesis]] that two populations are the same against an [[alternative hypothesis]], especially that a particular population tends to have larger values than the other.
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| It has greater [[asymptotic relative efficiency|efficiency]] than the [[t-test]] on non-normal distributions, such as a [[mixture distribution|mixture]] of [[normal distribution]]s, and it is nearly as efficient as the t-test on normal distributions.
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| ==Assumptions and formal statement of hypotheses==
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| Although [[Henry Mann|Mann]] and [[Donald Ransom Whitney|Whitney]]<ref name="mannwhitney1947" /> developed the MWW test under the assumption of [[Continuous probability distribution|continuous]] responses with the [[alternative hypothesis]] being that one distribution is [[stochastic dominance|stochastically greater]] than the other, there are many other ways to formulate the [[null hypothesis|null]] and alternative hypotheses such that the MWW test will give a valid test.<ref>{{cite journal |last=Fay |first=Michael P. |last2=Proschan |first2=Michael 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>
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| A very general formulation is to assume that:
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| # All the observations from both groups are [[statistical independence|independent]] of each other,
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| # The responses are [[ordinal measurement|ordinal]] (i.e. one can at least say, of any two observations, which is the greater),
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| # The distributions of both groups are equal under the null hypothesis, so that the probability of an observation from one population (''X'') exceeding an observation from the second population (''Y'') equals the probability of an observation from ''Y'' exceeding an observation from ''X''. That is, there is a symmetry between populations with respect to probability of random drawing of a larger observation.
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| # Under the alternative hypothesis, the probability of an observation from one population (''X'') exceeding an observation from the second population (''Y'') (after exclusion of ties) is not equal to 0.5. The alternative may also be stated in terms of a one-sided test, for example: P(''X'' > ''Y'') + 0.5 P(''X'' = ''Y'') > 0.5.
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| 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''<sub>1</sub>(''x'') = ''F''<sub>2</sub>(''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.
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| ==Calculations==
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| The test involves the calculation of a [[statistic]], usually called ''U'', whose distribution under the [[null hypothesis]] is known. In the case of small samples, the distribution is tabulated, but for sample sizes above ~20 approximation using the [[normal distribution]] is fairly good. Some books tabulate statistics equivalent to ''U'', such as the sum of [[Rank (set theory)|rank]]s in one of the samples, rather than ''U'' itself.
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| The ''U'' test is included in most modern [[List of statistical packages|statistical packages]]. It is also easily calculated by hand, especially for small samples. There are two ways of doing this.
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| First, arrange all the observations into a single ranked series. That is, rank all the observations without regard to which sample they are in.
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| '''Method one:'''
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| For small samples a direct method is recommended. It is very quick, and gives an insight into the meaning of the ''U'' statistic.
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| # Choose the sample for which the ranks seem to be smaller (The only reason to do this is to make computation easier). Call this "sample 1," and call the other sample "sample 2."
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| # For each observation in sample 1, count the number of observations in sample 2 that have a smaller rank (count a half for any that are equal to it). The sum of these counts is ''U''.
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| '''Method two:'''
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| For larger samples, a formula can be used:
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| # Add up the ranks for the observations which came from sample 1. Where there are tied groups, take the rank to be equal to the midpoint of the group. The sum of ranks in sample 2 is now determinate, since the sum of all the ranks equals ''N''(''N'' + 1)/2 where ''N'' is the total number of observations.
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| # ''U'' is then given by:
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| :::<math>U_1=R_1 - {n_1(n_1+1) \over 2} \,\!</math>
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| ::where ''n''<sub>1</sub> is the sample size for sample 1, and ''R''<sub>1</sub> is the sum of the ranks in sample 1.
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| ::Note that it doesn't matter which of the two samples is considered sample 1. An equally valid formula for ''U'' is
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| :::<math>U_2=R_2 - {n_2(n_2+1) \over 2}. \,\!</math>
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| ::The smaller value of ''U''<sub>1</sub> and ''U''<sub>2</sub> is the one used when consulting significance tables. The sum of the two values is given by
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| :::<math>U_1 + U_2 = R_1 - {n_1(n_1+1) \over 2} + R_2 - {n_2(n_2+1) \over 2}. \,\!</math>
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| :: Knowing that ''R''<sub>1</sub> + ''R''<sub>2</sub> = ''N''(''N'' + 1)/2 and ''N'' = ''n''<sub>1</sub> + ''n''<sub>2</sub> , and doing some algebra, we find that the sum is
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| :::<math>U_1 + U_2 = n_1n_2. \,\!</math>
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| ==Properties==
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| The maximum value of ''U'' is the product of the sample sizes for the two samples. In such a case, the "other" ''U'' would be 0.
