factor analysis - wikipedia, the free encyclopedia

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https://en.wikipedia.org/wiki/Factor_analysis#Exploratory_factor_analysis_versus_principal_components_analysis 1/20

FactoranalysisFromWikipedia,thefreeencyclopedia

Factoranalysisisastatisticalmethodusedtodescribevariabilityamongobserved,correlatedvariablesintermsofapotentiallylowernumberofunobservedvariablescalledfactors.Forexample,itispossiblethatvariationsinfourobservedvariablesmainlyreflectthevariationsintwounobservedvariables.Factoranalysissearchesforsuchjointvariationsinresponsetounobservedlatentvariables.Theobservedvariablesaremodelledaslinearcombinationsofthepotentialfactors,plus"error"terms.Theinformationgainedabouttheinterdependenciesbetweenobservedvariablescanbeusedlatertoreducethesetofvariablesinadataset.Computationallythistechniqueisequivalenttolowrankapproximationofthematrixofobservedvariables.Factoranalysisoriginatedinpsychometricsandisusedinbehavioralsciences,socialsciences,marketing,productmanagement,operationsresearch,andotherappliedsciencesthatdealwithlargequantitiesofdata.

Factoranalysisisrelatedtoprincipalcomponentanalysis(PCA),butthetwoarenotidentical.Latentvariablemodels,includingfactoranalysis,useregressionmodellingtechniquestotesthypothesesproducingerrorterms,whilePCAisadescriptivestatisticaltechnique.[1]Therehasbeensignificantcontroversyinthefieldovertheequivalenceorotherwiseofthetwotechniques(seeexploratoryfactoranalysisversusprincipalcomponentsanalysis).

Contents

1Statisticalmodel1.1Definition1.2Example1.3Mathematicalmodelofthesameexample1.4Geometricinterpretation

2Practicalimplementation2.1Typeoffactoranalysis2.2Typesoffactoring2.3Terminology2.4Criteriafordeterminingthenumberoffactors2.5Rotationmethods

3Factoranalysisinpsychometrics3.1History3.2Applicationsinpsychology3.3Advantages3.4Disadvantages

4Exploratoryfactoranalysisversusprincipalcomponentsanalysis4.1ArgumentscontrastingPCAandEFA



4.2Varianceversuscovariance4.3Differencesinprocedureandresults

5Factoranalysisinmarketing5.1Informationcollection5.2Analysis5.3Advantages5.4Disadvantages

6Factoranalysisinphysicalandbiologicalsciences7Factoranalysisinmicroarrayanalysis8Implementation9Seealso10Furtherreading11Externallinks

Statisticalmodel

Definition

Supposewehaveasetof observablerandomvariables, withmeans .

Supposeforsomeunknownconstants and unobservedrandomvariables ,whereand ,where ,wehave

Here,the areindependentlydistributederrortermswithzeromeanandfinitevariance,whichmaynotbethesameforall .Let ,sothatwehave

Inmatrixterms,wehave

Ifwehave observations,thenwewillhavethedimensions , ,and .Eachcolumnofand denotevaluesforoneparticularobservation,andmatrix doesnotvaryacrossobservations.

Alsowewillimposethefollowingassumptionson :

1. and areindependent.2.



3. (tomakesurethatthefactorsareuncorrelated).

Anysolutionoftheabovesetofequationsfollowingtheconstraintsfor isdefinedasthefactors,andastheloadingmatrix.

Suppose .Thennotethatfromtheconditionsjustimposedon ,wehave

or

or

Notethatforanyorthogonalmatrix ,ifweset and ,thecriteriaforbeingfactorsandfactorloadingsstillhold.Henceasetoffactorsandfactorloadingsisidenticalonlyuptoorthogonaltransformation.

Example

Thefollowingexampleisforexpositorypurposes,andshouldnotbetakenasbeingrealistic.Supposeapsychologistproposesatheorythattherearetwokindsofintelligence,"verbalintelligence"and"mathematicalintelligence",neitherofwhichisdirectlyobserved.Evidenceforthetheoryissoughtintheexaminationscoresfromeachof10differentacademicfieldsof1000students.Ifeachstudentischosenrandomlyfromalargepopulation,theneachstudent's10scoresarerandomvariables.Thepsychologist'stheorymaysaythatforeachofthe10academicfields,thescoreaveragedoverthegroupofallstudentswhosharesomecommonpairofvaluesforverbalandmathematical"intelligences"issomeconstanttimestheirlevelofverbalintelligenceplusanotherconstanttimestheirlevelofmathematicalintelligence,i.e.,itisacombinationofthosetwo"factors".Thenumbersforaparticularsubject,bywhichthetwokindsofintelligencearemultipliedtoobtaintheexpectedscore,arepositedbythetheorytobethesameforallintelligencelevelpairs,andarecalled"factorloadings"forthissubject.Forexample,thetheorymayholdthattheaveragestudent'saptitudeinthefieldoftaxonomyis

{10thestudent'sverbalintelligence}+{6thestudent'smathematicalintelligence}.

Thenumbers10and6arethefactorloadingsassociatedwithtaxonomy.Otheracademicsubjectsmayhavedifferentfactorloadings.

Twostudentshavingidenticaldegreesofverbalintelligenceandidenticaldegreesofmathematicalintelligencemayhavedifferentaptitudesintaxonomybecauseindividualaptitudesdifferfromaverageaptitudes.Thatdifferenceiscalledthe"error"astatisticaltermthatmeanstheamountbywhichanindividualdiffersfromwhatisaverageforhisorherlevelsofintelligence(seeerrorsandresidualsinstatistics).



Theobservabledatathatgointofactoranalysiswouldbe10scoresofeachofthe1000students,atotalof10,000numbers.Thefactorloadingsandlevelsofthetwokindsofintelligenceofeachstudentmustbeinferredfromthedata.

Mathematicalmodelofthesameexample

Inthefollowing,matriceswillbeindicatedbyindexedvariables."Subject"indiceswillbeindicatedusinglettersa,bandc,withvaluesrunningfrom1to whichisequalto10intheaboveexample."Factor"indiceswillbeindicatedusinglettersp,qandr,withvaluesrunningfrom1to whichisequalto2intheaboveexample."Instance"or"sample"indiceswillbeindicatedusinglettersi,jandk,withvaluesrunningfrom1to .Intheexampleabove,ifasampleof studentsrespondedtothequestions,theithstudent'sscorefortheathquestionaregivenby .Thepurposeoffactoranalysisistocharacterizethecorrelationsbetweenthevariables ofwhichthe areaparticularinstance,orsetofobservations.Inorderthatthevariablesbeonequalfooting,theyarestandardized:

wherethesamplemeanis:

andthesamplevarianceisgivenby:

Thefactoranalysismodelforthisparticularsampleisthen:

or,moresuccinctly:

where

istheithstudent's"verbalintelligence",istheithstudent's"mathematicalintelligence",arethefactorloadingsfortheathsubject,forp=1,2.

