倾向评分配对简介培训课件

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

5倾向评分配对简介WhyPSM?(2)Theindependentvariablewisusuallycorrelatedwiththeerrorterm.Theconsequenceisinconsistentandbiasedestimateaboutthetreatmenteffect.6倾向评分配对简介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倾向评分配对简介WhyPSM?(4)Heckman&Smith(1995)FourImportantQuestions:

Whataretheeffectsoffactorssuchassubsidies,advertising,locallabormarkets,familyincome,race,andsexonprogramapplicationdecision?Whataretheeffectsofbureaucraticperformancestandards,locallabormarketsandindividualcharacteristicsonadministrativedecisionstoacceptapplicantsandplacetheminspecificprograms?Whataretheeffectsoffamilybackground,subsidiesandlocalmarketconditionsondecisionstodropoutfromaprogramandonthelengthoftimetakentocompleteaprogram?Whatarethecostsofvariousalternativetreatments?8倾向评分配对简介HistoryandDevelopmentofPSMThelandmarkpaper:Rosenbaum&Rubin(1983).Heckman’searlyworkinthelate1970sonselectionbiasandhiscloselyrelatedworkondummyendogenousvariables(Heckman,1978)addressthesameissueofestimatingtreatmenteffectswhenassignmentisnonrandom.Heckman’sworkonthedummyendogenousvariableproblemandtheselectionmodelcanbeunderstoodasageneralizationofthepropensity-scoreapproach(Winship&Morgan,1999).Inthe1990s,Heckmanandhiscolleaguesdevelopeddifference-in-differencesapproach,whichisasignificantcontributiontoPSM.Ineconomics,theDIDapproachanditsrelatedtechniquesaremoregenerallycallednonexperimentalevaluation,oreconometricsofmatching.9倾向评分配对简介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倾向评分配对简介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倾向评分配对简介FundamentalAssumption

Rosenbaum&Rubin(1983)Differentversions:“unconfoundedness”&“ignorabletreatmentassignment”(Rosenbaum&Robin,1983),“selectiononobservables”(Barnow,Cain,&Goldberger,1980),“conditionalindependence”(Lechner1999,2002),and“exogeneity”(Imbens,2004)12倾向评分配对简介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倾向评分配对简介NearestNeighborandCaliperMatchingNearestneighbor:ThenonparticipantwiththevalueofPjthatisclosesttoPiisselectedasthematch.Caliper:Avariationofnearestneighbor:Amatchforpersoniisselectedonlyifwhereisapre-specifiedtolerance.Recommendedcalipersize:.25p1-to-1Nearestneighborwithincaliper(Theisacommonpractice)1-to-nNearestneighborwithincaliper14倾向评分配对简介MahalanobisMetricMatching:(withorwithoutreplacement)

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

Part1.Rosenbaum&Rubin’sMethods

(Linktofile“Day1c.doc”)20倾向评分配对简介ProblemswiththeConventional(PriortoHeckman’sDID)ApproachesEqualweightisgiventoeachnonparticipant,thoughwithincaliper,inconstructingthecounterfactualmean.Lossofsamplecasesdueto1-to-1match.Whatdoestheresamplerepresent?Externalvalidity.It’sadilemmabetweeninexactmatchandincompletematch:whiletryingtomaximizeexactmatches,casesmaybeexcludedduetoincompletematching;whiletryingtomaximizecases,inexactmatchingmayresult.21倾向评分配对简介WeightsW(i.,j)(distancebetweeniandj)canbedeterminedbyusingoneoftwomethods:Kernelmatching:whereG(.)isakernelfunctionandnisabandwidthparameter.

Heckman’sDifference-in-DifferencesMatchingEstimator(2)23倾向评分配对简介Locallinearweightingfunction(lowess):

Heckman’sDifference-in-DifferencesMatchingEstimator(3)24倾向评分配对简介AReviewofNonparametricRegression

(CurveSmoothingEstimators)IamgratefultoJohnFox,theauthorofthetwoSagegreenbooksonnonparametricregression(2000),forhisprovisionoftheRcodetoproducetheillustratingexample.25倾向评分配对简介WhyNonparametric?WhyParametricRegressionDoesn’tWork?26倾向评分配对简介Focalx(120)

The120thorderedxSaintLucia:x=3183y=74.8Thewindow,calledspan,contains.5N=95observationsTheTask:DeterminingtheY-valueforaFocalPointX(120)27倾向评分配对简介TricubekernelweightsWeightswithintheSpanCanBeDeterminedbytheTricubeKernelFunction28倾向评分配对简介TheY-valueatFocalX(120)IsaWeightedMeanWeightedmean=71.1130129倾向评分配对简介TheNonparametricRegressionLineConnectsAll190AveragedYValues30倾向评分配对简介ReviewofKernelFunctionsTricubeisthedefaultkernelinpopularpackages.Gaussiannormalkernel:Epanechnikovkernel–parabolicshapewithsupport[-1,1].Butthekernelisnotdifferentiableatz=+1.Rectangularkernel(acrudemethod).31倾向评分配对简介LocalLinearRegression

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