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| ==Examples==
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| ===Illustration of calculation methods===
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| Suppose that [[Aesop]] is dissatisfied with his [[The Tortoise and the Hare|classic experiment]] in which one [[tortoise]] was found to beat one [[hare]] in a race, and decides to carry out a significance test to discover whether the results could be extended to tortoises and hares in general. He collects a sample of 6 tortoises and 6 hares, and makes them all run his race at once. The reversed order in which they reach the finishing post (their reversed rank order, from last to first crossing the finish line) is as follows, writing T for a tortoise and H for a hare:
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| :T H H H H H T T T T T H
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| What is the value of ''U''?
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| * Using the direct method, we take each tortoise in turn, and count the number of hares it is beaten by, getting 1, 1, 1, 1, 1, 6, which means ''U'' = 11. Alternatively, we could take each hare in turn, and count the number of tortoises it is beaten by. In this case, we get 0, 5, 5, 5, 5, 5. So ''U'' = 0 + 5 + 5 + 5 + 5 + 5 = 25. Note that the sum of these two values for ''U'' is 36, which is 6 × 6.
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| * Using the indirect method:
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| : the sum of the ranks achieved by the tortoises is 1 + 7 + 8 + 9 + 10 + 11 = 46.
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| :: Therefore ''U'' = 46 − (6×7)/2 = 46 − 21 = 25.
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| :: the sum of the ranks achieved by the hares is 2 + 3 + 4 + 5 + 6 + 12 = 32, leading to ''U'' = 32 − 21 = 11.
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| ===Illustration of object of test===
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| A second example race, with 19 participants of each species, in which the outcomes are as follows:
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| :H H H H H H H H H T T T T T T T T T T H H H H H H H H H H T T T T T T T T T
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| The median tortoise here comes in at position 19, and thus actually beats the median hare, which comes in at position 20.
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| However, the value of ''U'' (for hares) is 100.
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| (9 Hares beaten by (x) 0 tortoises) + (10 hares beaten by (x) 10 tortoises) = 0 + 100 = 100
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| Value of ''U''(for tortoises) is 261.
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| (10 tortoises beaten by 9 hares) + (9 tortoises beaten by 19 hares) = 90 + 171 = 261
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| Consulting tables, or using the approximation below, shows that this ''U'' value gives significant evidence that hares tend to do better than tortoises (''p'' < 0.05, two-tailed). Obviously this is an extreme distribution that would be spotted easily, but in a larger sample something similar could happen without it being so apparent. Notice that the problem here is not that the two distributions of ranks have different [[variance]]s; they are mirror images of each other, so their variances are the same, but they have very different means.
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| ==Normal approximation==
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| For large samples, ''U'' is approximately [[normal distribution|normally distributed]]. In that case, the [[Standard score|standardized value]]
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| :<math>z = \frac{ U - m_U }{ \sigma_U }, \, </math>
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| where ''m''<sub>''U''</sub> and ''σ''<sub>''U''</sub> are the mean and standard deviation of ''U'', is approximately a standard normal deviate whose significance can be checked in tables of the normal distribution. ''m''<sub>''U''</sub> and σ<sub>''U''</sub> are given by
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| :<math>m_U = \frac{n_1 n_2}{2}. \, </math>
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| :<math>\sigma_U=\sqrt{n_1 n_2 (n_1 + n_2+1) \over 12}. \, </math>
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| The formula for the standard deviation is more complicated in the presence of tied ranks; the full formula is given in the text books referenced below{{citation needed|date=December 2012}}. However, if the number of ties is small (and especially if there are no large tie bands) ties can be ignored when doing calculations by hand. The computer statistical packages will use the correctly adjusted formula as a matter of routine.