Inmatrixnotation,wehave



Observethatbydoublingthescaleonwhich"verbalintelligence"thefirstcomponentineachcolumnofFismeasured,andsimultaneouslyhalvingthefactorloadingsforverbalintelligencemakesnodifferencetothemodel.Thus,nogeneralityislostbyassumingthatthestandarddeviationofverbalintelligenceis1.Likewiseformathematicalintelligence.Moreover,forsimilarreasons,nogeneralityislostbyassumingthetwofactorsareuncorrelatedwitheachother.Inotherwords:

where istheKroneckerdelta(0when and1when ).Theerrorsareassumedtobeindependentofthefactors:

Notethat,sinceanyrotationofasolutionisalsoasolution,thismakesinterpretingthefactorsdifficult.Seedisadvantagesbelow.Inthisparticularexample,ifwedonotknowbeforehandthatthetwotypesofintelligenceareuncorrelated,thenwecannotinterpretthetwofactorsasthetwodifferenttypesofintelligence.Eveniftheyareuncorrelated,wecannottellwhichfactorcorrespondstoverbalintelligenceandwhichcorrespondstomathematicalintelligencewithoutanoutsideargument.

ThevaluesoftheloadingsL,theaverages,andthevariancesofthe"errors"mustbeestimatedgiventheobserveddataXandF(theassumptionaboutthelevelsofthefactorsisfixedforagivenF).The"fundamentaltheorem"maybederivedfromtheaboveconditions:

Thetermontheleftisjustthecorrelationmatrixoftheobserveddata,andits diagonalelementswillbe1's.Thelasttermontherightwillbeadiagonalmatrixwithtermslessthanunity.Thefirsttermontherightisthe"reducedcorrelationmatrix"andwillbeequaltothecorrelationmatrixexceptforitsdiagonalvalueswhichwillbelessthanunity.Thesediagonalelementsofthereducedcorrelationmatrixarecalled"communalities":

Thesampledata willnot,ofcourse,exactlyobeythefundamentalequationgivenaboveduetosamplingerrors,inadequacyofthemodel,etc.Thegoalofanyanalysisoftheabovemodelistofindthefactors andloadings which,insomesense,givea"bestfit"tothedata.Infactoranalysis,thebestfitisdefinedastheminimumofthemeansquareerrorintheoffdiagonalresidualsofthecorrelationmatrix:[2]



GeometricinterpretationofFactorAnalysisparametersfor3respondentstoquestion"a".The"answer"isrepresentedbytheunitvector ,whichisprojectedontoaplanedefinedbytwoorthonormalvectors and .Theprojectionvectoris andtheerror isperpendiculartotheplane,sothat .Theprojectionvector mayberepresentedintermsofthefactorvectorsas

.Thesquareofthelengthoftheprojectionvectoristhecommunality: .Ifanotherdatavector wereplotted,thecosineoftheanglebetween and wouldbe :the(a,b)entryinthecorrelationmatrix.(AdaptedfromHarmanFig.4.3)[2]

Thisisequivalenttominimizingtheoffdiagonalcomponentsoftheerrorcovariancewhich,inthemodelequationshaveexpectedvaluesofzero.Thisistobecontrastedwithprincipalcomponentanalysiswhichseekstominimizethemeansquareerrorofallresiduals.[2]Beforetheadventofhighspeedcomputers,considerableeffortwasdevotedtofindingapproximatesolutionstotheproblem,particularlyinestimatingthecommunalitiesbyothermeans,whichthensimplifiestheproblemconsiderablybyyieldingaknownreducedcorrelationmatrix.Thiswasthenusedtoestimatethefactorsandtheloadings.Withtheadventofhighspeedcomputers,theminimizationproblemcanbesolvedquicklyanddirectly,andthecommunalitiesarecalculatedintheprocess,ratherthanbeingneededbeforehand.TheMinResalgorithmisparticularlysuitedtothisproblem,butishardlytheonlymeansoffindinganexactsolution.

Geometricinterpretation

Theparametersandvariablesoffactoranalysiscanbegivenageometricalinterpretation.Thedata( ),thefactors( )andtheerrors( )canbeviewedasvectorsinan dimensionalEuclideanspace(samplespace),representedas

, and respectively.Sincethedataisstandardized,thedatavectorsareofunitlength( ).Thefactorvectorsdefinean

dimensionallinearsubspace(i.e.ahyperplane)inthisspace,uponwhichthedatavectorsareprojectedorthogonally.Thisfollowsfromthemodelequation

andtheindependenceofthefactorsandtheerrors: .Intheaboveexample,thehyperplaneisjusta2dimensionalplanedefinedbythetwofactorvectors.Theprojectionofthedatavectorsontothehyperplaneisgivenby

andtheerrorsarevectorsfromthatprojectedpointtothedatapointandareperpendiculartothehyperplane.Thegoaloffactoranalysisistofindahyperplanewhichisa"bestfit"tothedatainsomesense,soitdoesn'tmatterhowthefactorvectorswhichdefinethishyperplanearechosen,aslongastheyareindependentandlieinthehyperplane.Wearefreetospecifythemasbothorthogonalandnormal(

)withnolossofgenerality.Afterasuitablesetoffactorsarefound,theymayalsobearbitrarilyrotatedwithinthehyperplane,sothatanyrotationofthefactorvectorswilldefinethesamehyperplane,andalsobeasolution.Asaresult,intheaboveexample,inwhichthefittinghyperplaneistwodimensional,ifwedonotknowbeforehandthatthetwotypesofintelligenceareuncorrelated,thenwe



cannotinterpretthetwofactorsasthetwodifferenttypesofintelligence.Eveniftheyareuncorrelated,wecannottellwhichfactorcorrespondstoverbalintelligenceandwhichcorrespondstomathematicalintelligence,orwhetherthefactorsarelinearcombinationsofboth,withoutanoutsideargument.

Thedatavectors haveunitlength.Thecorrelationmatrixforthedataisgivenby .Thecorrelationmatrixcanbegeometricallyinterpretedasthecosineoftheanglebetweenthetwodatavectors

and .Thediagonalelementswillclearlybe1'sandtheoffdiagonalelementswillhaveabsolutevalueslessthanorequaltounity.The"reducedcorrelationmatrix"isdefinedas

.

