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Titlestata.
commvtestcorrelations—MultivariatetestsofcorrelationsDescriptionQuickstartMenuSyntaxOptionsformultiple-sampletestsOptionsforone-sampletestsRemarksandexamplesStoredresultsMethodsandformulasReferencesAlsoseeDescriptionmvtestcorrelationsperformsone-sampleandmultiple-sampletestsoncorrelations.
Thesetestsassumemultivariatenormality.
See[MV]mvtestformoremultivariatetests.
QuickstartLawley'stestthatthecorrelationsofvariablesv1,v2,v3,andv4areequalmvtestcorrelationsv1v2v3v4Jennrich'stestthatthecorrelationmatrixequalshypothesizedmatrixmymatmvtestcorrelationsv1v2v3v4,equals(mymat)Jennrich'stestforequalcorrelationswithsamplesdenedbycatvarmvtestcorrelationsv1v2v3v4v5,by(catvar)MenuStatistics>Multivariateanalysis>MANOVA,multivariateregression,andrelated>Multivariatetestofmeans,covariances,andnormality12mvtestcorrelations—MultivariatetestsofcorrelationsSyntaxMultiple-sampletestsmvtestcorrelationsvarlistifinweight,by(groupvars)multisampleoptionsOne-sampletestsmvtestcorrelationsvarlistifinweight,one-sampleoptionsmultisampleoptionsDescriptionModelby(groupvars)comparesubsampleswithsamevaluesingroupvarsmissingtreatmissingvaluesingroupvarsasordinaryvaluesby(groupvars)isrequired.
one-sampleoptionsDescriptionOptionscompoundtestthatcorrelationmatrixiscompoundsymmetric(equalcorrelations);thedefaultequals(C)testthatcorrelationmatrixequalsmatrixCbootstrap,by,jackknife,rolling,andstatsbyareallowed;see[U]11.
1.
10Prexcommands.
Weightsarenotallowedwiththebootstrapprex;see[R]bootstrap.
aweightsarenotallowedwiththejackknifeprex;see[R]jackknife.
aweightsandfweightsareallowed;see[U]11.
1.
6weight.
Optionsformultiple-sampletestsModelby(groupvars)isrequiredwiththemultiple-sampleversionofthetest.
Observationswiththesamevaluesingroupvarsformeachsample.
Observationswithmissingvaluesingroupvarsareignored,unlessthemissingoptionisspecied.
AWaldtestduetoJennrich(1970)isdisplayed.
missingspeciesthatmissingvaluesingroupvarsaretreatedlikeordinaryvalues.
Optionsforone-sampletestsOptionscompound,thedefault,teststhehypothesisthatthecorrelationmatrixofthevariablesiscompoundsymmetric,thatis,thatthecorrelationsofallvariablesinvarlistarethesame.
Lawley's(1963)chi-squaredtestisdisplayed.
equals(C)teststhehypothesisthatthecorrelationmatrixofvarlistisC.
ThematrixCshouldbek*k,symmetric,andpositivedenite.
Cisconvertedtoacorrelationmatrixifneeded.
TherowandcolumnnamesofCareimmaterial.
AWaldtestduetoJennrich(1970)isdisplayed.
mvtestcorrelations—Multivariatetestsofcorrelations3Remarksandexamplesstata.
comRemarksarepresentedunderthefollowingheadings:One-sampletestsforcorrelationmatricesAmultiple-sampletestforcorrelationmatricesOne-sampletestsforcorrelationmatricesBothone-sampleandmultiple-sampletestsofcorrelationmatricesareprovidedwiththemvtestcorrelationscommand.
Theone-sampletestsincludeLawley's(1963)testthatthecorrelationmatrixiscompoundsymmetric(thatis,allcorrelationsareequal),andtheWaldtestproposedbyJennrich(1970)thatthecorrelationmatrixequalsagivencorrelationmatrix.
Example1Thegasoline-poweredmilk-truckdatasetintroducedinexample1of[MV]mvtestmeanshaspricepermileforfuel,repair,andcapital.
