tests whether the vectors v1 and v2 are independent.


tests whether the matrices m1 and m2 are independent.


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
    MaxIterationsAutomaticmax iterations for multivariate test
    MethodAutomaticthe method to use for computing -values
    SignificanceLevel0.05cutoff 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.


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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:

Introduced in 2012