Thegoaloffactoranalysisistochoosethefittinghyperplanesuchthatthereducedcorrelationmatrixreproducesthecorrelationmatrixasnearlyaspossible,exceptforthediagonalelementsofthecorrelationmatrixwhichareknowntohaveunitvalue.Inotherwords,thegoalistoreproduceasaccuratelyaspossiblethecrosscorrelationsinthedata.Specifically,forthefittinghyperplane,themeansquareerrorintheoffdiagonalcomponents

istobeminimized,andthisisaccomplishedbyminimizingitwithrespecttoasetoforthonormalfactorvectors.Itcanbeseenthat

Thetermontherightisjustthecovarianceoftheerrors.Inthemodel,theerrorcovarianceisstatedtobeadiagonalmatrixandsotheaboveminimizationproblemwillinfactyielda"bestfit"tothemodel:Itwillyieldasampleestimateoftheerrorcovariancewhichhasitsoffdiagonalcomponentsminimizedinthemeansquaresense.Itcanbeseenthatsincethe areorthogonalprojectionsofthedatavectors,theirlengthwillbelessthanorequaltothelengthoftheprojecteddatavector,whichisunity.Thesquareoftheselengthsarejustthediagonalelementsofthereducedcorrelationmatrix.Thesediagonalelementsofthereducedcorrelationmatrixareknownas"communalities":

Largevaluesofthecommunalitieswillindicatethatthefittinghyperplaneisratheraccuratelyreproducingthecorrelationmatrix.Itshouldbenotedthatthemeanvaluesofthefactorsmustalsobeconstrainedtobezero,fromwhichitfollowsthatthemeanvaluesoftheerrorswillalsobezero.

Practicalimplementation

Typeoffactoranalysis

Exploratoryfactoranalysis(EFA)isusedtoidentifycomplexinterrelationshipsamongitemsandgroupitemsthatarepartofunifiedconcepts.[3]Theresearchermakesno"apriori"assumptionsaboutrelationshipsamongfactors.[3]



Confirmatoryfactoranalysis(CFA)isamorecomplexapproachthatteststhehypothesisthattheitemsareassociatedwithspecificfactors.[3]CFAusesstructuralequationmodelingtotestameasurementmodelwherebyloadingonthefactorsallowsforevaluationofrelationshipsbetweenobservedvariablesandunobservedvariables.[3]Structuralequationmodelingapproachescanaccommodatemeasurementerror,andarelessrestrictivethanleastsquaresestimation.[3]Hypothesizedmodelsaretestedagainstactualdata,andtheanalysiswoulddemonstrateloadingsofobservedvariablesonthelatentvariables(factors),aswellasthecorrelationbetweenthelatentvariables.[3]

Typesoffactoring

Principalcomponentanalysis(PCA):PCAisawidelyusedmethodforfactorextraction,whichisthefirstphaseofEFA.[3]Factorweightsarecomputedinordertoextractthemaximumpossiblevariance,withsuccessivefactoringcontinuinguntilthereisnofurthermeaningfulvarianceleft.[3]Thefactormodelmustthenberotatedforanalysis.[3]

Canonicalfactoranalysis,alsocalledRao'scanonicalfactoring,isadifferentmethodofcomputingthesamemodelasPCA,whichusestheprincipalaxismethod.Canonicalfactoranalysisseeksfactorswhichhavethehighestcanonicalcorrelationwiththeobservedvariables.Canonicalfactoranalysisisunaffectedbyarbitraryrescalingofthedata.

Commonfactoranalysis,alsocalledprincipalfactoranalysis(PFA)orprincipalaxisfactoring(PAF),seekstheleastnumberoffactorswhichcanaccountforthecommonvariance(correlation)ofasetofvariables.

Imagefactoring:basedonthecorrelationmatrixofpredictedvariablesratherthanactualvariables,whereeachvariableispredictedfromtheothersusingmultipleregression.

Alphafactoring:basedonmaximizingthereliabilityoffactors,assumingvariablesarerandomlysampledfromauniverseofvariables.Allothermethodsassumecasestobesampledandvariablesfixed.

Factorregressionmodel:acombinatorialmodeloffactormodelandregressionmodeloralternatively,itcanbeviewedasthehybridfactormodel,[4]whosefactorsarepartiallyknown.

Terminology

Factorloadings:Thefactorloadings,alsocalledcomponentloadingsinPCA(notsureinFactorAnalysis),arethecorrelationcoefficientsbetweenthecases(rows)andfactors(columns).AnalogoustoPearson'sr,thesquaredfactorloadingisthepercentofvarianceinthatindicatorvariableexplainedbythefactor.Togetthepercentofvarianceinallthevariablesaccountedforbyeachfactor,addthesumofthesquaredfactorloadingsforthatfactor(column)anddividebythenumberofvariables.(Notethenumberofvariablesequalsthesumoftheirvariancesasthevarianceofastandardizedvariableis1.)Thisisthesameasdividingthefactor'seigenvaluebythenumberofvariables.

Interpretingfactorloadings:Byoneruleofthumbinconfirmatoryfactoranalysis,loadingsshouldbe.7orhighertoconfirmthatindependentvariablesidentifiedaprioriarerepresentedbyaparticularfactor,ontherationalethatthe.7levelcorrespondstoabouthalfofthevarianceintheindicatorbeingexplainedbythefactor.However,the.7standardisahighoneandreallifedatamaywellnotmeetthiscriterion,which



iswhysomeresearchers,particularlyforexploratorypurposes,willusealowerlevelsuchas.4forthecentralfactorand.25forotherfactors.Inanyevent,factorloadingsmustbeinterpretedinthelightoftheory,notbyarbitrarycutofflevels.

Inobliquerotation,onegetsbothapatternmatrixandastructurematrix.Thestructurematrixissimplythefactorloadingmatrixasinorthogonalrotation,representingthevarianceinameasuredvariableexplainedbyafactoronbothauniqueandcommoncontributionsbasis.Thepatternmatrix,incontrast,containscoefficientswhichjustrepresentuniquecontributions.Themorefactors,thelowerthepatterncoefficientsasarulesincetherewillbemorecommoncontributionstovarianceexplained.Forobliquerotation,theresearcherlooksatboththestructureandpatterncoefficientswhenattributingalabeltoafactor.Principlesofobliquerotationcanbederivedfrombothcrossentropyanditsdualentropy.[5]

Communality:Thesumofthesquaredfactorloadingsforallfactorsforagivenvariable(row)isthevarianceinthatvariableaccountedforbyallthefactors,andthisiscalledthecommunality.Thecommunalitymeasuresthepercentofvarianceinagivenvariableexplainedbyallthefactorsjointlyandmaybeinterpretedasthereliabilityoftheindicator.

Spurioussolutions:Ifthecommunalityexceeds1.0,thereisaspurioussolution,whichmayreflecttoosmallasampleortheresearcherhastoomanyortoofewfactors.

Uniquenessofavariable:Thatis,uniquenessisthevariabilityofavariableminusitscommunality.