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| Note that since ''U''<sub>1</sub> + ''U''<sub>2</sub> = ''n''<sub>1</sub> ''n''<sub>2</sub>, the mean ''n''<sub>1</sub> ''n''<sub>2</sub>/2 used in the normal approximation is the mean of the two values of ''U''. Therefore, the absolute value of the ''z'' statistic calculated will be same whichever value of ''U'' is used.
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| ==Relation to other tests==
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| ===Comparison to Student's t-test===
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| The ''U'' test is more widely applicable than [[Student's_t-test#Independent_two-sample_t-test|independent samples]] [[Student's t-test|Student's ''t''-test]], and the question arises of which should be preferred.
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| ;Ordinal data: ''U'' remains the logical choice when the data are [[Level of measurement#Ordinal scale|ordinal]] but not interval scaled, so that the spacing between adjacent values cannot be assumed to be constant.
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| ;Robustness: As it compares the sums of ranks,<ref name="Motulsky 2007">Motulsky, Harvey J.; ''Statistics Guide'', San Diego, CA: GraphPad Software, 2007, p. 123</ref> the Mann–Whitney test is less likely than the ''t''-test to spuriously indicate significance because of the presence of [[outlier]]s – i.e. Mann–Whitney is more [[Robust statistics|robust]].{{Clarify|date=September 2009}}{{Citation needed|date=September 2009}}
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| ;Efficiency: When normality holds, ''MWW'' has an (asymptotic) [[Efficiency (statistics)|efficiency]] of <math>3/\pi</math> or about 0.95 when compared to the ''t'' test.<ref name="Lehmann 1999">Lehamnn, Erich L.; ''Elements of Large Sample Theory'', Springer, 1999, p. 176</ref> For distributions sufficiently far from normal and for sufficiently large sample sizes, the ''MWW'' is considerably more efficient than the ''t''.<ref name="Conover 1980">Conover, William J.; [http://kecubung.webfactional.com/ebook/practical-nonparametric-statistics-conover-download-pdf.pdf ''Practical Nonparametric Statistics''], John Wiley & Sons, 1980 (2nd Edition), pp. 225–226</ref>
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| Overall, the robustness makes the ''MWW'' more widely applicable than the ''t'' test, and for large samples from the normal distribution, the efficiency loss compared to the ''t'' test is only 5%, so one can recommend ''MWW'' as the default test for comparing interval or ordinal measurements ''with similar distributions''.{{citation needed|date=February 2012}}
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| The relation between [[efficiency (statistics)|efficiency]] and [[statistical power|power]] in concrete situations isn't trivial though. For small sample sizes one should investigate the power of the ''MWW'' vs ''t''.
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| MWW 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 J. |last1=Conover |first2=Ronald L. |last2=Iman |authorlink2=Ronald L. Iman |journal=[[The American Statistician]] |volume=35 |issue=3 |year=1981 |pages=124–129 |jstor=2683975 }}</ref>
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| ===Area-under-curve (AUC) statistic for ROC curves===
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| The U statistic is equivalent to the '''area under the [[receiver operating characteristic]] curve''' that can be readily calculated.<ref name="Hanley">{{cite journal
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| |last1=Hanley |first1=James A. |last2=McNeil |first2=Barbara J.
