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BlomqvistBetaTest

BlomqvistBetaTest

BlomqvistBetaTest[v₁,v₂]

tests whether the vectors v₁ and v₂ are independent.

BlomqvistBetaTest[m₁,m₂]

tests whether the matrices m₁ and m₂ are independent.

BlomqvistBetaTest[…,"property"]

returns the value of "property".

Details and Options

BlomqvistBetaTest 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.
BlomqvistBetaTest is based on Blomqvist's medial correlation coefficient β computed by BlomqvistBeta[v₁,v₂].
For testing matrices, the test statistic is based on inner standardized spatial signs and asymptotically follows a ChiSquareDistribution[r*s] where r and s are the dimensions of m₁ and m₂, respectively. The test statistic is invariant under affine transformations.
BlomqvistBetaTest[v₁,v₂,"HypothesisTestData"] returns a HypothesisTestData object htd that can be used to extract additional test results and properties using the form htd["property"].
BlomqvistBetaTest[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

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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 a multivariate test:

By default, is used:

Lowering the setting can shorten computing 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 (4)

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

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

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

BlomqvistBetaTest is one of the available tests:

BlomqvistBetaTest only detects monotonic dependence:

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

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