Wetestifthecorrelationsbetweenthesethreevariablesareequal(thatis,thecorrelationmatrixiscompoundsymmetric)usingthecompoundoptionofmvtestcorrelations.
.
usehttps://www.
stata-press.
com/data/r16/milktruck(Milktransportationcostsfor25gasolinetrucks(JohnsonandWichern2007)).
mvtestcorrelationsfuelrepaircapital,compoundTestthatcorrelationmatrixiscompoundsymmetric(allcorrelationsequal)Lawleychi2(2)=7.
75Prob>chi2=0.
0208Werejectthenullhypothesisandconcludethatthereareprobablydifferencesinthecorrelationsofthethreecostvariables.
Example2Usingtheequals()optionofmvtestcorrelations,wetestthehypothesisthatfuelandrepaircostshaveacorrelationof0.
75,whilethecorrelationbetweencapitalandthesetwovariablesiszero.
.
matrixC=(1,0.
75,0\0.
75,1,0\0,0,1).
matrixlistCsymmetricC[3,3]c1c2c3r11r2.
751r3001.
mvtestcorrelationsfuelrepaircapital,equals(C)TestthatcorrelationmatrixequalsspecifiedpatternCJennrichchi2(3)=4.
55Prob>chi2=0.
2077Wefailtorejectthisnullhypothesis.
4mvtestcorrelations—MultivariatetestsofcorrelationsAmultiple-sampletestforcorrelationmatricesAmultiple-sampletestofequalityofcorrelationmatricesisprovidedbythemvtestcorrelationscommandwiththeby()optiondeningthemultiplesamples(groups).
Example3Psychologicaltestscoredataareintroducedinexample2of[MV]mvtestcovariances.
Wetestwhetherthecorrelationmatricesforthefourtestscoresarethesameformalesandfemales.
.
usehttps://www.
stata-press.
com/data/r16/genderpsych(FourPsychologicalTestScores,RencherandChristensen(2012)).
mvtestcorrelationsy1y2y3y4,by(gender)TestofequalityofcorrelationmatricesacrosssamplesJennrichchi2(6)=5.
01Prob>chi2=0.
5422Wefailtorejectthenullhypothesisofequalcorrelationmatricesformalesandfemales.
Storedresultsmvtestcorrelationsstoresthefollowinginr():Scalarsr(chi2)χ2statisticr(df)degreesoffreedomforχ2testr(pchi2)p-valueforχ2testMacrosr(chi2type)typeofmodelχ2testMethodsandformulasMethodsandformulasarepresentedunderthefollowingheadings:One-sampletestsforcorrelationmatricesAmultiple-sampletestforcorrelationmatricesOne-sampletestsforcorrelationmatricesLetthesampleconsistofNi.
i.
d.
observationsfromak-variatemultivariatenormaldistributionMVNk(,Σ),withsamplecorrelationmatrixR.
Totestthatacorrelationmatrixequalsagivenmatrix,R0,mvtestcorrelationscomputesaWaldtestproposedbyJennrich(1970):χ2ocf=12trace(ZZ)diagonal(Z)I+R0R101diagonal(Z)whereZ=√NR10(RR0)anddenotestheHadamardproduct.
χ2ocfisasymptoticallyχ2distributedwithk(k1)/2degreesoffreedom.
mvtestcorrelations—Multivariatetestsofcorrelations5Totestthatthecorrelationmatrixiscompoundsymmetric,thatis,totestthatallcorrelationsareequal,thelikelihood-ratiotestissomewhatcumbersome.
Lawley(1963)offersanasymptoticallyequivalenttestthatiscomputationallysimple(JohnsonandWichern2007,457–458):χ2occ=N1(1R)2ki=2i1j=1(RijR)2ukh=1(RhR)2whereR=2k(k1)ki=2i1j=1RijRh=1k1ki=1;i=hRihu=(k1)21(1R)2k(k2)(1R)2andRijdenoteselement(i,j)ofthek*kcorrelationmatrixR.
χ2occisasymptoticallyχ2distributedwith(k2)(k+1)/2degreesoffreedom.
