傾向評分配對簡介培訓(xùn)課件

傾向評分配對簡介OutlineDay1Overview:WhyPSM?HistoryanddevelopmentofPSMCounterfactualframeworkThefundamentalassumptionGeneralprocedureSoftwarepackagesReview&illustrationofthebasicmethodsdevelopedbyRosenbaumandRubin2傾向評分配對簡介Outline(continued)ReviewandillustrationofHeckman’sdifference-in-differencesmethodProblemswiththeRosenbaum&Rubin’smethodDifference-in-differencesmethodNonparametricregressionBootstrappingDay2Practicalissues,concerns,andstrategiesQuestionsanddiscussions3傾向評分配對簡介PSMReferencesCheckwebsite:/VRC/Lectures/index.htm(Linktofile“Day1b.doc”)4傾向評分配對簡介WhyPSM?(1)Need1:AnalyzecausaleffectsoftreatmentfromobservationaldataObservationaldata-thosethatarenotgeneratedbymechanismsofrandomizedexperiments,suchassurveys,administrativerecords,andcensusdata.Toanalyzesuchdata,anordinaryleastsquare(OLS)regressionmodelusingadichotomousindicatoroftreatmentdoesnotwork,becauseinsuchmodeltheerrortermiscorrelatedwithexplanatoryvariable.

5傾向評分配對簡介WhyPSM?(2)Theindependentvariablewisusuallycorrelatedwiththeerrorterm.Theconsequenceisinconsistentandbiasedestimateaboutthetreatmenteffect.6傾向評分配對簡介WhyPSM?(3)Need2:RemovingSelectionBiasinProgramEvaluationFisher’srandomizationidea.Whethersocialbehavioralresearchcanreallyaccomplishrandomizedassignmentoftreatment?ConsiderE(Y1|W=1)–E(Y0|W=0).AddandsubtractE(Y0|W=1),wehave{E(Y1|W=1)–E(Y0|W=1)}+{E(Y0|W=1)-E(Y0|W=0)}

Crucial:E(Y0|W=1)E(Y0|W=0)

Thedebateamongeducationresearchers:theimpactofCatholicschoolsvis-à-vispublicschoolsonlearning.TheCatholicschooleffectisthestrongestamongthoseCatholicstudentswhoarelesslikelytoattendCatholicschools(Morgan,2001).7傾向評分配對簡介WhyPSM?(4)Heckman&Smith(1995)FourImportantQuestions:

Whataretheeffectsoffactorssuchassubsidies,advertising,locallabormarkets,familyincome,race,andsexonprogramapplicationdecision?Whataretheeffectsofbureaucraticperformancestandards,locallabormarketsandindividualcharacteristicsonadministrativedecisionstoacceptapplicantsandplacetheminspecificprograms?Whataretheeffectsoffamilybackground,subsidiesandlocalmarketconditionsondecisionstodropoutfromaprogramandonthelengthoftimetakentocompleteaprogram?Whatarethecostsofvariousalternativetreatments?8傾向評分配對簡介HistoryandDevelopmentofPSMThelandmarkpaper:Rosenbaum&Rubin(1983).Heckman’searlyworkinthelate1970sonselectionbiasandhiscloselyrelatedworkondummyendogenousvariables(Heckman,1978)addressthesameissueofestimatingtreatmenteffectswhenassignmentisnonrandom.Heckman’sworkonthedummyendogenousvariableproblemandtheselectionmodelcanbeunderstoodasageneralizationofthepropensity-scoreapproach(Winship&Morgan,1999).Inthe1990s,Heckmanandhiscolleaguesdevelopeddifference-in-differencesapproach,whichisasignificantcontributiontoPSM.Ineconomics,theDIDapproachanditsrelatedtechniquesaremoregenerallycallednonexperimentalevaluation,oreconometricsofmatching.9傾向評分配對簡介TheCounterfactualFrameworkCounterfactual:whatwouldhavehappenedtothetreatedsubjects,hadtheynotreceivedtreatment?Thekeyassumptionofthecounterfactualframeworkisthatindividualsselectedintotreatmentandnontreatmentgroupshavepotentialoutcomesinbothstates:theoneinwhichtheyareobservedandtheoneinwhichtheyarenotobserved(Winship&Morgan,1999).Forthetreatedgroup,wehaveobservedmeanoutcomeundertheconditionoftreatmentE(Y1|W=1)andunobservedmeanoutcomeundertheconditionofnontreatmentE(Y0|W=1).Similarly,forthenontreatedgroupwehavebothobservedmeanE(Y0|W=0)andunobservedmeanE(Y1|W=0).10傾向評分配對簡介TheCounterfactualFramework(Continued)Underthisframework,anevaluationofE(Y1|W=1)-E(Y0|W=0)canbethoughtasaneffortthatusesE(Y0|W=0)toestimatethecounterfactualE(Y0|W=1).ThecentralinterestoftheevaluationisnotinE(Y0|W=0),butinE(Y0|W=1).Therealdebateabouttheclassicalexperimentalapproachcentersonthequestion:whetherE(Y0|W=0)reallyrepresentsE(Y0|W=1)?