Eigenvalues:/Characteristicroots:Theeigenvalueforagivenfactormeasuresthevarianceinallthevariableswhichisaccountedforbythatfactor.Theratioofeigenvaluesistheratioofexplanatoryimportanceofthefactorswithrespecttothevariables.Ifafactorhasaloweigenvalue,thenitiscontributinglittletotheexplanationofvariancesinthevariablesandmaybeignoredasredundantwithmoreimportantfactors.Eigenvaluesmeasuretheamountofvariationinthetotalsampleaccountedforbyeachfactor.

Extractionsumsofsquaredloadings:Initialeigenvaluesandeigenvaluesafterextraction(listedbySPSSas"ExtractionSumsofSquaredLoadings")arethesameforPCAextraction,butforotherextractionmethods,eigenvaluesafterextractionwillbelowerthantheirinitialcounterparts.SPSSalsoprints"RotationSumsofSquaredLoadings"andevenforPCA,theseeigenvalueswilldifferfrominitialandextractioneigenvalues,thoughtheirtotalwillbethesame.

Factorscores(alsocalledcomponentscoresinPCA):arethescoresofeachcase(row)oneachfactor(column).Tocomputethefactorscoreforagivencaseforagivenfactor,onetakesthecase'sstandardizedscoreoneachvariable,multipliesbythecorrespondingloadingsofthevariableforthegivenfactor,andsumstheseproducts.Computingfactorscoresallowsonetolookforfactoroutliers.Also,factorscoresmaybeusedasvariablesinsubsequentmodeling.(ExplainedfromPCAnotfromFactorAnalysisperspective).

Criteriafordeterminingthenumberoffactors

Usingoneormoreofthemethodsbelow,theresearcherdeterminesanappropriaterangeofsolutionstoinvestigate.Methodsmaynotagree.Forinstance,theKaisercriterionmaysuggestfivefactorsandthescreetestmaysuggesttwo,sotheresearchermayrequest3,4,and5factorsolutionsdiscusseachintermsoftheirrelationtoexternaldataandtheory.



Comprehensibility:Apurelysubjectivecriterionwouldbetoretainthosefactorswhosemeaningiscomprehensibletotheresearcher.Thisisnotrecommended.

Kaisercriterion:TheKaiserruleistodropallcomponentswitheigenvaluesunder1.0thisbeingtheeigenvalueequaltotheinformationaccountedforbyanaveragesingleitem.TheKaisercriterionisthedefaultinSPSSandmoststatisticalsoftwarebutisnotrecommendedwhenusedasthesolecutoffcriterionforestimatingthenumberoffactorsasittendstooverextractfactors.[6]Avariationofthismethodhasbeencreatedwherearesearchercalculatesconfidenceintervalsforeacheigenvalueandretainsonlyfactorswhichhavetheentireconfidenceintervalgreaterthan1.0.[7][8]

Varianceexplainedcriteria:Someresearcherssimplyusetheruleofkeepingenoughfactorstoaccountfor90%(sometimes80%)ofthevariation.Wheretheresearcher'sgoalemphasizesparsimony(explainingvariancewithasfewfactorsaspossible),thecriterioncouldbeaslowas50%

Screeplot:TheCattellscreetestplotsthecomponentsastheXaxisandthecorrespondingeigenvaluesastheYaxis.Asonemovestotheright,towardlatercomponents,theeigenvaluesdrop.Whenthedropceasesandthecurvemakesanelbowtowardlesssteepdecline,Cattell'sscreetestsaystodropallfurthercomponentsaftertheonestartingtheelbow.Thisruleissometimescriticisedforbeingamenabletoresearchercontrolled"fudging".Thatis,aspickingthe"elbow"canbesubjectivebecausethecurvehasmultipleelbowsorisasmoothcurve,theresearchermaybetemptedtosetthecutoffatthenumberoffactorsdesiredbyhisorherresearchagenda.

Horn'sParallelAnalysis(PA):AMonteCarlobasedsimulationmethodthatcomparestheobservedeigenvalueswiththoseobtainedfromuncorrelatednormalvariables.Afactororcomponentisretainediftheassociatedeigenvalueisbiggerthanthe95thofthedistributionofeigenvaluesderivedfromtherandomdata.PAisoneofthemostrecommendablerulesfordeterminingthenumberofcomponentstoretain,butonlyfewprogramsincludethisoption.[9]

However,beforedroppingafactorbelowone'scutoff,theanalyst(s)shouldcreateadatasetbasedonthefactorloadingsandcheckthescores'correlationwithanygivendependentvariable(s)ofinterest.Scoresbasedonafactorwithaverysmalleigenvaluecancorrelatestronglywithdependentvariables,inwhichcasedroppingsuchafactorfromatheoreticalmodelmayreduceitspredictivevalidity.

Velicers(1976)MAPtest[10]involvesacompleteprincipalcomponentsanalysisfollowedbytheexaminationofaseriesofmatricesofpartialcorrelations(p.397).ThesquaredcorrelationforStep0(seeFigure4)istheaveragesquaredoffdiagonalcorrelationfortheunpartialedcorrelationmatrix.OnStep1,thefirstprincipalcomponentanditsassociateditemsarepartialedout.Thereafter,theaveragesquaredoffdiagonalcorrelationforthesubsequentcorrelationmatrixisthencomputedforStep1.OnStep2,thefirsttwoprincipalcomponentsarepartialedoutandtheresultantaveragesquaredoffdiagonalcorrelationisagaincomputed.Thecomputationsarecarriedoutforkminusonestep(krepresentingthetotalnumberofvariablesinthematrix).Thereafter,alloftheaveragesquaredcorrelationsforeachsteparelinedupandthestepnumberintheanalysesthatresultedinthelowestaveragesquaredpartialcorrelationdeterminesthenumberofcomponentsorfactorstoretain(Velicer,1976).Bythismethod,componentsaremaintainedaslongasthevarianceinthecorrelationmatrixrepresentssystematicvariance,asopposedtoresidualorerrorvariance.Althoughmethodologicallyakintoprincipalcomponentsanalysis,theMAPtechniquehasbeenshowntoperformquitewellindeterminingthenumberoffactorstoretaininmultiplesimulationstudies.[8][11][12]ThisprocedureismadeavailablethroughSPSS'suserinterface.SeeCourtney(2013)[13]forguidance.



Rotationmethods

Theunrotatedoutputmaximisesvarianceaccountedforbythefirstandsubsequentfactors,andforcingthefactorstobeorthogonal.Thisdatacompressioncomesatthecostofhavingmostitemsloadontheearlyfactors,andusually,ofhavingmanyitemsloadsubstantiallyonmorethanonefactor.Rotationservestomaketheoutputmoreunderstandable,byseekingsocalled"SimpleStructure":Apatternofloadingswhereitemsloadmoststronglyononefactor,andmuchmoreweaklyontheotherfactors.Rotationscanbeorthogonaloroblique(allowingthefactorstocorrelate).

