MannWhitneyTest
MannWhitneyTest[{data_{1},data_{2}}]
tests whether the medians of data_{1} and data_{2} are equal.
MannWhitneyTest[dspec,μ_{0}]
tests the median difference against μ_{0}.
MannWhitneyTest[dspec,μ_{0},"property"]
returns the value of "property".
Details and Options
 MannWhitneyTest performs a hypothesis test on data_{1} and data_{2} with null hypothesis that the true median difference against that .
 By default, a probability value or value is returned.
 A small value suggests that it is unlikely that is true.
 The data in dspec can be univariate {x_{1},x_{2},…} or multivariate {{x_{1},y_{1},…},{x_{2},y_{2},…},…}.
 The argument μ_{0} can be a real number or a real vector with length equal to the dimension of the data.
 MannWhitneyTest assumes that the data is elliptically symmetric about a common spatial median in the multivariate case.
 MannWhitneyTest[dspec,μ_{0},"HypothesisTestData"] returns a HypothesisTestData object htd that can be used to extract additional test results and properties using the form htd["property"].
 MannWhitneyTest[dspec,μ_{0},"property"] can be used to directly give the value of "property".
 Properties related to the reporting of test results include:

"DegreesOfFreedom" the degrees of freedom used in a test "PValue" list of values "PValueTable" formatted table of values "ShortTestConclusion" a short description of the conclusion of a test "TestConclusion" a description of the conclusion of a test "TestData" list of pairs of test statistics and values "TestDataTable" formatted table of values and test statistics "TestStatistic" list of test statistics "TestStatisticTable" formatted table of test statistics  The following options can be used:

AlternativeHypothesis "Unequal" the inequality for the alternative hypothesis MaxIterations Automatic max iterations for multivariate median tests Method Automatic the method to use for computing values SignificanceLevel 0.05 cutoff for diagnostics and reporting  For univariate samples, MannWhitneyTest performs the Mann–Whitney test for median differences of independent samples. The following methods are available:

Automatic chooses depending on the size of the sample "Asymptotic" large sample statistic is asymptotically normally distributed "Exact" small sample values can be computed exactly "Permutation" simulation based  For the "Asymptotic" and "Permutation" methods, a correction for ties is applied; the test statistic is corrected for continuity, and it is assumed to follow a NormalDistribution.
 For multivariate samples, MannWhitneyTest performs an extension of the Mann–Whitney test using spatial ranks. The test statistic is assumed to asymptotically follow a ChiSquareDistribution[dim] where dim is the dimension of dspec.
 For the MannWhitneyTest, a cutoff is chosen such that is rejected only if . The value of used for the "TestConclusion" and "ShortTestConclusion" properties is controlled by the SignificanceLevel option. By default, is set to 0.05.
Examples
open allclose allBasic Examples (2)
Scope (9)
Testing (6)
The values are generally small when the locations are not equal:
The values are generally large when the locations are equal:
The order of the datasets affects the test results:
Test whether the median difference vector of two multivariate populations is the zero vector:
Alternatively, test against {1,0,1,0}:
Create a HypothesisTestData object for repeated property extraction:
The properties available for extraction:
Extract some properties from a HypothesisTestData object:
The value and test statistic:
Generalizations & Extensions (2)
The Mann–Whitney test ignores the time stamps when the input is a TimeSeries:
The Mann–Whitney test recognizes the path structure of a TemporalData with exactly two paths:
Options (12)
AlternativeHypothesis (4)
A twosided test is performed by default:
Perform a twosided test or a onesided alternative:
Perform tests with onesided alternatives when μ_{0} is given:
The twosided test is symmetric under the exchange of the datasets:
The onesided tests are not symmetric under the exchange of the datasets:
However, for the permutation method even the (default) twosided test is not symmetric:
MaxIterations (2)
Applications (3)
Test whether the medians of some populations are equal:
The medians of the first two populations are similar:
The median of the third population is different from the first:
It has been observed that the duration of Old Faithful geyser eruptions is proportional to the time elapsed since the previous eruption:
Assuming one hour is a long wait for an eruption, test the statement that long waits lead to long eruption durations:
Two hundred Australian crabs were collected, and five morphological measures were taken for each crab. The data is organized by type and gender:
Determine if there is a difference in the first four morphological measures for the two varieties:
Properties & Relations (4)
For univariate data, the test statistic follows a NormalDistribution[0,1] under :
A large sample approximation to the NormalDistribution:
For multivariate data, the test statistic follows a ChiSquareDistribution[dim] under :
The test statistic is computed by pooling and ranking the data:
In the absence of ties, Ordering can compute the ranks:
The exact value matches the frequencies of the test statistic when considering every possible permutation for a given size of the datasets:
Compute the statistics and values for all the permutation pairs:
Compute frequencies by normalizing the counts of the statistic:
The statistic is an integer between and the product of the lengths of the data inputs—here :
Since the value is the cumulative mass function (CDF) of the test distribution, it needs to be compared to the cumulative frequencies:
By convention, the values start at :
The exact value is computed using the following recursion relation:
Possible Issues (1)
Text
Wolfram Research (2010), MannWhitneyTest, Wolfram Language function, https://reference.wolfram.com/language/ref/MannWhitneyTest.html.
CMS
Wolfram Language. 2010. "MannWhitneyTest." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/MannWhitneyTest.html.
APA
Wolfram Language. (2010). MannWhitneyTest. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MannWhitneyTest.html