FactorAnalysis.docx
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FactorAnalysis.docx
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FactorAnalysis
Factoranalysis
FromWikipedia,the freeencyclopedia
Factoranalysis isa statistical methodusedtodescribe variability amongobserved,correlated variables intermsofapotentiallylowernumberofunobservedvariablescalled factors.Forexample,itispossiblethatvariationsinfourobservedvariablesmainlyreflectthevariationsintwounobservedvariables. Factoranalysis searchesforsuchjointvariationsinresponsetounobserved latentvariables.Theobservedvariablesaremodelledas linearcombinations ofthepotentialfactors,plus"error"terms.Theinformationgainedabouttheinterdependenciesbetweenobservedvariablescanbeusedlatertoreducethesetofvariablesinadataset.Computationallythistechniqueisequivalentto lowrankapproximation ofthematrixofobservedvariables.Factoranalysisoriginatedinpsychometrics,andisusedin behavioralsciences, socialsciences, marketing, productmanagement, operationsresearch,andother appliedsciences thatdealwithlargequantitiesof data.
Factoranalysisisrelatedto principalcomponentanalysis (PCA),butthetwoarenotidentical. Latentvariablemodels,includingfactoranalysis,use regressionmodelling techniquestotesthypothesesproducingerrorterms,whilePCAisadescriptivestatisticaltechnique.[1] Therehasbeensignificantcontroversyinthefieldovertheequivalenceorotherwiseofthetwotechniques(seeexploratoryfactoranalysisversusprincipalcomponentsanalysis).[citationneeded]
Contents
[hide]
∙1 Statisticalmodel
o1.1 Definition
o1.2 Example
o1.3 Mathematicalmodelofthesameexample
∙2 Practicalimplementation
o2.1 Typeoffactoranalysis
o2.2 Typesoffactoring
o2.3 Terminology
o2.4 Criteriafordeterminingthenumberoffactors
o2.5 Rotationmethods
∙3 Factoranalysisinpsychometrics
o3.1 History
o3.2 Applicationsinpsychology
o3.3 Advantages
o3.4 Disadvantages
∙4 Exploratoryfactoranalysisversusprincipalcomponentsanalysis
o4.1 ArgumentscontrastingPCAandEFA
o4.2 Varianceversuscovariance
o4.3 Differencesinprocedureandresults
∙5 Factoranalysisinmarketing
o5.1 Informationcollection
o5.2 Analysis
o5.3 Advantages
o5.4 Disadvantages
∙6 Factoranalysisinphysicalsciences
∙7 Factoranalysisinmicroarrayanalysis
∙8 Implementation
∙9 Seealso
∙10 References
∙11 Furtherreading
∙12 Externallinks
Statisticalmodel[edit]
Definition[edit]
Supposewehaveasetof
observable randomvariables,
withmeans
.
Supposeforsomeunknownconstants
and
unobservedrandomvariables
where
and
where
wehave
Here,the
areindependentlydistributederrortermswithzeromeanandfinitevariance,whichmaynotbethesameforall
.Let
sothatwehave
Inmatrixterms,wehave
Ifwehave
observations,thenwewillhavethedimensions
and
.Eachcolumnof
and
denotevaluesforoneparticularobservation,andmatrix
doesnotvaryacrossobservations.
Alsowewillimposethefollowingassumptionson
.
1.
and
areindependent.
2.
3.
(tomakesurethatthefactorsareuncorrelated)
Anysolutionoftheabovesetofequationsfollowingtheconstraintsfor
isdefinedasthe factors,and
asthe loadingmatrix.
Suppose
.Thennotethatfromtheconditionsjustimposedon
wehave
or
or
Notethatforany orthogonalmatrix
ifweset
and
thecriteriaforbeingfactorsandfactorloadingsstillhold.Henceasetoffactorsandfactorloadingsisidenticalonlyuptoorthogonaltransformation.
Example[edit]
Thefollowingexampleisforexpositorypurposes,andshouldnotbetakenasbeingrealistic.Supposeapsychologistproposesatheorythattherearetwokindsof intelligence,"verbalintelligence"and"mathematicalintelligence",neitherofwhichisdirectlyobserved. Evidence forthetheoryissoughtintheexaminationscoresfromeachof10differentacademicfieldsof1000students.Ifeachstudentischosenrandomlyfromalarge population,theneachstudent's10scoresarerandomvariables.Thepsychologist'stheorymaysaythatforeachofthe10academicfields,thescoreaveragedoverthegroupofallstudentswhosharesomecommonpairofvaluesforverbalandmathematical"intelligences"issome constant timestheirlevelofverbalintelligenceplusanother constant timestheirlevelofmathematicalintelligence,i.e.,itisacombinationofthosetwo"factors".Thenumbersforaparticularsubject,bywhichthetwokindsofintelligencearemultipliedtoobtaintheexpectedscore,arepositedbythetheorytobethesameforallintelligencelevelpairs,andarecalled "factorloadings" forthissubject.Forexample,thetheorymayholdthattheaveragestudent'saptitudeinthefieldof taxonomy is
{10×thestudent'sverbalintelligence}+{6×thestudent'smathematicalintelligence}.