Varimaxrotationisanorthogonalrotationofthefactoraxestomaximizethevarianceofthesquaredloadingsofafactor(column)onallthevariables(rows)inafactormatrix,whichhastheeffectofdifferentiatingtheoriginalvariablesbyextractedfactor.Eachfactorwilltendtohaveeitherlargeorsmallloadingsofanyparticularvariable.Avarimaxsolutionyieldsresultswhichmakeitaseasyaspossibletoidentifyeachvariablewithasinglefactor.Thisisthemostcommonrotationoption.However,theorthogonality(i.e.,independence)offactorsisoftenanunrealisticassumption.Obliquerotationsareinclusiveoforthogonalrotation,andforthatreason,obliquerotationsareapreferredmethod.[14]

Quartimaxrotationisanorthogonalalternativewhichminimizesthenumberoffactorsneededtoexplaineachvariable.Thistypeofrotationoftengeneratesageneralfactoronwhichmostvariablesareloadedtoahighormediumdegree.Suchafactorstructureisusuallynothelpfultotheresearchpurpose.

EquimaxrotationisacompromisebetweenVarimaxandQuartimaxcriteria.

Directobliminrotationisthestandardmethodwhenonewishesanonorthogonal(oblique)solutionthatis,oneinwhichthefactorsareallowedtobecorrelated.Thiswillresultinhighereigenvaluesbutdiminishedinterpretabilityofthefactors.Seebelow.

Promaxrotationisanalternativenonorthogonal(oblique)rotationmethodwhichiscomputationallyfasterthanthedirectobliminmethodandthereforeissometimesusedforverylargedatasets.

Factoranalysisinpsychometrics

History

CharlesSpearmanpioneeredtheuseoffactoranalysisinthefieldofpsychologyandissometimescreditedwiththeinventionoffactoranalysis.Hediscoveredthatschoolchildren'sscoresonawidevarietyofseeminglyunrelatedsubjectswerepositivelycorrelated,whichledhimtopostulatethatageneralmentalability,org,underliesandshapeshumancognitiveperformance.Hispostulatenowenjoysbroadsupportinthefieldofintelligenceresearch,whereitisknownasthegtheory.

RaymondCattellexpandedonSpearman'sideaofatwofactortheoryofintelligenceafterperforminghisowntestsandfactoranalysis.Heusedamultifactortheorytoexplainintelligence.Cattell'stheoryaddressedalternatefactorsinintellectualdevelopment,includingmotivationandpsychology.Cattellalsodevelopedseveralmathematicalmethodsforadjustingpsychometricgraphs,suchashis"scree"testandsimilaritycoefficients.Hisresearchledtothedevelopmentofhistheoryoffluidandcrystallizedintelligence,aswellashis16PersonalityFactorstheoryofpersonality.Cattellwasastrongadvocateoffactoranalysisandpsychometrics.Hebelievedthatalltheoryshouldbederivedfromresearch,whichsupportsthecontinueduseofempiricalobservationandobjectivetestingtostudyhumanintelligence.



Applicationsinpsychology

Factoranalysisisusedtoidentify"factors"thatexplainavarietyofresultsondifferenttests.Forexample,intelligenceresearchfoundthatpeoplewhogetahighscoreonatestofverbalabilityarealsogoodonotherteststhatrequireverbalabilities.Researchersexplainedthisbyusingfactoranalysistoisolateonefactor,oftencalledcrystallizedintelligenceorverbalintelligence,whichrepresentsthedegreetowhichsomeoneisabletosolveproblemsinvolvingverbalskills.

Factoranalysisinpsychologyismostoftenassociatedwithintelligenceresearch.However,italsohasbeenusedtofindfactorsinabroadrangeofdomainssuchaspersonality,attitudes,beliefs,etc.Itislinkedtopsychometrics,asitcanassessthevalidityofaninstrumentbyfindingiftheinstrumentindeedmeasuresthepostulatedfactors.

Advantages

Reductionofnumberofvariables,bycombiningtwoormorevariablesintoasinglefactor.Forexample,performanceatrunning,ballthrowing,batting,jumpingandweightliftingcouldbecombinedintoasinglefactorsuchasgeneralathleticability.Usually,inanitembypeoplematrix,factorsareselectedbygroupingrelateditems.IntheQfactoranalysistechnique,thematrixistransposedandfactorsarecreatedbygroupingrelatedpeople:Forexample,liberals,libertarians,conservativesandsocialists,couldformseparategroups.Identificationofgroupsofinterrelatedvariables,toseehowtheyarerelatedtoeachother.Forexample,CarrollusedfactoranalysistobuildhisThreeStratumTheory.Hefoundthatafactorcalled"broadvisualperception"relatestohowgoodanindividualisatvisualtasks.Healsofounda"broadauditoryperception"factor,relatingtoauditorytaskcapability.Furthermore,hefoundaglobalfactor,called"g"orgeneralintelligence,thatrelatestoboth"broadvisualperception"and"broadauditoryperception".Thismeanssomeonewithahigh"g"islikelytohavebothahigh"visualperception"capabilityandahigh"auditoryperception"capability,andthat"g"thereforeexplainsagoodpartofwhysomeoneisgoodorbadinbothofthosedomains.

Disadvantages

"...eachorientationisequallyacceptablemathematically.Butdifferentfactorialtheoriesprovedtodifferasmuchintermsoftheorientationsoffactorialaxesforagivensolutionasintermsofanythingelse,sothatmodelfittingdidnotprovetobeusefulindistinguishingamongtheories."

(Sternberg,1977[15]).Thismeansallrotationsrepresentdifferentunderlyingprocesses,butallrotationsareequallyvalidoutcomesofstandardfactoranalysisoptimization.Therefore,itisimpossibletopicktheproperrotationusingfactoranalysisalone.Factoranalysiscanbeonlyasgoodasthedataallows.Inpsychology,whereresearchersoftenhavetorelyonlessvalidandreliablemeasuressuchasselfreports,thiscanbeproblematic.



Interpretingfactoranalysisisbasedonusinga"heuristic",whichisasolutionthatis"convenient

evenifnotabsolutelytrue".[16]Morethanoneinterpretationcanbemadeofthesamedatafactoredthesameway,andfactoranalysiscannotidentifycausality.

