BUILT-IN MATHEMATICA SYMBOL

# IndependenceTest

IndependenceTest[v1, v2]
tests whether the vectors and are independent.

IndependenceTest[m1, m2]
tests whether the matrices and are independent.

IndependenceTest[..., "property"]
returns the value of .

## Details and OptionsDetails and Options

• IndependenceTest performs a hypothesis test on and 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 and can be any real-valued vectors or matrices of equal length.
• IndependenceTest[v1, v2] will choose the most powerful test that applies to and .
• IndependenceTest[v1, v2, All] will choose all tests that apply to and .
• IndependenceTest[v1, v2, "test"] reports the -value according to .
• Some of the available tests assume normality for the . Some tests are restricted to vectors. Most tests are restricted to tests of monotonic or linear independence.
• The following tests can be used:
•  "BlomqvistBeta" monotonic based on Blomqvist's "GoodmanKruskalGamma" monotonic, vector based on the -coefficient "HoeffdingD" vector based on Hoeffding's "KendallTau" monotonic based on Kendall's "PearsonCorrelation" linear, normality, vector based on Pearson product-moment "PillaiTrace" normality, linear based on Pillai's trace "SpearmanRank" monotonic based on Spearman's "WilksW" normality, linear based on Wilks'
• IndependenceTest[v1, v2, "HypothesisTestData"] returns a HypothesisTestData object htd that can be used to extract additional test results and properties using the form htd["property"].
• IndependenceTest[v1, v2, "property"] can be used to directly give the value of .
• Properties related to the reporting of test results include:
•  "AllTests" list of all applicable tests "AutomaticTest" test chosen if Automatic is used "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 nonparametric tests Method Automatic the method to use for computing -values SignificanceLevel 0.05 cutoff for diagnostics and reporting VerifyTestAssumptions Automatic what assumptions to verify
• For tests of independence, a cutoff is chosen such that is rejected only if . The value of used for the and properties is controlled by the SignificanceLevel option. This value is also used in diagnostic tests of normality. By default, is set to .
• Named settings for VerifyTestAssumptions in IndependenceTest include:
•  "Normality" verify that all data is normally distributed

## ExamplesExamplesopen allclose all

### Basic Examples (2)Basic Examples (2)

Test whether two vectors are independent:

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Test whether two matrices are independent:

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

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