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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 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 
-valuesSignificanceLevel 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
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