Exploratoryfactoranalysisversusprincipalcomponentsanalysis

Whileexploratoryfactoranalysisandprincipalcomponentanalysisaretreatedassynonymoustechniquesinsomefieldsofstatistics,thishasbeencriticised(e.g.Fabrigaretal.,1999[17]Suhr,2009[18]).Infactoranalysis,theresearchermakestheassumptionthatanunderlyingcausalmodelexists,whereasPCAissimplyavariablereductiontechnique.[19]Researchershavearguedthatthedistinctionsbetweenthetwotechniquesmaymeanthatthereareobjectivebenefitsforpreferringoneovertheotherbasedontheanalyticgoal.Ifthefactormodelisincorrectlyformulatedortheassumptionsarenotmet,thenfactoranalysiswillgiveerroneousresults.Factoranalysishasbeenusedsuccessfullywhereadequateunderstandingofthesystempermitsgoodinitialmodelformulations.Principalcomponentanalysisemploysamathematicaltransformationtotheoriginaldatawithnoassumptionsabouttheformofthecovariancematrix.TheaimofPCAistodetermineafewlinearcombinationsoftheoriginalvariablesthatcanbeusedtosummarizethedatasetwithoutlosingmuchinformation.[20]

ArgumentscontrastingPCAandEFA

Fabrigaretal.(1999)[17]addressanumberofreasonsusedtosuggestthatprincipalcomponentsanalysisisnotequivalenttofactoranalysis:

1. Itissometimessuggestedthatprincipalcomponentsanalysisiscomputationallyquickerandrequiresfewerresourcesthanfactoranalysis.Fabrigaretal.suggestthatthereadyavailabilityofcomputerresourceshaverenderedthispracticalconcernirrelevant.

2. PCAandfactoranalysiscanproducesimilarresults.ThispointisalsoaddressedbyFabrigaretal.incertaincases,wherebythecommunalitiesarelow(e.g.,.40),thetwotechniquesproducedivergentresults.Infact,Fabrigaretal.arguethatincaseswherethedatacorrespondtoassumptionsofthecommonfactormodel,theresultsofPCAareinaccurateresults.

3. Therearecertaincaseswherefactoranalysisleadsto'Heywoodcases'.Theseencompasssituationswhereby100%ormoreofthevarianceinameasuredvariableisestimatedtobeaccountedforbythemodel.Fabrigaretal.suggestthatthesecasesareactuallyinformativetotheresearcher,indicatingamisspecifiedmodeloraviolationofthecommonfactormodel.ThelackofHeywoodcasesinthePCAapproachmaymeanthatsuchissuespassunnoticed.

4. ResearchersgainextrainformationfromaPCAapproach,suchasanindividualsscoreonacertaincomponentsuchinformationisnotyieldedfromfactoranalysis.However,asFabrigaretal.contend,thetypicalaimoffactoranalysisi.e.todeterminethefactorsaccountingforthestructureofthecorrelationsbetweenmeasuredvariablesdoesnotrequireknowledgeoffactorscoresandthusthisadvantageisnegated.Itisalsopossibletocomputefactorscoresfromafactoranalysis.



Varianceversuscovariance

Factoranalysistakesintoaccounttherandomerrorthatisinherentinmeasurement,whereasPCAfailstodoso.ThispointisexemplifiedbyBrown(2009),[21]whoindicatedthat,inrespecttothecorrelationmatricesinvolvedinthecalculations:

"InPCA,1.00sareputinthediagonalmeaningthatallofthevarianceinthematrixistobeaccountedfor(includingvarianceuniquetoeachvariable,variancecommonamongvariables,anderrorvariance).Thatwould,therefore,bydefinition,includeallofthevarianceinthevariables.Incontrast,inEFA,thecommunalitiesareputinthediagonalmeaningthatonlythevariancesharedwithothervariablesistobeaccountedfor(excludingvarianceuniquetoeachvariableanderrorvariance).Thatwould,therefore,bydefinition,includeonlyvariancethatiscommonamongthevariables."

Brown(2009),PrincipalcomponentsanalysisandexploratoryfactoranalysisDefinitions,differencesandchoices

Forthisreason,Brown(2009)recommendsusingfactoranalysiswhentheoreticalideasaboutrelationshipsbetweenvariablesexist,whereasPCAshouldbeusedifthegoaloftheresearcheristoexplorepatternsintheirdata.

Differencesinprocedureandresults

ThedifferencesbetweenprincipalcomponentsanalysisandfactoranalysisarefurtherillustratedbySuhr(2009):

PCAresultsinprincipalcomponentsthataccountforamaximalamountofvarianceforobserved

variablesFAaccountforcommonvarianceinthedata.[18]

PCAinsertsonesonthediagonalsofthecorrelationmatrixFAadjuststhediagonalsofthe

correlationmatrixwiththeuniquefactors.[18]

PCAminimizesthesumofsquaredperpendiculardistancetothecomponentaxisFAestimates

factorswhichinfluenceresponsesonobservedvariables.[18]

ThecomponentscoresinPCArepresentalinearcombinationoftheobservedvariablesweightedbyeigenvectorstheobservedvariablesinFAarelinearcombinationsoftheunderlyingandunique

factors.[18]

InPCA,thecomponentsyieldedareuninterpretable,i.e.theydonotrepresentunderlyingconstructsinFA,theunderlyingconstructscanbelabeledandreadilyinterpreted,givenan

accuratemodelspecification.[18]

Factoranalysisinmarketing



Thebasicstepsare:

Identifythesalientattributesconsumersusetoevaluateproductsinthiscategory.Usequantitativemarketingresearchtechniques(suchassurveys)tocollectdatafromasampleofpotentialcustomersconcerningtheirratingsofalltheproductattributes.Inputthedataintoastatisticalprogramandrunthefactoranalysisprocedure.Thecomputerwillyieldasetofunderlyingattributes(orfactors).Usethesefactorstoconstructperceptualmapsandotherproductpositioningdevices.

Informationcollection

Thedatacollectionstageisusuallydonebymarketingresearchprofessionals.Surveyquestionsasktherespondenttorateaproductsampleordescriptionsofproductconceptsonarangeofattributes.Anywherefromfivetotwentyattributesarechosen.Theycouldincludethingslike:easeofuse,weight,accuracy,durability,colourfulness,price,orsize.Theattributeschosenwillvarydependingontheproductbeingstudied.Thesamequestionisaskedaboutalltheproductsinthestudy.ThedataformultipleproductsiscodedandinputintoastatisticalprogramsuchasR,SPSS,SAS,Stata,STATISTICA,JMP,andSYSTAT.

Analysis

Theanalysiswillisolatetheunderlyingfactorsthatexplainthedatausingamatrixofassociations.[22]Factoranalysisisaninterdependencetechnique.Thecompletesetofinterdependentrelationshipsisexamined.Thereisnospecificationofdependentvariables,independentvariables,orcausality.Factoranalysisassumesthatalltheratingdataondifferentattributescanbereduceddowntoafewimportantdimensions.Thisreductionispossiblebecausesomeattributesmayberelatedtoeachother.Theratinggiventoanyoneattributeispartiallytheresultoftheinfluenceofotherattributes.Thestatisticalalgorithmdeconstructstherating(calledarawscore)intoitsvariouscomponents,andreconstructsthepartialscoresintounderlyingfactorscores.Thedegreeofcorrelationbetweentheinitialrawscoreandthefinalfactorscoreiscalledafactorloading.

