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TheStataJournal(2008)8,Number3,pp.354–373
AStatapackagefortheestimationofthedose–responsefunctionthroughadjustmentfor
thegeneralizedpropensityscore
MichelaBia
LaboratorioRiccardoRevelliCentreforEmploymentStudies
CollegioCarloAlbertoMoncalieri,Italy
michela.bia@laboratoriorevelli.it
AlessandraMatteiDepartmentofStatisticsUniversityofFlorence
Florence,Italymattei@ds.uni?.it
Abstract.Inthisarticle,webrie?yreviewtheroleofthepropensityscoreinestimatingdose–responsefunctionsasdescribedinHiranoandImbens(2004,Ap-pliedBayesianModelingandCausalInferencefromIncomplete-DataPerspectives,73–84).ThenwepresentasetofStataprogramsthatestimatethepropensityscoreinasettingwithacontinuoustreatment,testthebalancingpropertyofthegeneralizedpropensityscore,andestimatethedose–responsefunction.Weillus-tratetheseprogramsbyusingadatasetcollectedbyImbens,Rubin,andSacerdote(2001,AmericanEconomicReview91:778–794).
Keywords:st0150,gpscore,doseresponse,doseresponsemodel,biasremoval,dose–responsefunction,generalizedpropensityscore,weakunconfoundedness
1Introduction
Muchoftheworkonpropensity-scoreanalysishasfocusedoncaseswherethetreat-mentisbinary.Matchingestimatorsforcausale?ectsofabinarytreatmentbasedonpropensityscoreshavealsobeenimplementedinStata(e.g.,BeckerandIchino[2002]andLeuvenandSianesi[2003]).
Inmanyobservationalstudies,thetreatmentmaynotbebinaryorevencategorical.Insuchacase,onemaybeinterestedinestimatingthedose–responsefunctionwherethetreatmentmighttakeonacontinuumofvalues.Forexample,ineconomics,animportantquantityofinterestisthee?ectofaidto?rms(e.g.,BiaandMattei[2007]).Insocioeconomicstudies,onemaybeinterestedinthee?ectoftheamountofalotteryprizeonsubsequentlaborearnings(e.g.,HiranoandImbens[2004]).
HiranoandImbens(2004)developedanextensiontothepropensity-scoremethodinasettingwithacontinuoustreatment.FollowingRosenbaumandRubin(1983)andmostoftheliteratureonpropensity-scoreanalysis,theymakeanunconfoundednessassumption,whichallowsthemtoremoveallbiasesincomparisonsbytreatmentstatusbyadjustingfordi?erencesinasetofcovariates.Thentheyde?neageneralizationofthepropensityscoreforthebinarycase—henceforthlabeledgeneralizedpropensityscore(GPS)—whichhasmanyoftheattractivepropertiesofthebinary-treatmentpropensityscore.
c2008StataCorpLP??
st0150
M.BiaandA.Mattei355
Inthisarticle,webrie?yreviewthemethoddevelopedbyHiranoandImbens(2004),andweprovideasetofStataprogramsthatestimatetheGPS,assesstheadequacyoftheunderlyingassumptionsonthedistributionofthetreatmentvariable,testwhethertheestimatedGPSsatis?esthebalancingproperty,andestimatethedose–responsefunction.FollowingHiranoandImbens(2004),ourStataprogramsaddresstheproblemofestimationandinferencebyusingparametricmodels.
WeillustratetheseprogramswithadatasetcollectedfromImbens,Rubin,andSac-erdote(2001).ThepopulationconsistsofindividualswhowontheMegabuckslotteryinMassachusettsinthemid-1980s.Weapplyourprogramstoestimatetheaveragepo-tentialpost-winninglaborearningsforeachlevelofthelotteryprize(thedose–responsefunction).Althoughtheassignmentoftheprizeisobviouslyrandom,substantialitemandunitnonresponseledtoaselectedsamplewheretheamountoftheprizeisnolongerindependentofbackgroundcharacteristics.Inusingtheseprograms,rememberthattheyonlyallowyoutoreduce,nottoeliminate,thebiasgeneratedbyunobservableconfoundingfactors.Asinthebinary-treatmentcase,theextenttowhichthisbiasisreduceddependscruciallyontherichnessandqualityofthecontrolvariables,onwhichtheGPSiscomputed.
2Thepropensityscorewithcontinuoustreatments
SupposewehavearandomsampleofsizeNfromalargepopulation.Foreachunitiinthesample,weobserveap×1vectorofpretreatmentcovariates,Xi;thetreatmentreceived,Ti;andthevalueoftheoutcomevariableassociatedwiththistreatment,Yi.UsingtheRubincausalmodel(Holland1986)asaframeworkforcausalinference,wede?neasetofpotentialoutcomes,{Yi(t)}t∈T,i=1,...,N,whereTisacontinuoussetofpotentialtreatmentvalues,andYi(t)isarandomvariablethatmapsaparticu-larpotentialtreatment,t,toapotentialoutcome.HiranoandImbens(2004)referto{Yi(t)}t∈Tastheunit-leveldose–responsefunction.Weareinterestedintheaveragedose–responsefunction,μ(t)=E{Yi(t)}.FollowingHiranoandImbens(2004),weas-sumethat{Yi(t)}t∈T,Ti,andXi,i=1,...,N,arede?nedonacommonprobabilityspace;thatTiiscontinuouslydistributedwithrespecttotheLebesguemeasureonT;andthatYi=Yi(Ti)isawell-de?nedrandomvariable.Tosimplifythenotation,wewilldroptheisubscriptinthesequel.
