permutation procedure for testing the equality of pattern hypotheses across groups involving correlation or covariance matrices

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1、StatisticsandComputing(2000)10,253–257ApermutationprocedurefortestingtheequalityofpatternhypothesesacrossgroupsinvolvingcorrelationorcovariancematricesBILLSHIPLEYD´epartementdeBiologie,Universit´edeSherbrooke,Sherbrooke(Qc),Canada,J1K2R1bshipley@courrier.usherb.c

2、aReceivedandAcceptedThispaperdescribesapermutationproceduretotestfortheequalityofselectedelementsofacovari-anceorcorrelationmatrixacrossgroups.Itinvolveseithercentringorstandardisingeachvariablewithineachgroupbeforerandomlypermutingobservationsbetweengroups.Since

3、theassumptionofexchangeabilityofobservationsbetweengroupsdoesnotstrictlyholdfollowingsuchtransforma-tions,MonteCarlosimulationswereusedtocompareexpectedandempiricalrejectionlevelsasafunctionofgroupsize,thenumberofgroupsanddistributiontype(Normal,mixturesofNormals

4、andGammawithvariousvaluesoftheshapeparameter).TheMonteCarlostudyshowedthattheestimatedprobabilitylevelsareclosetothosethatwouldbeobtainedwithanexacttestexceptatverysmallsamplesizes(5or10observationspergroup).Thetestappearsrobustagainstnon-normaldata,differentnumb

5、ersofgroupsorvariablespergroupandunequalsamplesizespergroup.Powerwasincreasedwithincreasingsamplesize,effectsizeandthenumberofelementsinthematrixandpowerwasdecreasedwithincreasinglyunequalnumbersofobservationspergroup.Keywords:permutationtest,equalityofcorrelatio

6、nmatrices,covariancematrices1.Introductionsizes.Bollen(1989)describeshowtouseStructuralEquationModellingtotestsuchhypotheses,butthesameassumptionsResearchersoftenobtainavectorofpobservationsXTDapply.Testsapplicabletocorrelationmatricesarenotwellknown(X1;:::;Xp)ma

7、deonpvariablesfromeachofnDn1Candtheprocessofdatastandardisationinvolvedinobtainingthen2C¢¢¢Cngindividualsthatcanbeassignedtooneofgcorrelationmatrixviolatestheassumptionsunderlyinglikeli-groups.Occasionallytheresearchermightwishtodeterminehoodratiotests(Krzanowski

8、1993).ManlyandRayner(1987)whetherthemultivariatepatternofcovariation,orcorrelation,describealikelihoodratiotestforthedecompositionofthesam-orpartialcorrelation