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| |journal=Radiology |number=1 |pages=29–36
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| |title=The Meaning and Use of the Area under a Receiver Operating (ROC) Curve Characteristic
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| |volume=143
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| |year=1982
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| |pmid=7063747
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| }}</ref><ref name="Mason">{{cite journal |last1=Mason |first1=Simon J. |last2=Graham |first2=Nicholas E.|year=2002 |title=Areas beneath the relative operating characteristics (ROC) and relative operating levels (ROL) curves: Statistical significance and interpretation |journal=Quarterly Journal of the Royal Meteorological Society |issue=128 |pages=2145–2166 |url=http://www.inmet.gov.br/documentos/cursoI_INMET_IRI/Climate_Information_Course/References/Mason+Graham_2002.pdf }}</ref>
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| ::<math>AUC_1 = {U_1 \over n_1n_2}</math>
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| Because of its probabilistic form, the U statistic can be generalised to a measure of a classifier's separation power for more than two classes:<ref>{{cite journal |title=A Simple Generalisation of the Area Under the ROC Curve for Multiple Class Classification Problems |first1=David J. |last1=Hand |first2=Robert J. |last2=Till |journal=Machine Learning |volume=45 |number=2 |pages=171–186 |year=2001 |doi=10.1023/A:1010920819831 |url=http://www.springerlink.com/index/nn141j42838n7u21.pdf |type=pdf }}</ref>
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| ::<math>M = {1 \over c(c-1)} \sum AUC_{k,l}</math>
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| Where ''c'' is the number of classes, and the <math>R_{k,l}</math> term of <math>AUC_{k,l}</math> considers only the ranking of the items belonging to classes ''k'' and ''l'' (i.e., items belonging to all other classes are ignored) according to the classifier's estimates of the probability of those items belonging to class ''k''. <math>AUC_{k,k}</math> will always be zero but, unlike in the two-class case, generally <math>AUC_{k,l} \ne AUC_{l,k}</math>, which is why the <math>M</math> measure sums over all (''k'', ''l'') pairs, in effect using the average of <math>AUC_{k,l}</math> and <math>AUC_{l,k}</math>.
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| ===Different distributions===
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| If one is only interested in stochastic ordering of the two populations (i.e., the concordance probability P(''Y'' > ''X'')), the U test can be used even if the shapes of the distributions are different. The concordance probability is exactly equal to the area under the [[receiver operating characteristic]] curve (ROC) that is often used in the context.{{Citation needed|date=November 2009}}
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| ====Alternatives====
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| If one desires a simple shift interpretation, the ''U'' test should ''not'' be used when the distributions of the two samples are very different, as it can give erroneously significant results.{{citation needed|date=February 2012}} In that situation, the [[Student's_t-test#Unequal_sample_sizes.2C_unequal_variance|unequal variances]] version of the ''t'' test is likely to give more reliable results, but only ''if normality holds.''{{citation needed|date=February 2012}}
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| Alternatively, some authors (e.g. Conover{{full|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.
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| The [[Brown–Forsythe test]] has been suggested as an appropriate non-parametric equivalent to the [[F test]] for equal variances.{{citation needed|date=February 2012}}
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| ==History==
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| The statistic appeared in a 1914 article <ref name="Kruskal57">{{cite journal |url=http://www.jstor.org/stable/2280906 |title=Historical Notes on the Wilcoxon Unpaired Two-Sample Test |last=Kruskal |first=William H. |journal=Journal of the American Statistical Association |date=September 1957 |volume=52 |issue=279 |pages=356–360 }}</ref> by the German [[Gustav Deuchler]] (with a missing term in the variance).
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| As a one-sample statistic, the signed rank was proposed by [[Frank Wilcoxon]] in 1945,<ref name="wilcoxon1945">{{cite journal |doi=10.2307/3001968 |last=Wilcoxon |first=Frank |authorlink=Frank Wilcoxon |year=1945 |title=Individual comparisons by ranking methods |journal=[[Biometrics Bulletin]] |volume=1 |issue=6 |pages=80–83 |jstor=3001968 }}</ref> with some discussion of a two-sample variant for equal sample sizes, in a [[test of significance]] with a point null-hypothesis against its complementary alternative (that is, equal versus not equal).
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| A thorough analysis of the statistic, which included a recurrence allowing the computation of tail probabilities for arbitrary sample sizes and tables for sample sizes of eight or less appeared in the article by [[Henry Mann]] and his student<!-- source: Olson, cited with url link in Mann article --> Donald Ransom Whitney in 1947.<ref name="mannwhitney1947">{{cite journal |first1=Henry B. |last1=Mann |authorlink=Henry Mann |first2=Donald R. |last2=Whitney |title=On a Test of Whether one of Two Random Variables is Stochastically Larger than the Other |journal=[[Annals of Mathematical Statistics]] |volume=18 |issue=1 |year=1947 |pages=50–60 |doi=10.1214/aoms/1177730491 |mr=22058 |zbl=0041.26103 }}</ref> This article discussed alternative hypotheses, including a [[stochastic ordering]] (where the [[cumulative distribution function]]s satisfied the pointwise inequality <math> F_X(t) < F_Y(t) </math>). This paper also computed the first four moments and established the limiting normality of the statistic under the null hypothesis, so establishing that it is asymptotically distribution-free.