Aitkin,Nelson,andReinfurt(1968)studythequalityofthisχ2approximationforkuptosixandvariouscorrelations,andconcludethattheapproximationisadequateforNassmallas25.
Amultiple-sampletestforcorrelationmatricesLettherebem≥2independentsampleswiththejthsamplecontainingNji.
i.
d.
observationsfromak-variatemultivariatenormaldistribution,MVNk(j,Σj),withsamplecorrelationmatrixRj,j=1,m.
LetN=mj=1Nj.
Totestfortheequalityofcorrelationmatricesacrossmindependentsamples,mvtestcorrela-tionscomputesaWaldtestproposedbyJennrich(1970):χ2mc=mj=112traceZ2jdiagonal(Zj)I+RR11diagonal(Zj)whereR=1/Nmj=1NjRj,Zj=NjR1RjR,anddenotestheHadamardproduct.
χ2mcisasymptoticallyχ2distributedwith(m1)k(k1)/2degreesoffreedom.
ReferencesAitkin,M.
A.
,W.
C.
Nelson,andK.
H.
Reinfurt.
1968.
Testsforcorrelationmatrices.
Biometrika55:327–334.
Jennrich,R.
I.
1970.
Anasymptoticχ2testfortheequalityoftwocorrelationmatrices.
JournaloftheAmericanStatisticalAssociation65:904–912.
Johnson,R.
A.
,andD.
W.
Wichern.
2007.
AppliedMultivariateStatisticalAnalysis.
6thed.
EnglewoodCliffs,NJ:PrenticeHall.
Lawley,D.
N.
1963.
Ontestingasetofcorrelationcoefcientsforequality.
AnnalsofMathematicalStatistics34:149–151.
Rencher,A.
C.
,andW.
F.
Christensen.
2012.
MethodsofMultivariateAnalysis.
3rded.
Hoboken,NJ:Wiley.
6mvtestcorrelations—MultivariatetestsofcorrelationsAlsosee[MV]canon—Canonicalcorrelations[R]correlate—Correlationsofvariables
commvtestcorrelations—MultivariatetestsofcorrelationsDescriptionQuickstartMenuSyntaxOptionsformultiple-sampletestsOptionsforone-sampletestsRemarksandexamplesStoredresultsMethodsandformulasReferencesAlsoseeDescriptionmvtestcorrelationsperformsone-sampleandmultiple-sampletestsoncorrelations.
Thesetestsassumemultivariatenormality.
See[MV]mvtestformoremultivariatetests.
QuickstartLawley'stestthatthecorrelationsofvariablesv1,v2,v3,andv4areequalmvtestcorrelationsv1v2v3v4Jennrich'stestthatthecorrelationmatrixequalshypothesizedmatrixmymatmvtestcorrelationsv1v2v3v4,equals(mymat)Jennrich'stestforequalcorrelationswithsamplesdenedbycatvarmvtestcorrelationsv1v2v3v4v5,by(catvar)MenuStatistics>Multivariateanalysis>MANOVA,multivariateregression,andrelated>Multivariatetestofmeans,covariances,andnormality12mvtestcorrelations—MultivariatetestsofcorrelationsSyntaxMultiple-sampletestsmvtestcorrelationsvarlistifinweight,by(groupvars)multisampleoptionsOne-sampletestsmvtestcorrelationsvarlistifinweight,one-sampleoptionsmultisampleoptionsDescriptionModelby(groupvars)comparesubsampleswithsamevaluesingroupvarsmissingtreatmissingvaluesingroupvarsasordinaryvaluesby(groupvars)isrequired.
one-sampleoptionsDescriptionOptionscompoundtestthatcorrelationmatrixiscompoundsymmetric(equalcorrelations);thedefaultequals(C)testthatcorrelationmatrixequalsmatrixCbootstrap,by,jackknife,rolling,andstatsbyareallowed;see[U]11.
1.
10Prexcommands.
Weightsarenotallowedwiththebootstrapprex;see[R]bootstrap.
aweightsarenotallowedwiththejackknifeprex;see[R]jackknife.
aweightsandfweightsareallowed;see[U]11.
