人大陈卫教授手把手教你统计15 MCA

Lecture13

MultipleClassificationAnalysis(MCA)1ThisLectureCoversTheMCAastheequivalentofamultipleregressionanalysisMCAadaptedtologisticregression2

Multipleclassificationanalysis(MCA)isusedtoexaminetheeffectofeachindependentvariableonthedependentvariablewhilecontrollingfortheeffectsoftheotherindependentvariables,whenthedependentvariableisanquantitativevariableandtheindependentvariablesarecategorical.3

MCAismosteasilyexplainedasmultipleregressionwithdummyvariables.Thus,thedependentvariableisaquantitativevariable,andtheindependentvariablesarecategoricalvariables,representedbydummyvariables.4

MCAwithonecategoricalpredictorvariableisequivalenttoone-wayanalysisofvariance;MCAwithtwocategoricalpredictorvariablesisequivalenttotwo-wayanalysisofvariance;andsoon.5ControlvariablesmaybeaddedtotheMCAmodel.Whenquantitativevariables,inadditiontocategoricalvariables,areincludedamongthepredictorvariables,MCAisequivalenttowhatiscommonlycalledanalysisofcovariance.6

MCAcanbeextendedbeyondmultipleregression,forexample,itcanbeextendedtologisticregressionandCoxregression.7MCAspecifiesthelinearmodelasfollows:8MCAisastatisticaloptionwiththeANOVAprocedure9AnexampleToexaminetheeffectsofplaceofresidence,ethnicityandeducationlevelonwomen’sageatfirstmarriageANOVAVARIABLES=afmBYuandr(1,2)ethnic(1,2)educat(1,5)/MAXORDERSNONE/STATISTICSMEANMCA/METHODEXPERIM.10111213TheMCAoutputshowstheestimated(orpredicted)meansofthedependentvariableforeachcategoryoftheexplanatoryvariables,unadjustedandadjustedfortheeffectsoftheotherexplanatoryvariablesinthemodel.Italsoshowstheunadjustedandadjusteddeviationsfromthegrandmean.Therefore,thepredictedmeanminusthedeviationforeachcategoryisalwaysthesameandisequaltothegrandmean.

predictedmean–deviation=grandmeanThe“deviationsadjustedforfactors”areequivalenttotheb1,b2,…bncoefficientsaftercontrollingfortheeffectoftheotherexplanatoryvariables.14Eachcombinationofthebcoefficientswillgivetheestimatedvalueofthedependentvariableforarespondentwiththecorrespondingcharacteristics.15Acomparisonoftheunadjustedandadjustedmeansforeachcategoryoftheindependentvariablesshowswhathappenswhenanadjustmentismadefortheeffectsoftheothervariables.Thelargertherangeinthedeviationsamongthecategoriesofeachexplanatoryvariable,thegreaterthesignificanceofthatfactorinaffectingthedependentvariable.16Ifwerunmultipleregression1718

Multipleclassificationanalysisisanextensionofmultipleregressionthatallowsustouseregressioncoefficientstopredictthemeanvaluecontrollingfortheeffectofotherpredictorsinthemodel.MCAissimplyawayofsolvingtheregressionequation.

19UnadjustedmeansUnadjustedmeanscanbeobtainedfromsimplelinearregression:“unadjusted”means“withoutcontrols”:20AdjustedmeansAdjustedmeansareobtainedfrommultiplelinearregression:“adjusted”means“withcontrols”:21AdjustedmeansWhenwecalculatemeanageatfirstmarriagebyresidence,ethnicityandeducationarethecontrolvariables;Whenwecalculatemeanageatfirstmarriagebyethnicity,residenceandeducationarethecontrolvariables;Whenwecalculatemeanageatfirstmarriagebyeducation,residenceandethnicityarethecontrolvariables.22AdjustedmeansIngeneral,whenweconsideradjustedmeansforonepredictorvariable,alltheothervariablesarethecontrolvariables.Statisticalcontrolsareintroducedbyholdingthecontrolvariablesconstantattheirmeanvalues.23HoldingthecontrolvariablesconstantattheirmeanvaluesForacontinuousvariable,suchasage,themeanvalueseemsanappropriatemeasure.Inessencewearecontrollingfortheaverageexperienceorperson.However,theuseofameanforadichotomous‘dummy’variableseemsartificialaseachindividualcanhaveavalueof0or1butnoonecanhaveavalueinbetween0and1.However,ifwethinkofthemeanofadichotomousvariableasaproportionitsusemakesmoresense.Herewearecontrollingfortheproportionofmales(forsex)ortheproportionofpeoplewithcollegeeducationetc.