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| ==Related test statistics==
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| ===Kendall's τ===
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| The ''U'' test is related to a number of other non-parametric statistical procedures. For example, it is equivalent to [[Kendall tau rank correlation coefficient|Kendall's τ]] correlation coefficient if one of the variables is binary (that is, it can only take two values).{{citation needed|date=February 2012}}
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| ===''ρ'' statistic===
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| A statistic called ''ρ'' that is linearly related to ''U'' and widely used in studies of categorization ([[discrimination learning]] involving [[concept]]s){{citation needed|date=February 2012}}, and elsewhere,<ref name="H1976" /> is calculated by dividing ''U'' by its maximum value for the given sample sizes, which is simply ''n''<sub>1</sub> × ''n''<sub>2</sub>. ''ρ'' is thus a non-parametric measure of the overlap between two distributions; it can take values between 0 and 1, and it is an estimate of P(''Y'' > ''X'') + 0.5 P(''Y'' = ''X''), where ''X'' and ''Y'' are randomly chosen observations from the two distributions. Both extreme values represent complete separation of the distributions, while a ρ of 0.5 represents complete overlap. The usefulness of the ρ statistic can be seen in the case of the odd example used above, where two distributions that were significantly different on a ''U''-test nonetheless had nearly identical medians: the ρ value in this case is approximately 0.723 in favour of the hares, correctly reflecting the fact that even though the median tortoise beat the median hare, the hares collectively did better than the tortoises collectively.{{citation needed|date=February 2012}}.
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| ==Example statement of results==
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| In reporting the results of a Mann–Whitney test, it is important to state:
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| *A measure of the central tendencies of the two groups (means or medians; since the Mann–Whitney is an ordinal test, medians are usually recommended)
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| *The value of ''U''
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| *The sample sizes
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| *The significance level.
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| In practice some of this information may already have been supplied and common sense should be used in deciding whether to repeat it. A typical report might run,
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| :"Median latencies in groups E and C were 153 and 247 ms; the distributions in the two groups differed significantly (Mann–Whitney ''U'' = 10.5, ''n''<sub>1</sub> = ''n''<sub>2</sub> = 8, ''P'' < 0.05 two-tailed)."
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| A statement that does full justice to the statistical status of the test might run,
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| :"Outcomes of the two treatments were compared using the Wilcoxon–Mann–Whitney two-sample rank-sum test. The treatment effect (difference between treatments) was quantified using the Hodges–Lehmann (HL) estimator, which is consistent with the Wilcoxon test.<ref>{{cite book |title= Nonparametric Statistical Methods |authors= Myles Hollander and Douglas A. Wolfe |publisher= Wiley-Interscience |edition=2 |year=1999 |ISBN= 978-0471190455}}</ref> This estimator (HLΔ) is the median of all possible differences in outcomes between a subject in group B and a subject in group A. A non-parametric 0.95 confidence interval for HLΔ accompanies these estimates as does ρ, an estimate of the probability that a randomly chosen subject from population B has a higher weight than a randomly chosen subject from population A. The median [quartiles] weight for subjects on treatment A and B respectively are 147 [121, 177] and 151 [130, 180] kg. Treatment A decreased weight by HLΔ = 5 kg (0.95 CL [2, 9] kg, 2''P'' = 0.02, ρ = 0.58)."
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| However it would be rare to find so extended a report in a document whose major topic was not statistical inference.
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| == Implementations ==
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| In many software packages, the Mann–Whitney test (of the hypothesis of equal distributions against appropriate alternatives) has been poorly documented. Some packages incorrectly treat ties or fail to document asymptotic techniques (e.g., correction for continuity). A 2000 review discussed versions of the following packages:<ref>{{cite journal |title=Different Outcomes of the Wilcoxon-Mann-Whitney Test from Different Statistics Packages |first=Reinhard |last=Bergmann |first2=John |last2=Ludbrook |first2=Will P. J. M. |last3=Spooren |journal=The American Statistician |volume=54 |issue=1 |year=2000 |pages=72–77 |jstor=2685616 }}</ref>
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| * [[Matlab]] has [http://www.mathworks.co.uk/help/stats/ranksum.html ranksum] within the its Statistics Toolbox.