1.
6weight.
Optionsformultiple-sampletestsModelby(groupvars)isrequiredwiththemultiple-sampleversionofthetest.
Observationswiththesamevaluesingroupvarsformeachsample.
Observationswithmissingvaluesingroupvarsareignored,unlessthemissingoptionisspecied.
AWaldtestduetoJennrich(1970)isdisplayed.
missingspeciesthatmissingvaluesingroupvarsaretreatedlikeordinaryvalues.
Optionsforone-sampletestsOptionscompound,thedefault,teststhehypothesisthatthecorrelationmatrixofthevariablesiscompoundsymmetric,thatis,thatthecorrelationsofallvariablesinvarlistarethesame.
Lawley's(1963)chi-squaredtestisdisplayed.
equals(C)teststhehypothesisthatthecorrelationmatrixofvarlistisC.
ThematrixCshouldbek*k,symmetric,andpositivedenite.
Cisconvertedtoacorrelationmatrixifneeded.
TherowandcolumnnamesofCareimmaterial.
AWaldtestduetoJennrich(1970)isdisplayed.
mvtestcorrelations—Multivariatetestsofcorrelations3Remarksandexamplesstata.
comRemarksarepresentedunderthefollowingheadings:One-sampletestsforcorrelationmatricesAmultiple-sampletestforcorrelationmatricesOne-sampletestsforcorrelationmatricesBothone-sampleandmultiple-sampletestsofcorrelationmatricesareprovidedwiththemvtestcorrelationscommand.
Theone-sampletestsincludeLawley's(1963)testthatthecorrelationmatrixiscompoundsymmetric(thatis,allcorrelationsareequal),andtheWaldtestproposedbyJennrich(1970)thatthecorrelationmatrixequalsagivencorrelationmatrix.
Example1Thegasoline-poweredmilk-truckdatasetintroducedinexample1of[MV]mvtestmeanshaspricepermileforfuel,repair,andcapital.
Wetestifthecorrelationsbetweenthesethreevariablesareequal(thatis,thecorrelationmatrixiscompoundsymmetric)usingthecompoundoptionofmvtestcorrelations.
.
usehttps://www.
stata-press.
com/data/r16/milktruck(Milktransportationcostsfor25gasolinetrucks(JohnsonandWichern2007)).
mvtestcorrelationsfuelrepaircapital,compoundTestthatcorrelationmatrixiscompoundsymmetric(allcorrelationsequal)Lawleychi2(2)=7.
75Prob>chi2=0.
0208Werejectthenullhypothesisandconcludethatthereareprobablydifferencesinthecorrelationsofthethreecostvariables.
Example2Usingtheequals()optionofmvtestcorrelations,wetestthehypothesisthatfuelandrepaircostshaveacorrelationof0.
75,whilethecorrelationbetweencapitalandthesetwovariablesiszero.
.
matrixC=(1,0.
75,0\0.
75,1,0\0,0,1).
matrixlistCsymmetricC[3,3]c1c2c3r11r2.
751r3001.
mvtestcorrelationsfuelrepaircapital,equals(C)TestthatcorrelationmatrixequalsspecifiedpatternCJennrichchi2(3)=4.
55Prob>chi2=0.
2077Wefailtorejectthisnullhypothesis.
4mvtestcorrelations—MultivariatetestsofcorrelationsAmultiple-sampletestforcorrelationmatricesAmultiple-sampletestofequalityofcorrelationmatricesisprovidedbythemvtestcorrelationscommandwiththeby()optiondeningthemultiplesamples(groups).
Example3Psychologicaltestscoredataareintroducedinexample2of[MV]mvtestcovariances.
Wetestwhetherthecorrelationmatricesforthefourtestscoresarethesameformalesandfemales.
.
usehttps://www.
stata-press.
com/data/r16/genderpsych(FourPsychologicalTestScores,RencherandChristensen(2012)).
mvtestcorrelationsy1y2y3y4,by(gender)TestofequalityofcorrelationmatricesacrosssamplesJennrichchi2(6)=5.