24TosolvetheregressionequationusingExcelThefollowingexamplepresentsawayofsettingupanMCAcalculationtableinExcel.

Weneedthecoefficientsoftheregressionmodel.ThesecanbecopiedstraightfromtheSPSSoutputintoExcel.ThenweneedtogeneratethemeanvalueofeachindependentvariableandcopythesevaluesintoExcel.Wegivetheconstantavalueof1inthemeancolumnsothatwhenwemultiplythecolumnsitremainsattheoriginalvalue.

2526IncludingintervalindependentvariablesinMCAIntervalscalevariablescanbeincludedinMCAascovariates.Theirroleinthemodelcanbeviewedintwowaysconceptually:

●asacontrolvariable

●asanotherexplanatoryvariable27SPSSSyntax28AnexampleToexaminetheeffectsofplaceofresidence,ethnicityandeducationlevelonwomen’sageatfirstmarriage,whencontrollingforwomen’sageANOVAVARIABLES=afmBYuandr(1,2)ethnic(1,2)educat(1,5)WITHage/COVARIATESWITH/MAXORDERSNONE/STATISTICSMEANMCAREG/METHODEXPERIM.29303132333435Thethreeequationsarethesame:36MCAAdaptedto

LogisticRegressionWhenMCAisadaptedtologisticregression,bothunadjustedandadjustedvaluesoftheresponsevariablecanbecalculated,justasinordinaryMCA.37Theunadjustedvaluesarebasedonlogisticregressionthatincorporateonepredictorvariableatatime,andtheadjustedvaluesarebasedonthecompletemodelincludingallpredictorvariablessimultaneously.38UnfortunatelySPSSdoesnotincludeMCAprogramsforlogisticregression,wehavetoconstructtheMCAtablesfromtheunderlyinglogisticregressions.39AnIllustrativeExampleToillustratehowMCAcanbeadaptedtologisticregression,weconsidertheabortionexample:theabortionuseeffectofage,pregnancy,residence,ethnicity,andeducation.40VariablesP:estimatedprobabilityofabortionuseAgeisclassifiedintotwobroadgroups:15-34and35-49Numberofpregnanciesinthreecategories:1,2,and3andoverResidence,ethnicity,andeducationcategorizedasformerly4142Whencalculatingthevalueoffromthefittedregressionforeachcategoryofaparticularindependentvariable,dummyvaluesofthatvariablearesettobecombinationsofonesandzeroswhileallothervariablesarecontrolledbyholdingthemconstantattheirmeanvalues.

43AdjustedPExponentiationofthevaluesofyieldstheadjustedvaluesof()

AdjustedvaluesofParecalculatedas:44AnimportantpointUnlikeOLSregressioninwhichsubstitutionofmeanvaluesfortheindependentvariablesalwaysyieldsthemeanvalueofthedependentvariable,thisisgenerallynotthecaseforlogisticregression.45Inlinearregression,substitutionofmeanvaluesfortheindependentvariablesalwaysyieldsthemeanvalueofthedependentvariable46Thisiscalculatedfromtheregressionequation47OverallmeanageatfirstmarriagecalculatedfromthesampleThisisdirectlycalculatedfromthesample48However,thevalueofoverallPproducedfromthelogisticregressionequationisgenerallynot