Thenumbers10and6arethefactorloadingsassociatedwithtaxonomy.Otheracademicsubjectsmayhavedifferentfactorloadings.
Twostudentshavingidenticaldegreesofverbalintelligenceandidenticaldegreesofmathematicalintelligencemayhavedifferentaptitudesintaxonomybecauseindividualaptitudesdifferfromaverageaptitudes.Thatdifferenceiscalledthe"error"—astatisticaltermthatmeanstheamountbywhichanindividualdiffersfromwhatisaverageforhisorherlevelsofintelligence(see errorsandresidualsinstatistics).
Theobservabledatathatgointofactoranalysiswouldbe10scoresofeachofthe1000students,atotalof10,000numbers.Thefactorloadingsandlevelsofthetwokindsofintelligenceofeachstudentmustbeinferredfromthedata.
Mathematicalmodelofthesameexample[edit]
Intheexampleabove,for i =1,...,1,000the ithstudent'sscoresare
where
∙xk,i isthe ithstudent'sscoreforthe kthsubject
∙
isthemeanofthestudents'scoresforthe kthsubject(assumedtobezero,forsimplicity,intheexampleasdescribedabove,whichwouldamounttoasimpleshiftofthescaleused)
∙vi isthe ithstudent's"verbalintelligence",
∙mi isthe ithstudent's"mathematicalintelligence",
∙
arethefactorloadingsforthe kthsubject,for j =1,2.
∙εk,i isthedifferencebetweenthe ithstudent'sscoreinthe kthsubjectandtheaveragescoreinthe kthsubjectofallstudentswhoselevelsofverbalandmathematicalintelligencearethesameasthoseofthe ithstudent,
In matrix notation,wehave
∙N is1000students
∙X isa10×1,000matrixof observable randomvariables,
∙μisa10×1columnvectorof unobservable constants(inthiscase"constants"arequantitiesnotdifferingfromoneindividualstudenttothenext;and"randomvariables"arethoseassignedtoindividualstudents;therandomnessarisesfromtherandomwayinwhichthestudentsarechosen).Notethat,
isan outerproduct ofμwitha1×1000rowvectorofones,yieldinga10×1000matrixoftheelementsofμ,
∙L isa10×2matrixoffactorloadings(unobservable constants,tenacademictopics,eachwithtwointelligenceparametersthatdeterminesuccessinthattopic),
∙F isa2×1,000matrixof unobservable randomvariables(twointelligenceparametersforeachof1000students),
∙εisa10×1,000matrixof unobservable randomvariables.
Observethatbydoublingthescaleonwhich"verbalintelligence"—thefirstcomponentineachcolumnof F—ismeasured,andsimultaneouslyhalvingthefactorloadingsforverbalintelligencemakesnodifferencetothemodel.Thus,nogeneralityislostbyassumingthatthestandarddeviationofverbalintelligenceis1.Likewiseformathematicalintelligence.Moreover,forsimilarreasons,nogeneralityislostbyassumingthetwofactorsare uncorrelated witheachother.The"errors"εaretakentobeindependentofeachother.Thevariancesofthe"errors"associatedwiththe10differentsubjectsarenotassumedtobeequal.
Notethat,sinceanyrotationofasolutionisalsoasolution,thismakesinterpretingthefactorsdifficult.Seedisadvantagesbelow.Inthisparticularexample,ifwedonotknowbeforehandthatthetwotypesofintelligenceareuncorrelated,thenwecannotinterpretthetwofactorsasthetwodifferenttypesofintelligence.Eveniftheyareuncorrelated,wecannottellwhichfactorcorrespondstoverbalintelligenceandwhichcorrespondstomathematicalintelligencewithoutanoutsideargument.
Thevaluesoftheloadings L,theaveragesμ,andthe variances ofthe"errors"εmustbeestimatedgiventheobserveddata X and F (theassumptionaboutthelevelsofthefactorsisfixedforagiven F).
Practicalimplementation[edit]
Thissection needsadditionalcitationsfor verification. Pleasehelp improvethisarticle by addingcitationstoreliablesources.Unsourcedmaterialmaybechallengedandremoved. (April2012)
Typeoffactoranalysis[edit]
Exploratoryfactoranalysis (EFA) isusedtoidentifycomplexinterrelationshipsamongitemsandgroupitemsthatarepartofunifiedconcepts.[2] Theresearchermakesno"apriori"assumptionsaboutrelationshipsamongfactors.[2]
Confirmatoryfactoranalysis (CFA) isamorecomplexapproachthatteststhehypothesisthattheitemsareassociatedwithspecificfactors.[2] CFAuses structuralequationmodeling to test ameasurementmodelwherebyloadingonthefactorsallowsforevaluationofrelationshipsbetweenobservedvariablesandunobservedvariables.[2] Structuralequationmodelingapproachescanaccommodatemeasurementerror,andarelessrestrictivethan least-squaresestimation.[2] Hypothesizedmodelsaretestedagainstactualdata,andtheanalysiswoulddemonstrateloadingsofobservedvariablesonthelatentvariables(factors),aswellasthecorrelationbetweenthelatentvariables.[2]
Typesoffactoring[edit]
Pr
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