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SpearmanRankTest

SpearmanRankTest[v₁,v₂]

tests whether the vectors v₁ and v₂ are independent.

SpearmanRankTest[m₁,m₂]

tests whether the matrices m₁ and m₂ are independent.

SpearmanRankTest[…,"property"]

returns the value of "property".

Details and Options

SpearmanRankTest performs a hypothesis test on v₁ and v₂ with null hypothesis that the vectors are independent, and alternative hypothesis that they are not.
By default a probability value or -value is returned.
A small -value suggests that it is unlikely that is true.
The arguments v₁ and v₂ can be any real-valued vectors or matrices of equal length.
SpearmanRankTest is based on Spearman's rank correlation computed by SpearmanRho[v₁,v₂].
For testing matrices the test statistic is based on inner standardized spatial ranks and asymptotically follows a ChiSquareDistribution[r*s] where r and s are the dimension of m₁ and m₂, respectively. The test is invariant under affine transformations.
SpearmanRankTest[v₁,v₂,"HypothesisTestData"] returns a HypothesisTestData object htd that can be used to extract additional test results and properties using the form htd["property"].
SpearmanRankTest[v₁,v₂,"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 the test
	"PValue"	the -value of the test
	"PValueTable"	formatted table containing the -value
	"ShortTestConclusion"	a short description of the conclusion of the test
	"TestConclusion"	a description of the conclusion of the test
	"TestData"	a list containing the test statistic and -value
	"TestDataTable"	formatted table of the -value and test statistic
	"TestStatistic"	the test statistic
	"TestStatisticTable"	formatted table containing the test statistic

The following options can be used:

	AlternativeHypothesis	"Unequal"	the inequality for the alternative hypothesis
	MaxIterations	Automatic	max iterations for multivariate test
	Method	Automatic	the method to use for computing -values
	SignificanceLevel	0.05	cutoff for diagnostics and reporting

For tests of independence, 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 all

Basic Examples (2)

Test whether two vectors are independent:

Test whether two matrices are independent:

At the 0.05 level there is insufficient evidence to reject independence:

Scope (8)

Testing (5)

Test whether two vectors are independent:

The -values are typically large when the vectors are independent:

The -values are typically small when there are dependencies:

Test whether two matrices are independent:

The -values are typically small for dependent matrices:

The -values are typically large when matrices are independent:

Create a HypothesisTestData object for repeated property extraction:

The properties available for extraction:

Extract some properties from the HypothesisTestData object:

The -value and test statistic from the test:

Extract any number of properties simultaneously:

The -value and test statistic from the test:

Reporting (3)

Tabulate the results from the test:

A table of the test results:

Retrieve the entries from a test table for customized reporting:

Tabulate the -value or test statistic:

The -value from the table:

The test statistic from the table:

Options (9)

AlternativeHypothesis (3)

A two-sided test is performed by default:

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

A two-sided test:

The two one-sided alternatives:

The multivariate test is inherently two-sided:

This is due to the shape of the null distribution:

MaxIterations (1)

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

By default is used:

Lowering the setting can shorten compute times but may result in failed convergence:

Method (4)

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

The -value can be obtained using permutation methods:

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":

Properties & Relations (9)

For vector-to-vector comparisons the test statistic is computed as SpearmanRho:

The test statistic follows a StudentTDistribution[n-2] under :

In higher dimensions the test statistic follows a ChiSquareDistribution[r*s]:

For matrix comparisons the test statistic is invariant under affine transformations:

To test a particular value of Spearman's use CorrelationTest:

Test against :

IndependenceTest can be used to select an appropriate test of independence:

SpearmanRankTest is one of the available tests:

SpearmanRankTest only detects monotonic dependence:

HoeffdingDTest can be used to detect a wider variety of dependence structures:

The Spearman rank test works with the values only when the input is a TimeSeries:

The Spearman rank test works with all the values together when the input is a TemporalData:

Test selected components of the temporal data explicitly:

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:

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