362EstimatingtheGPSandthedose–responsefunction
5.2Optional
ttransf(transformation)(gpscore)speci?esthetransformationofthetreatmentvari-ableusedinestimatingtheGPS.Thedefaulttransformationistheidentityfunction.Thesupportedtransformationsarethelogarithmictransformation,ttransf(ln);thezero-skewnesslogtransformation,ttransf(lnskew0);thezero-skewnessBox–Coxtransformation,ttransf(bcskew0);andtheBox–Coxtransformation,ttransf(boxcox).TheBox–Coxtransformation?ndsthemaximumlikelihoodestimatesoftheparametersoftheBox–Coxtransformregressingthetreatmentvariablet(varname)onthecontrolvariableslistedintheinputvariablelist.3normaltest(test)(gpscore)speci?esthegoodness-of-?ttestthatgpscorewillper-formtoassessthevalidityoftheassumednormaldistributionmodelforthetreat-mentconditionalonthecovariates.Bydefault,gpscoreperformstheKolmogorov–Smirnovtest(normaltest(ksmirnov)).PossiblealternativesaretheShapiro–Franciatest,normaltest(sfrancia);theShapiro–Wilktest,normaltest(swilk);andtheStataskewnessandkurtosistestfornormality,normaltest(sktest).normlevel(#)(gpscore)setsthesigni?cancelevelofthegoodness-of-?ttestfornor-mality.Thedefaultisnormlevel(0.05).testvarlist(varlist)(gpscore)speci?esthattheextentofcovariatebalancinghastobeinspectedforeachvariableofvarlist.ThedefaultvarlistconsistsofthevariablesusedtoestimatetheGPS.Thisoptionisusefulwhentherearecategoricalvariablesamongthecovariates.gpscore,whichisaregression-likecommand,requiresthatcategoricalvariablesareexpandedintoindicator(alsocalleddummy)variablesetsandthatonedummy-variablesetisdroppedinestimatingtheGPS.However,thebalancingtestshouldalsobeperformedontheomittedgroup.Thiscanbedonebyusingthetestvarlist(varlist)optionandbylistinginvarlistallthevariables,includingthecompletesetofindicatorvariablesforeachcategoricalcovariate.
3.Theproblemiswhetherthetreatmentvariabletakeszerovalue.Insuchacase,theprogramcontinues,forcingatransformationofthetreatmentvariabletotakeasuitablevalue.Speci?cally,weassumethatln(0)=0,ttransf(0)=?1/λifλ>0,andttransf(0)=ln(0)=0ifλ=0,forttransf=bcskew0orboxcox.Allowingforzerovaluesofthetreatmentimpliesthatuntreatedunitsmightbeincludedinthestudy.BecausetheGPSmethodsaredesignedforanalyzingthee?ectofatreatmentintensity,theyspeci?callyrefertothesubpopulationoftreatedunits.Thisimpliesthatincludinguntreatedunitsmightleadtomisleadingresults.
M.BiaandA.Mattei363
test(type)(gpscore)speci?eswhetherthebalancingpropertyhastobetestedusingeitherastandardtwo-sidedttest(thedefault)oraBayes-factor–basedmethod(test(Bayesfactor)).Theprograminformstheuserifthereissomeevidencethatthebalancingpropertyissatis?ed.Recallthatthetestisperformedforeachsinglevariableintestvarlist(varlist)andforeachtreatmentinterval.Speci?cally,letpbethenumberofcontrolvariablesintestvarlist(varlist),andletKbethenumberofthetreatmentintervals.We?rstcalculatep×Kvaluesoftheteststatistic;thenweselecttheworstvalue(thehighesttvalueinmodulus,orthelowestBayesfactor)andcompareitwithstandardvalues.Table1showsthe“orderofmagnitude”interpretationsoftheteststatisticsweconsider.
Table1.“Orderofmagnitude”interpretationsoftheteststatistics
tvalue|t|<1.282
1.282<|t|<1.645
Bayesfactor(BF)?Evidenceforthebalancingproperty(BP)√
BF>1.00
EvidencesupportstheBP
0.10 √ 1.645<|t|<1.9600.10 |t|>2.576 ? 0.01 BF<0.01 DecisiveevidenceagainsttheBP TheorderofmagnitudeinterpretationsoftheBayesfactorweappliedwereproposedbyJe?reys(1961). flag(#)(gpscore)speci?esthatgpscoreestimatestheGPSwithoutperformingeitheragoodness-of-?ttestfornormalityorabalancingtest.Thedefault#is1,meaningthatboththenormaldistributionmodelandthebalancingpropertyaretested;thedefaultlevelisrecommended.Weintroducedthisoptionforpracticalreasons.Recallthatdoseresponseestimatesthestandarderrorsofthedose–responsefunctionbyusingbootstrapmethods.Ineachbootstrapiteration,wewanttoreestimatetheGPSwithouttestingeitherthenormalityassumptionorthebalancingproperty.cmd(regressioncmd)(doseresponsemodel)de?nestheregressioncommandtobeusedforestimatingtheconditionalexpectationoftheoutcomegiventhetreatmentandtheGPS.Thedefaultfortheoutcomevariableiscmd(logit)whentherearetwodis-tinctvalues,cmd(mlogit)whenthereare3–5values,andcmd(regress)otherwise.Thesupportedregressioncommandsarelogit,probit,mlogit,mprobit,ologit,oprobit,andregress. 