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| * [[R (programming language)|R]]'s statistics base-package implements the test [http://stat.ethz.ch/R-manual/R-patched/library/stats/html/wilcox.test.html <code>wilcox.test</code>] and in its COIN package.
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| * [[SAS (software)|SAS]] implements the test in its PROC NPAR1WAY procedure.
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| * [[Stata]] implements the test in its [http://www.stata.com/help.cgi?ranksum ranksum] command.
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| * [[Python (programming language)]] has an implementation of this test provided by [[SciPy]]: [http://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.mannwhitneyu.html]
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| ==See also== | |
| * [[Kolmogorov–Smirnov test]]
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| * [[Wilcoxon signed-rank test]]
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| * [[Kruskal–Wallis one-way analysis of variance]]
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| ==Notes==
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| {{Reflist|refs=
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| * <ref name="H1976">{{cite journal |doi=10.1037/0097-7403.2.4.285 |last1=Herrnstein |first1=Richard J. |last2=Loveland |first2=Donald H. |last3=Cable |first3=Cynthia |year=1976 |title=Natural Concepts in Pigeons |journal=Journal of Experimental Psychology: Animal Behavior Processes |volume=2 |pages=285–302 }}</ref>
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| }}
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| ==References==
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| * Lehmann, Erich L. (1975); ''Nonparametrics: Statistical Methods Based on Ranks''.
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| ==Further reading==
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| * {{cite book|last1=Hettmansperger|first1=T. P.|last2=McKean|first2=J. W.|title=Robust nonparametric statistical methods| edition=First ed., rather than Taylor and Francis (2010) second|series=Kendall's Library of Statistics|volume=5|publisher=Edward Arnold|location=London|publisher2=John Wiley and Sons, Inc.|location2=New York|year=1998|pages=xiv+467|isbn=0-340-54937-8|mr=1604954|ref=harv}}
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| * {{cite journal|last1=Hodges|first1=J. L.|last2=Lehmann| first2=E. L.|authorlink2=Erich Leo Lehmann| year=1963| title=Estimation of location based on ranks|journal=[[Annals of Mathematical Statistics]]|volume=34|pages=598–611|ref=harv|url=http://projecteuclid.org/euclid.aoms/1177704172|mr=152070|doi=10.1214/aoms/1177704172|zbl=0203.21105|jstor=2238406|id={{Euclid|euclid.aoms/1177704172}}||issue=2}}
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| * {{cite book| last=Lehmann|first=Erich L.|authorlink=Erich Leo Lehmann|title=Nonparametrics: Statistical methods based on ranks|edition=Reprinting of 1988 revision of 1975 Holden-Day | publisher=Springer | location=New York|year=2006|pages=xvi+463|isbn=978-0-387-35212-1|mr=395032|ref=harv|others=With the special assistance of H. J. M. D'Abrera}}
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| *{{cite book|last=Oja|first=Hannu
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| |title=Multivariate nonparametric methods with ''R'': An approach based on spatial signs and ranks|series=Lecture Notes in Statistics|volume=199|publisher=Springer|location=New York|year=2010|pages=xiv+232|isbn=978-1-4419-0467-6|doi=10.1007/978-1-4419-0468-3|mr=2598854|ref=harv}}
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| * {{cite journal|doi=10.2307/2527532|last=Sen|first=Pranab Kumar|authorlink=Pranab K. Sen|year=1963|title=On the estimation of relative potency in dilution(-direct) assays by distribution-free methods|journal=Biometrics|volume=19|pages=532–552|issue=4|month=December|jstor=2527532|zbl=0119.15604<!-- save for links to future articles -->|ref=harv}}
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| ==External links==
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| * Table of critical values of ''U'' [http://math.usask.ca/~laverty/S245/Tables/wmw.pdf (pdf)]
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| * [http://faculty.vassar.edu/lowry/utest.html Interactive calculator] for ''U'' and its significance
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| {{statistics|inference}}
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| {{DEFAULTSORT:Mann-Whitney U}}
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| [[Category:Statistical tests]]
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| [[Category:Non-parametric statistics]]
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| [[Category:U-statistics]]
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