MannWhitneyTest

MannWhitneyTest[{data1,data2}]

tests whether the medians of data1 and data2 are equal.

MannWhitneyTest[dspec,μ0]

tests the median difference against μ0.

MannWhitneyTest[dspec,μ0,"property"]

returns the value of "property".

Details and Options

  • MannWhitneyTest performs a hypothesis test on data1 and data2 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 {x1,x2,} or multivariate {{x1,y1,},{x2,y2,},}.
  • The argument μ0 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,μ0,"HypothesisTestData"] returns a HypothesisTestData object htd that can be used to extract additional test results and properties using the form htd["property"].
  • MannWhitneyTest[dspec,μ0,"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
  • For univariate samples MannWhitneyTest performs the MannWhitney -test for median differences of independent samples. A correction for ties is applied for both asymptotic and permutation-based -values. By default, the test statistic is corrected for continuity and is assumed to follow a NormalDistribution.
  • For multivariate samples, MannWhitneyTest performs an extension of the MannWhitney -test using spatial ranks. The test statistic is assumed to follow a ChiSquareDistribution[dim] where dim is the dimension of dspec.
  • The following options can be used:
  • AlternativeHypothesis"Unequal"the inequality for the alternative hypothesis
    MaxIterationsAutomaticmax iterations for multivariate median tests
    MethodAutomaticthe method to use for computing -values
    SignificanceLevel0.05cutoff for diagnostics and reporting
  • 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

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Basic Examples  (2)

Test whether the medians of two independent populations differ:

The median difference :

At the 0.05 level, the medians are significantly different:

Compare the locations of multivariate populations:

The median difference vector :

At the 0.05 level, is not significantly different from {1,2}:

Scope  (9)

Testing  (6)

Test versus :

The -values are generally small when the locations are not equal:

The -values are generally large when the locations are equal:

Test versus :

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:

Extract any number of properties simultaneously:

The -value and test statistic from a MannWhitney test:

Reporting  (3)

Tabulate the test results:

Retrieve the entries from a test table for customized reporting:

Tabulate -values or test statistics:

The -value from the table:

The test statistic from the table:

Options  (10)

AlternativeHypothesis  (3)

A two-sided test is performed by default:

Test versus :

Perform a two-sided test or a one-sided alternative:

Test versus :

Test versus :

Test versus :

Perform tests with one-sided alternatives when μ0 is given:

Test versus :

Test versus :

MaxIterations  (2)

Set the maximum number of iterations to use for multivariate tests:

By default, 250 iterations are allowed:

Setting the maximum number of iterations may result in lack of convergence:

The -values are not equivalent:

Method  (4)

By default, -values are computed using asymptotic test statistic distributions:

Permutation methods can be used:

Set the number of permutations to use:

By default, random permutations are used:

Set the seed used for generating random permutations:

SignificanceLevel  (1)

The significance level is used for "TestConclusion" and "ShortTestConclusion":

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:

Compare the morphological measures across the genders:

Properties & Relations  (5)

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 MannWhitney test ignores the time stamps when the input is a TimeSeries:

The MannWhitney test recognizes the path structure of a TemporalData with exactly two paths:

Use the values directly:

Neat Examples  (1)

Compute the statistic when the null hypothesis is true:

The test statistic given a particular alternative:

Compare the distributions of the test statistics:

Introduced in 2010
 (8.0)