Advantages

Bothobjectiveandsubjectiveattributescanbeusedprovidedthesubjectiveattributescanbeconvertedintoscores.Factoranalysiscanidentifylatentdimensionsorconstructsthatdirectanalysismaynot.Itiseasyandinexpensive.

Disadvantages

Usefulnessdependsontheresearchers'abilitytocollectasufficientsetofproductattributes.Ifimportantattributesareexcludedorneglected,thevalueoftheprocedureisreduced.Ifsetsofobservedvariablesarehighlysimilartoeachotheranddistinctfromotheritems,factoranalysiswillassignasinglefactortothem.Thismayobscurefactorsthatrepresentmoreinteresting



WikimediaCommonshasmediarelatedtoFactoranalysis.

relationships.Namingfactorsmayrequireknowledgeoftheorybecauseseeminglydissimilarattributescancorrelatestronglyforunknownreasons.

Factoranalysisinphysicalandbiologicalsciences

Factoranalysishasalsobeenwidelyusedinphysicalsciencessuchasgeochemistry,ecology,hydrochemistry.,[23]astrophysics,cosmology,aswellasbiologicalsciencessuchasmolecularbiologyandbiochemistry.

Ingroundwaterqualitymanagement,itisimportanttorelatethespatialdistributionofdifferentchemicalparameterstodifferentpossiblesources,whichhavedifferentchemicalsignatures.Forexample,asulfidemineislikelytobeassociatedwithhighlevelsofacidity,dissolvedsulfatesandtransitionmetals.ThesesignaturescanbeidentifiedasfactorsthroughRmodefactoranalysis,andthelocationofpossiblesourcescanbesuggestedbycontouringthefactorscores.[24]

Ingeochemistry,differentfactorscancorrespondtodifferentmineralassociations,andthustomineralisation.[25]

Factoranalysisinmicroarrayanalysis

FactoranalysiscanbeusedforsummarizinghighdensityoligonucleotideDNAmicroarraysdataatprobelevelforAffymetrixGeneChips.Inthiscase,thelatentvariablecorrespondstotheRNAconcentrationinasample.[26]

Implementation

Factoranalysishasbeenimplementedinseveralstatisticalanalysisprogramssincethe1980s:SAS,BMDPandSPSS.[27]ItisalsoimplementedintheRprogramminglanguage(withthefactanalfunction),OpenOpt,andthestatisticalsoftwarepackageStata.RotationsareimplementedintheGPArotationRpackage.

Seealso

DesignofexperimentsFormalconceptanalysisHigherorderfactoranalysisIndependentcomponentanalysisNonnegativematrixfactorizationPerceptualmappingProductmanagementQmethodologyRecommendationsystem



[28]==References==

RecommendationsystemVarimaxrotationGeneralizedStructuredComponentAnalysis

1. ^Bartholomew,D.J.Steele,F.Galbraith,J.Moustaki,I.(2008).AnalysisofMultivariateSocialScienceData.StatisticsintheSocialandBehavioralSciencesSeries(2nded.).Taylor&Francis.ISBN1584889608.

2. âbcHarman,HarryH.(1976).ModernFactorAnalysis.UniversityofChicagoPress.pp.175,176.ISBN0226316521.

3. âbcdefghiPolitDFBeckCT(2012).NursingResearch:GeneratingandAssessingEvidenceforNursingPractice,9thed.Philadelphia,USA:WoltersKlowerHealth,LippincottWilliams&Wilkins.

4. ^Meng,J.(2011)."UncovercooperativegeneregulationsbymicroRNAsandtranscriptionfactorsinglioblastomausinganonnegativehybridfactormodel"(http://www.cmsworldwide.com/ICASSP2011/Papers/ViewPapers.asp?PaperNum=4439).InternationalConferenceonAcoustics,SpeechandSignalProcessing.

5. ^Liou,C.Y.Musicus,B.R.(2008)."CrossEntropyApproximationofStructuredGaussianCovarianceMatrices"(http://ieeexplore.ieee.org/xpl/articleDetails.jsp?reload=true&arnumber=4545272&contentType=Journals+%26+Magazines).IEEETransactionsonSignalProcessing56(7):33623367.doi:10.1109/TSP.2008.917878(https://dx.doi.org/10.1109%2FTSP.2008.917878).

6. ^Bandalos,D.L.BoehmKaufman,M.R.(2008)."Fourcommonmisconceptionsinexploratoryfactoranalysis"(http://books.google.com/books?id=KFAnkvqD8CgC&pg=PA61).InLance,CharlesE.Vandenberg,RobertJ.StatisticalandMethodologicalMythsandUrbanLegends:Doctrine,VerityandFableintheOrganizationalandSocialSciences.Taylor&Francis.pp.6187.ISBN9780805862379.

7. ^Larsen,R.,&Warne,R.T.(2010).Estimatingconfidenceintervalsforeigenvaluesinexploratoryfactoranalysis.BehaviorResearchMethods,42,871876.doi:10.3758/BRM.42.3.871

8. âbWarne,R.T.,&Larsen,R.(2014).EvaluatingaproposedmodificationoftheGuttmanrulefordeterminingthenumberoffactorsinanexploratoryfactoranalysis.PsychologicalTestandAssessmentModeling,56,104123.

9. ^*Ledesma,R.D.ValeroMora,P.(2007)."DeterminingtheNumberofFactorstoRetaininEFA:AneasytousecomputerprogramforcarryingoutParallelAnalysis"(http://pareonline.net/getvn.asp?v=12&n=2).PracticalAssessmentResearch&Evaluation12(2):111.

10. ^Velicer,W.F.(1976)."Determiningthenumberofcomponentsfromthematrixofpartialcorrelations".Psychometrika41:321327.doi:10.1007/bf02293557(https://dx.doi.org/10.1007%2Fbf02293557).

11. ^Ruscio,JohnRoche,B.(2012)."Determiningthenumberoffactorstoretaininanexploratoryfactoranalysisusingcomparisondataofknownfactorialstructure".PsychologicalAssessment24:282292.doi:10.1037/a0025697(https://dx.doi.org/10.1037%2Fa0025697).

12. ^Garrido,L.E.,&Abad,F.J.,&Ponsoda,V.(2012).AnewlookatHorn'sparallelanalysiswithordinalvariables.PsychologicalMethods.Advanceonlinepublication.doi:10.1037/a0030005

13. ^Courtney,M.G.R.(2013).DeterminingthenumberoffactorstoretaininEFA:UsingtheSPSSRMenuv2.0tomakemorejudiciousestimations.PracticalAssessment,ResearchandEvaluation,18(8).Availableonline:



tomakemorejudiciousestimations.PracticalAssessment,ResearchandEvaluation,18(8).Availableonline:http://pareonline.net/getvn.asp?v=18&n=8

14. ^Russell,D.W.(December2002)."Insearchofunderlyingdimensions:Theuse(andabuse)offactoranalysisinPersonalityandSocialPsychologyBulletin"(http://psp.sagepub.com/content/28/12/1629.short).PersonalityandSocialPsychologyBulletin28(12):162946.doi:10.1177/014616702237645(https://dx.doi.org/10.1177%2F014616702237645).