Thepropensityfunctionisde?nedbyHiranoandImbens(2004)astheconditionaldensityoftheactualtreatmentgiventheobservedcovariates.
De?nition2.1(GPS)Letr(t,x)betheconditionaldensityofthetreatmentgiventhecovariates:
r(t,x)=fT|X(t|x)
ThentheGPSisR=r(T,X).
356EstimatingtheGPSandthedose–responsefunction
TheGPShasabalancingpropertysimilartothatofthestandardpropensityscore;thatis,withinstratawiththesamevalueofr(t,x),theprobabilitythatT=tdoesnotdependonthevalueofX:
X⊥I(T=t)|r(t,x)
whereI(·)istheindicatorfunction.HiranoandImbens(2004)showthat,incombina-tionwithasuitableunconfoundednessassumption,thisbalancingpropertyimpliesthatassignmenttotreatmentisunconfounded,giventheGPS.
Theorem2.1(WeakunconfoundednessgiventheGPS)Supposethatassignmenttothetreatmentisweaklyunconfounded,givenpretreatmentvariablesX:
Y(t)⊥T|X
Then,foreveryt,
fT{t|r(t,X),Y(t)}=fT{t|r(t,X)}
Usingthistheorem,HiranoandImbens(2004)showthattheGPScanbeusedtoeliminateanybiasesassociatedwithdi?erencesinthecovariates.
Theorem2.2(BiasremovalwithGPS)Supposethatassignmenttothetreatmentisweaklyunconfounded,givenpretreatmentvariablesX.Then
β(t,r)=E{Y(t)|r(t,X)=r}=E(Y|T=t,R=r)
and
μ(t)=E[β{t,r(t,X)}]
forallt∈T
3Estimationandinference
TheimplementationoftheGPSmethodconsistsofthreesteps.Inthe?rststep,weestimatethescorer(t,x).Inthesecondstep,weestimatetheconditionalexpectationoftheoutcomeasafunctionoftwoscalarvariables,thetreatmentlevelTandtheGPSR:β(t,r)=E(Y|T=t,R=r).Inthethirdstep,weestimatethedose–responsefunction,μ(t)=E[β{t,r(t,X)}],t∈T,byaveragingtheestimatedconditionalexpectation,??{t,r(t,X)},overtheGPSateachlevelofthetreatmentweareinterestedin.β
M.BiaandA.Mattei357
3.1
Modelingtheconditionaldistributionofthetreatmentgiventhecovariates
The?rststepistoestimatetheconditionaldistributionofthetreatmentgiventhecovariates.Weassumethatthetreatment(oritstransformation)hasanormaldistri-butionconditionalonthecovariates:
????
g(Ti)|Xi～Nh(γ,Xi),σ2
(1)
whereg(Ti)isasuitabletransformationofthetreatmentvariable[g(·)maybetheidentityfunction],andh(γ,Xi)isafunctionofcovariateswithlinearandhigher-orderterms,whichdependsonavectorofparameters,γ.Thechoiceofthehigher-ordertermstoincludeisonlydeterminedbytheneedtoobtainanestimateoftheGPSthatsatis?esthebalancingproperty.
Theprogramgpscore.adoestimatestheGPSandteststhebalancingpropertyac-cordingtothefollowingalgorithm:
1.Estimatetheparametersγandσ2oftheconditionaldistributionofthetreatmentgiventhecovariates(1)bymaximumlikelihood.12.Assessthevalidityoftheassumednormaldistributionmodelbyoneofthefollow-inguser-speci?edgoodness-of-?ttests:theKolmogorov–Smirnov,theShapiro–Francia,theShapiro–Wilk,ortheStataskewnessandkurtosistestfornormality.
a.Ifthenormaldistributionmodelisstatisticallydisapproved,informtheuserthattheassumptionofnormalityisnotsatis?ed.Theuserisinvitedtouseadi?erenttransformationofthetreatmentvariableg(Ti).3.EstimatetheGPSas
1??i=√Rexp?2{g(Ti)?h(γ??,Xi)}2σ??2πσ??2whereγ??andσ??2aretheestimatedparametersinstep1.
4.Testthebalancingpropertyandinformtheuserwhetherandtowhatextent
thebalancingpropertyissupportedbythedata.FollowingHiranoandImbens(2004),theprogramgpscore.adotestsforbalancingofcovariatesaccordingtothefollowingscheme:
a.Dividethesetofpotentialtreatmentvalues,T,intoKintervalsaccordingtoauser-speci?edrule,whichshouldbede?nedonthebasisofthesampledis-tributionofthetreatmentvariable.LetG1,...,GKdenotetheKtreatmentintervals.
1.Themodel(1)isspeci?edintheauxiliaryado-?legpscoremodel.ado.
1
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