01Prob>chi2=0.
5422Wefailtorejectthenullhypothesisofequalcorrelationmatricesformalesandfemales.
Storedresultsmvtestcorrelationsstoresthefollowinginr():Scalarsr(chi2)χ2statisticr(df)degreesoffreedomforχ2testr(pchi2)p-valueforχ2testMacrosr(chi2type)typeofmodelχ2testMethodsandformulasMethodsandformulasarepresentedunderthefollowingheadings:One-sampletestsforcorrelationmatricesAmultiple-sampletestforcorrelationmatricesOne-sampletestsforcorrelationmatricesLetthesampleconsistofNi.
i.
d.
observationsfromak-variatemultivariatenormaldistributionMVNk(,Σ),withsamplecorrelationmatrixR.
Totestthatacorrelationmatrixequalsagivenmatrix,R0,mvtestcorrelationscomputesaWaldtestproposedbyJennrich(1970):χ2ocf=12trace(ZZ)diagonal(Z)I+R0R101diagonal(Z)whereZ=√NR10(RR0)anddenotestheHadamardproduct.
χ2ocfisasymptoticallyχ2distributedwithk(k1)/2degreesoffreedom.
mvtestcorrelations—Multivariatetestsofcorrelations5Totestthatthecorrelationmatrixiscompoundsymmetric,thatis,totestthatallcorrelationsareequal,thelikelihood-ratiotestissomewhatcumbersome.
Lawley(1963)offersanasymptoticallyequivalenttestthatiscomputationallysimple(JohnsonandWichern2007,457–458):χ2occ=N1(1R)2ki=2i1j=1(RijR)2ukh=1(RhR)2whereR=2k(k1)ki=2i1j=1RijRh=1k1ki=1;i=hRihu=(k1)21(1R)2k(k2)(1R)2andRijdenoteselement(i,j)ofthek*kcorrelationmatrixR.
χ2occisasymptoticallyχ2distributedwith(k2)(k+1)/2degreesoffreedom.
Aitkin,Nelson,andReinfurt(1968)studythequalityofthisχ2approximationforkuptosixandvariouscorrelations,andconcludethattheapproximationisadequateforNassmallas25.
Amultiple-sampletestforcorrelationmatricesLettherebem≥2independentsampleswiththejthsamplecontainingNji.
i.
d.
observationsfromak-variatemultivariatenormaldistribution,MVNk(j,Σj),withsamplecorrelationmatrixRj,j=1,m.
LetN=mj=1Nj.
Totestfortheequalityofcorrelationmatricesacrossmindependentsamples,mvtestcorrela-tionscomputesaWaldtestproposedbyJennrich(1970):χ2mc=mj=112traceZ2jdiagonal(Zj)I+RR11diagonal(Zj)whereR=1/Nmj=1NjRj,Zj=NjR1RjR,anddenotestheHadamardproduct.
χ2mcisasymptoticallyχ2distributedwith(m1)k(k1)/2degreesoffreedom.
ReferencesAitkin,M.
A.
,W.
C.
Nelson,andK.
H.
Reinfurt.
1968.
Testsforcorrelationmatrices.
Biometrika55:327–334.
Jennrich,R.
I.
1970.
Anasymptoticχ2testfortheequalityoftwocorrelationmatrices.
JournaloftheAmericanStatisticalAssociation65:904–912.
Johnson,R.
A.
,andD.
W.
Wichern.
2007.
AppliedMultivariateStatisticalAnalysis.
6thed.
EnglewoodCliffs,NJ:PrenticeHall.
Lawley,D.
N.
1963.
Ontestingasetofcorrelationcoefcientsforequality.
AnnalsofMathematicalStatistics34:149–151.
Rencher,A.
C.
,andW.
F.
Christensen.
2012.
MethodsofMultivariateAnalysis.
3rded.
Hoboken,NJ:Wiley.
6mvtestcorrelations—MultivariatetestsofcorrelationsAlsosee[MV]canon—Canonicalcorrelations[R]correlate—Correlationsofvariables
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