11傾向評分配對簡介FundamentalAssumption

Rosenbaum&Rubin(1983)Differentversions:“unconfoundedness”&“ignorabletreatmentassignment”(Rosenbaum&Robin,1983),“selectiononobservables”(Barnow,Cain,&Goldberger,1980),“conditionalindependence”(Lechner1999,2002),and“exogeneity”(Imbens,2004)12傾向評分配對簡介1-to-1or1-to-nMatch

NearestneighbormatchingCalipermatching

Mahalanobis

MahalanobiswithpropensityscoreaddedRunLogisticRegression:

Dependentvariable:Y=1,ifparticipate;Y=0,otherwise.Chooseappropriateconditioning(instrumental)variables.Obtainpropensityscore:predictedprobability(p)orlog[(1-p)/p].GeneralProcedureMultivariateanalysisbasedonnewsample

1-to-1or1-to-nmatchandthenstratification(subclassification)KernelorlocallinearweightmatchandthenestimateDifference-in-differences(Heckman)EitherOr13傾向評分配對簡介NearestNeighborandCaliperMatchingNearestneighbor:ThenonparticipantwiththevalueofPjthatisclosesttoPiisselectedasthematch.Caliper:Avariationofnearestneighbor:Amatchforpersoniisselectedonlyifwhereisapre-specifiedtolerance.Recommendedcalipersize:.25p1-to-1Nearestneighborwithincaliper(Theisacommonpractice)1-to-nNearestneighborwithincaliper14傾向評分配對簡介MahalanobisMetricMatching:(withorwithoutreplacement)

Mahalanobiswithoutp-score:Randomlyorderingsubjects,calculatethedistancebetweenthefirstparticipantandallnonparticipants.Thedistance,d(i,j)canbedefinedbytheMahalanobisdistance:whereuandvarevaluesofthematchingvariablesforparticipantiandnonparticipantj,andCisthesamplecovariancematrixofthematchingvariablesfromthefullsetofnonparticipants.Mahalanobismetricmatchingwithp-scoreadded(touandv).NearestavailableMahalandobismetricmatchingwithincalipersdefinedbythepropensityscore(needyourownprogramming).15傾向評分配對簡介Stratification(Subclassification)Matchingandbivariateanalysisarecombinedintooneprocedure(nostep-3multivariateanalysis):Groupsampleintofivecategoriesbasedonpropensityscore(quintiles).Withineachquintile,calculatemeanoutcomefortreatedandnontreatedgroups.Estimatethemeandifference(averagetreatmenteffects)forthewholesample(i.e.,allfivegroups)andvarianceusingthefollowingequations:16傾向評分配對簡介MultivariateAnalysisatStep-3Wecouldperformanykindofmultivariateanalysisweoriginallywishedtoperformontheunmatcheddata.Theseanalysesmayinclude:multipleregressiongeneralizedlinearmodelsurvivalanalysisstructuralequationmodelingwithmultiple-groupcomparison,andhierarchicallinearmodeling(HLM)Asusual,weuseadichotomousvariableindicatingtreatmentversuscontrolinthesemodels.17傾向評分配對簡介VeryUsefulTutorialforRosenbaum&Rubin’sMatchingMethodsD’Agostino,R.B.(1998).Propensityscoremethodsforbiasreductioninthecomparisonofatreatmenttoanon-randomizedcontrolgroup.StatisticsinMedicine17,2265-2281.18傾向評分配對簡介SoftwarePackagesThereiscurrentlynocommercialsoftwarepackagethatoffersformalprocedureforPSM.InSAS,LoriParsonsdevelopedseveralMacros(e.g.,theGREEDYmacrodoesnearestneighborwithincalipermatching).InSPSS,Dr.JohnPainterofJordanInstitutedevelopedaSPSSmacrotodosimilarworksasGREEDY(/VRC/Lectures/index.htm).WehaveinvestigatedseveralcomputingpackagesandfoundthatPSMATCH2(developedbyEdwinLeuvenandBarbaraSianesi[2003],asauser-suppliedroutineinSTATA)isthemostcomprehensivepackagethatallowsuserstofulfillmosttasksforpropensityscorematching,andtheroutineisbeingcontinuouslyimprovedandupdated.19傾向評分配對簡介DemonstrationofRunningSTATA/PSMATCH2:

Part1.Rosenbaum&Rubin’sMethods

(Linktofile“Day1c.doc”)20傾向評分配對簡介ProblemswiththeConventional(PriortoHeckman’sDID)ApproachesEqualweightisgiventoeachnonparticipant,thoughwithincaliper,inconstructingthecounterfactualmean.Lossofsamplecasesdueto1-to-1match.Whatdoestheresamplerepresent?Externalvalidity.It’sadilemmabetweeninexactmatchandincompletematch:whiletryingtomaximizeexactmatches,casesmaybeexcludedduetoincompletematching;whiletryingtomaximizecases,inexactmatchingmayresult.21傾向評分配對簡介WeightsW(i.,j)(distancebetweeniandj)canbedeterminedbyusingoneoftwomethods:Kernelmatching:whereG(.)isakernelfunctionandnisabandwidthparameter.

Heckman’sDifference-in-DifferencesMatchingEstimator(2)23傾向評分配對簡介Locallinearweightingfunction(lowess):

Heckman’sDifference-in-DifferencesMatchingEstimator(3)24傾向評分配對簡介AReviewofNonparametricRegression

(CurveSmoothingEstimators)IamgratefultoJohnFox,theauthorofthetwoSagegreenbooksonnonparametricregression(2000),forhisprovisionoftheRcodetoproducetheillustratingexample.25傾向評分配對簡介WhyNonparametric?WhyParametricRegressionDoesn’tWork?26傾向評分配對簡介Focalx(120)

The120thorderedxSaintLucia:x=3183y=74.8Thewindow,calledspan,contains.5N=95observationsTheTask:DeterminingtheY-valueforaFocalPointX(120)27傾向評分配對簡介TricubekernelweightsWeightswithintheSpanCanBeDeterminedbytheTricubeKernelFunction28傾向評分配對簡介TheY-valueatFocalX(120)IsaWeightedMeanWeightedmean=71.1130129傾向評分配對簡介TheNonparametricRegressionLineConnectsAll190AveragedYValues30傾向評分配對簡介ReviewofKernelFunctionsTricubeisthedefaultkernelinpopularpackages.Gaussiannormalkernel:Epanechnikovkernel–parabolicshapewithsupport[-1,1].Butthekernelisnotdifferentiableatz=+1.Rectangularkernel(acrudemethod).31傾向評分配對簡介LocalLinearRegression

(Alsoknownaslowessorloess)AmoresophisticatedwaytocalculatetheYvalues.Insteadofconstructingweightedaverage,itaimstoconstructasmoothlocallinearregressionwithestimated0and1thatminimizes:whereK(.)isakernelfunction,typicallytricube.32傾向評分配對簡介TheLocalAverageNowIsPredictedbyaRegressionLine,InsteadofaLineParalleltotheX-axis.33傾向評分配對簡介AsymptoticPropertiesoflowessFan(1992,1993)demonstratedadvantagesoflowessovermorestandardkernelestimators.Heprovedthatlowesshasnicesamplingpropertiesandhighminimaxefficiency.InHeckman’sworkspriorto1997,heandhisco-authorsusedthekernelweights.Butsince1997theyhaveusedlowess.Inpracticeit’sfairlycomplicatedtoprogramtheasymptoticproperties.NosoftwarepackagesprovideestimationoftheS.E.forlowess.Inpractice,oneusesS.E.estimatedbybootstrapping.34傾向評分配對簡介BootstrapStatisticsInference(1)Itallowstheusertomakeinferenceswithoutmakingstrongdistributionalassumptionsandwithouttheneedforanalyticformulasforthesamplingdistribution’sparameters.Basicidea:treatthesampleasifitisthepopulation,andapplyMonteCarlosamplingtogenerateanempiricalestimateofthestatistic’ssamplingdistribution.Thisisdonebydrawingalargenumberof“resamples”ofsizenfromthisoriginalsamplerandomlywithreplacement.AcloselyrelatedideaistheJackknife:“droponeout”.Thatis,itsystematicallydropsoutsubsetsofthedataoneatatimeandassessesthevariationinthesamplingdistributionofthestatisticsofinterest.35傾向評分配對簡介BootstrapStatisticsInference(2)Afterobtainingestimatedstandarderror(i.e.,thestandarddeviationofthesampling