FactorAnalysisWord文件下载.docx

FactorAnalysisWord文件下载.docx

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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 

o5.4 

∙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 

wehave

Here,the 

areindependentlydistributederrortermswithzeromeanandfinitevariance,whichmaynotbethesameforall 

.Let 

sothatwehave

Inmatrixterms,wehave

Ifwehave 

observations,thenwewillhavethedimensions 

and 

.Eachcolumnof 

denotevaluesforoneparticularobservation,andmatrix 

doesnotvaryacrossobservations.

Alsowewillimposethefollowingassumptionson 

1.

areindependent.

2.

3.

（tomakesurethatthefactorsareuncorrelated）

Anysolutionoftheabovesetofequationsfollowingtheconstraintsfor 

isdefinedasthe 

factors,and 

asthe 

loadingmatrix.

Suppose 

.Thennotethatfromtheconditionsjustimposedon 

or

Notethatforany 

orthogonalmatrix 

ifweset 

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 

timestheirlevelofmathematicalintelligence,i.e.,itisacombinationofthosetwo"

factors"

.Thenumbersforaparticularsubject,bywhichthetwokindsofintelligencearemultipliedtoobtaintheexpectedscore,arepositedbythetheorytobethesameforallintelligencelevelpairs,andarecalled 

"

factorloadings"

forthissubject.Forexample,thetheorymayholdthattheaveragestudent'

saptitudeinthefieldof 

taxonomy 

is

{10×

thestudent'

sverbalintelligence}+{6×

smathematicalintelligence}.

Thenumbers10and6arethefactorloadingsassociatedwithtaxonomy.Otheracademicsubjectsmayhavedifferentfactorloadings.

Twostudentshavingidenticaldegreesofverbalintelligenceandidenticaldegreesofmathematicalintelligencemayhavedifferentaptitudesintaxonomybecauseindividualaptitudesdifferfromaverageaptitudes.Thatdifferenceiscalledthe"

—astatisticaltermthatmeanstheamountbywhichanindividualdiffersfromwhatisaverageforhisorherlevelsofintelligence（see 

errorsandresidualsinstatistics）.

Theobservabledatathatgointofactoranalysiswouldbe10scoresofeachofthe1000students,atotalof10,000numbers.Thefactorloadingsandlevelsofthetwokindsofintelligenceofeachstudentmustbeinferredfromthedata.

Mathematicalmodelofthesameexample[edit]

Intheexampleabove,for 

i 

=1,...,1,000the 

ithstudent'

sscoresare

where

∙xk,i 

isthe 

sscoreforthe 

kthsubject

∙

isthemeanofthestudents'

scoresforthe 

kthsubject（assumedtobezero,forsimplicity,intheexampleasdescribedabove,whichwouldamounttoasimpleshiftofthescaleused）

∙vi 

s"

∙mi 

arethefactorloadingsforthe 

kthsubject,for 

j 

=1,2.

∙εk,i 

isthedifferencebetweenthe 

sscoreinthe 

kthsubjectandtheaveragescoreinthe 

kthsubjectofallstudentswhoselevelsofverbalandmathematicalintelligencearethesameasthoseofthe 

ithstudent,

In 

matrix 

notation,wehave

∙N 

is1000students

∙X 

isa10×

1,000matrixof 

randomvariables,

∙μisa10×

1columnvectorof 

unobservable 

constants（inthiscase"

constants"

arequantitiesnotdifferingfromoneindividualstudenttothenext;

randomvariables"

arethoseassignedtoindividualstudents;

therandomnessarisesfromtherandomwayinwhichthestudentsarechosen）.Notethat, 

isan 

outerproduct 

ofμwitha1×

1000rowvectorofones,yieldinga10×

1000matrixoftheelementsofμ,

∙L 

2matrixoffactorloadings（unobservable 

constants,tenacademictopics,eachwithtwointelligenceparametersthatdeterminesuccessinthattopic）,

∙F 

isa2×

randomvariables（twointelligenceparametersforeachof1000students）,

∙εisa10×

randomvariables.

Observethatbydoublingthescaleonwhich"

—thefirstcomponentineachcolumnof 

F—ismeasured,andsimultaneouslyhalvingthefactorloadingsforverbalintelligencemakesnodifferencetothemodel.Thus,nogeneralityislostbyassumingthatthestandarddeviationofverbalintelligenceis1.Likewiseformathematicalintelligence.Moreover,forsimilarreasons,nogeneralityislostbyassumingthetwofactorsare 

uncorrelated 

witheachother.The"

errors"

εaretakentobeindependentofeachother.Thevariancesofthe"

associatedwiththe10differentsubjectsarenotassumedtobeequal.

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

Thevaluesoftheloadings 

L,theaveragesμ,andthe 

variances 

ofthe"

εmustbeestimatedgiventheobserveddata 

X 

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