15. ^Sternberg,R.J.(1977).MetaphorsofMind:ConceptionsoftheNatureofIntelligence.NewYork:CambridgeUniversityPress.pp.85111.

16. ^RichardB.Darlington(2004)"FactorAnalysis"(http://comp9.psych.cornell.edu/Darlington/factor.htm).RetrievedJuly22,2004.

17. âbFabrigaretal.(1999)."Evaluatingtheuseofexploratoryfactoranalysisinpsychologicalresearch."(http://www.statpower.net/Content/312/Handout/Fabrigar1999.pdf).PsychologicalMethods.

18. âbcdefSuhr,Diane(2009)."Principalcomponentanalysisvs.exploratoryfactoranalysis"(http://www2.sas.com/proceedings/sugi30/20330.pdf).SUGI30Proceedings.Retrieved5April2012.

19. ^SASStatistics."PrincipalComponentsAnalysis"(http://support.sas.com/publishing/pubcat/chaps/55129.pdf).SASSupportTextbook.

20. ^Meglen,R.R.(1991)."ExaminingLargeDatabases:AChemometricApproachUsingPrincipalComponentAnalysis".JournalofChemometrics5(3):163179.doi:10.1002/cem.1180050305/(https://dx.doi.org/10.1002%2Fcem.1180050305%2F).

21. ^Brown,J.D.(January2009)."PrincipalcomponentsanalysisandexploratoryfactoranalysisDefinitions,differencesandchoices."(http://jalt.org/test/PDF/Brown29.pdf).Shiken:JALTTesting&EvaluationSIGNewsletter.Retrieved16April2012.

22. ^Ritter,N.(2012).Acomparisonofdistributionfreeandnondistributionfreemethodsinfactoranalysis.PaperpresentedatSouthwesternEducationalResearchAssociation(SERA)Conference2012,NewOrleans,LA(ED529153).

23. ^Subbarao,C.Subbarao,N.V.Chandu,S.N.(December1996)."Characterisationofgroundwatercontaminationusingfactoranalysis".EnvironmentalGeology28(4):175180.doi:10.1007/s002540050091(https://dx.doi.org/10.1007%2Fs002540050091).

24. ^Love,D.Hallbauer,D.K.Amos,A.Hranova,R.K.(2004)."Factoranalysisasatoolingroundwaterqualitymanagement:twosouthernAfricancasestudies".PhysicsandChemistryoftheEarth29:113543.doi:10.1016/j.pce.2004.09.027(https://dx.doi.org/10.1016%2Fj.pce.2004.09.027).

25. ^Barton,E.S.Hallbauer,D.K.(1996)."TraceelementandUPbisotopecompositionsofpyritetypesintheProterozoicBlackReef,TransvaalSequence,SouthAfrica:Implicationsongenesisandage".ChemicalGeology133:173199.doi:10.1016/S00092541(96)000757(https://dx.doi.org/10.1016%2FS00092541%2896%29000757).

26. ^Hochreiter,SeppClevert,DjorkArnObermayer,Klaus(2006)."Anewsummarizationmethodforaffymetrixprobeleveldata"(http://bioinformatics.oxfordjournals.org/content/22/8/943.full).Bioinformatics22(8):9439.doi:10.1093/bioinformatics/btl033(https://dx.doi.org/10.1093%2Fbioinformatics%2Fbtl033).PMID16473874(https://www.ncbi.nlm.nih.gov/pubmed/16473874).

27. ^MacCallum,Robert(June1983)."AcomparisonoffactoranalysisprogramsinSPSS,BMDP,andSAS".Psychometrika48(48).doi:10.1007/BF02294017(https://dx.doi.org/10.1007%2FBF02294017).

28. Îshida,E.E.O&deSouza,R.S..Hubbleparameterreconstructionfromaprincipalcomponentanalysis:



Furtherreading

Child,Dennis(2006).TheEssentialsofFactorAnalysis(http://books.google.com/books?id=rQ2vdJgohH0C)(3rded.).ContinuumInternational.ISBN9780826480002.Fabrigar,L.R.Wegener,D.T.MacCallum,R.C.Strahan,E.J.(September1999)."Evaluatingtheuseofexploratoryfactoranalysisinpsychologicalresearch"(http://psycnet.apa.org/journals/met/4/3/272/).PsychologicalMethods4(3):272299.doi:10.1037/1082989X.4.3.272(https://dx.doi.org/10.1037%2F1082989X.4.3.272).

Jennrich,RobertI.,"RotationtoSimpleLoadingsUsingComponentLossFunction:TheObliqueCase,"Psychometrika,Vol.71,No.1,pp.173191,March2006.

Katz,JeffreyOwen,andRohlf,F.James.Primaryproductfunctionplane:Anobliquerotationtosimplestructure.MultivariateBehavioralResearch,April1975,Vol.10,pp.219232.

Katz,JeffreyOwen,andRohlf,F.James.Functionplane:Anewapproachtosimplestructurerotation.Psychometrika,March1974,Vol.39,No.1,pp.3751.

Katz,JeffreyOwen,andRohlf,F.James.Functionpointclusteranalysis.SystematicZoology,September1973,Vol.22,No.3,pp.295301.

Thompson,B.(2004).Exploratoryandconfirmatoryfactoranalysis:Understandingconceptsandapplications.WashingtonDC:AmericanPsychologicalAssociation.ISBN1591470935.

Externallinks

FactorAnalysis.RetrievedJuly23,2004,RaymondCattell.RetrievedJuly22,2004,fromhttp://www.indiana.edu/~intell/rcattell.shtmlExploratoryFactorAnalysisABookManuscriptbyTucker,L.&MacCallumR.(1993).RetrievedJune8,2006,from:http://www.unc.edu/~rcm/book/factornew.htmGarson,G.David,"FactorAnalysis,"fromStatnotes:TopicsinMultivariateAnalysis.RetrievedonApril13,2009fromhttp://www2.chass.ncsu.edu/garson/pa765/statnote.htmFactorAnalysisat100(http://www.fa100.info/index.html)conferencematerialFARMSFactorAnalysisforRobustMicroarraySummarization,anRpackage(http://www.bioinf.jku.at/software/farms/farms.html)software

Retrievedfrom"http://en.wikipedia.org/w/index.php?title=Factor_analysis&oldid=639737268"

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