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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 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.
A very general formulation is to assume that:
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. F1(x) = F2(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.
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 ranks in one of the samples, rather than U itself.
The U test is included in most modern statistical packages. It is also easily calculated by hand, especially for small samples. There are two ways of doing this.
First, arrange all the observations into a single ranked series. That is, rank all the observations without regard to which sample they are in.
For small samples a direct method is recommended. It is very quick, and gives an insight into the meaning of the U statistic.
For larger samples, a formula can be used:
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.
Suppose that Aesop is dissatisfied with his 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:
What is the value of U?
A second example race, with 19 participants of each species, in which the outcomes are as follows:
The median tortoise here comes in at position 19, and thus actually beats the median hare, which comes in at position 20.
However, the value of U (for hares) is 100. (9 Hares beaten by (x) 0 tortoises) + (10 hares beaten by (x) 10 tortoises) = 0 + 100 = 100
Value of U(for tortoises) is 261. (10 tortoises beaten by 9 hares) + (9 tortoises beaten by 19 hares) = 90 + 171 = 261
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 variances; they are mirror images of each other, so their variances are the same, but they have very different means.
where mU and σU 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. mU and σU are given by
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. 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.
Note that since U1 + U2 = n1 n2, the mean n1 n2/2 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.
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.
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:
Where c is the number of classes, and the term of 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. will always be zero but, unlike in the two-class case, generally , which is why the measure sums over all (k, l) pairs, in effect using the average of and .
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.
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. In that situation, the unequal variances version of the t test is likely to give more reliable results, but only if normality holds.
Alternatively, some authors (e.g. Conover[full citation needed]) 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 statistic appeared in a 1914 article Gustav Deuchler (with a missing term in the variance).
As a one-sample statistic, the signed rank was proposed by Frank Wilcoxon in 1945, 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).
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 Donald Ransom Whitney in 1947. This article discussed alternative hypotheses, including a stochastic ordering (where the cumulative distribution functions satisfied the pointwise inequality ). 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.
The U test is related to a number of other non-parametric statistical procedures. For example, it is equivalent to Kendall's τ correlation coefficient if one of the variables is binary (that is, it can only take two values).
A statistic called ρ that is linearly related to U and widely used in studies of categorization (discrimination learning involving concepts), and elsewhere, is calculated by dividing U by its maximum value for the given sample sizes, which is simply n1 × n2. 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..
In reporting the results of a Mann–Whitney test, it is important to state:
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,
A statement that does full justice to the statistical status of the test might run,
However it would be rare to find so extended a report in a document whose major topic was not statistical inference.
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:
wilcox.testand in its COIN package.
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