# CorrelationTest

CorrelationTest[{{x1,y1},{x2,y2},}]

tests whether the correlation coefficient for a bivariate population is zero.

CorrelationTest[{{x1,y1},{x2,y2},},ρ0]

tests whether the correlation coefficient is ρ0.

CorrelationTest[{{x1,y1},{x2,y2},},{{u1,v1},{u2,v2},}]

tests whether the correlation coefficients for two populations are equal.

CorrelationTest[,"property"]

returns the value of "property".

# Details and Options   • CorrelationTest tests the null hypothesis against the alternative hypothesis :
•     {{{x1,y1},{x2,y2},…},{{u1,v1},{u2,v2},…}}  • where ρx,y is the population correlation coefficient for {{x1,y1},{x2,y2},}.
• By default, a probability value or -value is returned.
• A small -value suggests that it is unlikely that is true.
• The data {{x1,y1},} and {{u1,v1},} can be any real-valued bivariate matrices.
• The argument ρ0 can be any real number such that .,
• CorrelationTest[{{x1,y1},{x2,y2},},ρ0] will choose the most powerful test that applies to the data.
• will choose all tests that apply to the data.
• CorrelationTest[,"test"] reports the -value according to "test".
• The following tests can be used:
•  "PearsonCorrelation" based on Pearson product-moment "SpearmanRank" based on Spearman's • The "PearsonCorrelation" test assumes the data was drawn from a normal distribution.
• CorrelationTest[,"HypothesisTestData"] returns a HypothesisTestData object htd that can be used to extract additional test results and properties using the form htd["property"].
• CorrelationTest[,"property"] can be used to directly give the value of "property".
• Properties related to the reporting of test results include:
•  "AllTests" list of all applicable tests "AutomaticTest" test chosen if Automatic is used "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 SignificanceLevel 0.05 cutoff for diagnostics and reporting VerifyTestAssumptions Automatic what assumptions to verify
• For tests of correlation, 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. This value is also used in diagnostic tests of normality. By default, is set to 0.05.
• Named settings for VerifyTestAssumptions in CorrelationTest include:
•  "Normality" verify that all data is normally distributed

# Examples

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## Basic Examples(3)

Test whether the population correlation coefficient is zero:

 In:= In:= Out= In:= Out= Test whether the population correlation coefficient is 0.25:

 In:= In:= Out= Compare two correlation coefficients:

 In:= In:= Out= In:= Out= Report the test statistics and -values in a table:

 In:= Out= ## Possible Issues(3)

Introduced in 2012
(9.0)