364EstimatingtheGPSandthedose–responsefunction regtypet(type)(doseresponsemodel)de?nesthemaximumpowerofthetreatmentvariableinthepolynomialfunctionusedtoapproximatethepredictorforthecon-ditionalexpectationoftheoutcomegiventhetreatmentandtheGPS.Thedefault ??;α),isalinearfunctionofthetypeislinear,meaningthatthepredictor,ψ(T,R treatment.Alternatively,typecanbequadraticorcubic.regtypegps(type)(doseresponsemodel)de?nesthemaximumpoweroftheesti-matedGPSinthepolynomialfunctionusedtoapproximatethepredictorfortheconditionalexpectationoftheoutcomegiventhetreatmentandtheGPS.Thede-??;α),isalinearfunctionoffaulttypeislinear,meaningthatthepredictor,ψ(T,R theestimatedGPS.Alternatively,typecanbequadraticorcubic.interaction(#)(doseresponsemodel)speci?eswhetherthemodelforthecondi-tionalexpectationoftheoutcomegiventhetreatmentandtheGPShastheinterac-tionbetweentreatmentandGPS.Thedefault#is1,meaningthattheinteractionisincluded.tpoints(vector)speci?esthatdoseresponseestimatestheaveragepotentialoutcomeforeachlevelofthetreatmentinvector.Bydefault,doseresponsecreatesavectorwiththeithelementequaltotheithobservedtreatmentvalue.Thisoptioncannotbeusedwiththenpoints(#)option(seebelow).npoints(#)speci?esthatdoseresponseestimatestheaveragepotentialoutcomeforeachlevelofthetreatmentbelongingtoasetofevenlyspacedvalues,t0,t1,...,t#,thatcovertherangeoftheobservedtreatment.Thisoptioncannotbeusedwiththetpoints(vector)option(seeabove).delta(#)speci?esthatdoseresponsealsoestimatesthetreatment-e?ectfunctioncon-sideringa#-treatmentgap,whichisde?nedasμ(t+#)?μ(t).Thedefault#is0,meaningthatdoseresponseestimatesonlythedose–responsefunction,μ(t).filename(?lename)speci?esthatthetreatmentlevelsspeci?edthroughthe tpoints(vector)optionorthenpoints(#)option,theestimateddose–responsefunction,and,eventually,theestimatedtreatment-e?ectfunction,alongwiththeirstandarderrors(ifcalculated),bestoredtoanew?lecalled?lename.bootstrap(string)speci?estheuseofbootstrapmethodstoderivestandarderrorsandcon?denceintervals.Bydefault,doseresponsedoesnotapplybootstraptechniques.Insuchacase,nostandarderroriscalculated.Toactivatethisoption,stringshouldbesettoyes.bootreps(#)speci?esthenumberofbootstrapreplicationstobeperformed.Thedefaultisbootreps(50).Thisoptionproducesane?ectonlyifthebootstrap()optionissettoyes. M.BiaandA.Mattei365 analysis(string)speci?esthatdoseresponseplotstheestimateddose–responsefunc-tion(s)and,eventually,theestimatedtreatment-e?ectfunction(s),alongwiththecorrespondingcon?denceintervalsiftheyarecalculatedwithbootstrapping.Bydefault,doseresponseplotsonlytheestimateddose–responseandtreatmentfunc-tion(s).Inordertoplotcon?denceintervals,stringhastobesettoyes.Iftheusertypesanalysis(no),noplotisshown.analysislevel(#)setsthecon?dencelevelofthecon?denceintervals.Thedefaultisanalysislevel(0.95).graph(?lename)storestheplotsoftheestimateddose–responsefunctionandtheesti-matedtreatmente?ectstoanew?lecalled?lename.Whentheoutcomevariableiscategorical,doseresponsecreatesanew?leforeachcategoryioftheoutcomevariableandnamesit?lenamei.detail(gpscore)displaysmoredetailedoutput.Speci?cally,thisoptionspeci?esthatgpscoreshowstheresultsofthegoodness-of-?ttestfornormality,somesummarystatisticsofthedistributionoftheGPSevaluatedattherepresentativepointofeachtreatmentinterval,andtheresultsofthebalancingtestwithineachtreatmentinterval.Whenthisoptionisspeci?edfordoseresponse,theresultsoftheregressionoftheoutcomeonthetreatmentandtheGPSarealsoshown. 6 Example:TheImbens–Rubin–Sacerdotelotterysam-ple WeusedatafromthesurveyofMassachusettslotterywinners;thedataaredescribedindetailinImbens,Rubin,andSacerdote(2001).Weareinterestedinestimatingthee?ectoftheprizeamountonsubsequentlaborearnings(fromU.S.SocialSecurityrecords).Althoughthelotteryprizeisobviouslyrandomlyassigned,substantialunitanditemnonresponseledtoaselectedsample,wheretheamountoftheprizeispotentiallycorrelatedwithbackgroundcharacteristicsandpotentialoutcomes.Toremovesuchbiases,wemaketheweakunconfoundednessassumptionspecifyingthat,conditionalonthecovariates,thelotteryprizeisindependentofthepotentialoutcomes.4 Thesampleweuseinthisanalysisisthe“winners”sampleof237individualswhowonamajorprizeinthelottery.Theoutcomeofinterestisyear6(earningssixyearsafterwinningthelottery),andthetreatmentisprize,theprizeamount.Controlvariablesareage,gender,yearsofhighschool,yearsofcollege,winningyear,numberofticketsbought,workstatusafterwinning,andearningssyearsbeforewinningthelottery(withs=1,2,...,6). WetriedtoreplicatetheresultsproducedbyHiranoandImbens(2004)buthavenotbeenabletonumericallyreplicatealltheirestimatesbecauseofrestrictionsofour 4.Inthiscontext,thenonignorabilityoftheassignmentmechanismisduetothepresenceofnon-response.Therefore,sayingthattheunconfoundednessassumptionallowsustoremoveallbiasesassociatedwithdi?erencesintheobservedcovariatesmeansthatweareimplicitlyassumingthattheoutcomevariableismissingatrandom(Rubin1976).