tests whether the vectors and are independent.

tests whether the matrices and are independent.

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"monotonicbased on Blomqvist's
    "GoodmanKruskalGamma"monotonic, vectorbased on the -coefficient
    "HoeffdingD"vectorbased on Hoeffding's
    "KendallTau"monotonicbased on Kendall's
    "PearsonCorrelation"linear, normality, vectorbased on Pearson product-moment
    "PillaiTrace"normality, linearbased on Pillai's trace
    "SpearmanRank"monotonicbased on Spearman's
    "WilksW"normality, linearbased 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
    MaxIterationsAutomaticmax iterations for multivariate nonparametric tests
    MethodAutomaticthe method to use for computing -values
    SignificanceLevel0.05cutoff for diagnostics and reporting
    VerifyTestAssumptionsAutomaticwhat 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:

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

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Introduced in 2012