Statistics And Causal Inference Author(s): Paul W. Holland Source .
Statistics and Causal InferenceAuthor(s): Paul W. HollandSource: Journal of the American Statistical Association, Vol. 81, No. 396 (Dec., 1986), pp. 945-960Published by: Taylor & Francis, Ltd. on behalf of the American Statistical AssociationStable URL: http://www.jstor.org/stable/2289064Accessed: 12-03-2015 13:41 UTCYour use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available rms.jspJSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of contentin a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship.For more information about JSTOR, please contact support@jstor.org.Taylor & Francis, Ltd. and American Statistical Association are collaborating with JSTOR to digitize, preserve and extendaccess to Journal of the American Statistical Association.http://www.jstor.orgThis content downloaded from 128.59.22.37 on Thu, 12 Mar 2015 13:41:49 UTCAll use subject to JSTOR Terms and Conditions
Statisticsand Causal InferencePAULW. rawnfroma carefullydesignedexperimentareoftenvalid.Whatcan a statisticalmodelsayaboutcausation?Thisquestionis addressedbyusinga 983;Rubin1974)to sconcernedwithmeasurement,hascontributionsto make.It is myopinionthatan emphasison theeffectsofcausesratherthanon ceofbringingstatisticalreasoningto bearon ionalanalysesofcausation.KEY WORDS: Causal ality.2. MODEL FOR ationalinferenceissimplythe standardstatisticalmodelthatrelatestwovariables overa population.For clarityand edin thenext1. INTRODUCTIONI fThereactionofmanystatisticianswhenconfrontedwith I seemoverlyexplicitin describingthemodelit is teto a becauseI wishto be absolutelyclearon thefundamentaldiscussionof causationis immediatelyto denythatthere elementsofthetheorypresentedhere.is anysuchpossibility."Thatcorrelationis notcausation The modelbeginswitha populationor universeU ofis perhapsthefirstthingthatmustbe said"(Barnard1982, "units."A unitin U willbe denotedbyu. Unitsare thep. 387). Possiblythisevasiveactionis inresponseto all of tsthoseneedlinglittleheadlinesthatpop up in the most are ectedplaces,forexample,"If thestatisticscannot andplotsofland.A variableis simplya real-valuedfuncrelatecauseandeffect,theycan certainlyadd to therhet- tion-thatis definedon everyunitin U. The value of aoric"(Smith1980,p. 998).variablefora givenunitu is thenumberassignedbysomeOne need onlyrecallthata well-designedrandomized measurementprocessto u. A populationofunitsandvarican be a powerfulexperimentaid in investigatingcausal ablesdefinedon theseunitsare thebasicelementsoftherelationsto questiontheneedforsucha defensiveposture by statisticians.Randomizedexperimentshave trans- Theycorrespondtothemathematicalconceptsofa setandformedofscience,andtheearlyproponents real-valuedmanybranchesfunctionsdefinedon theelementsof theset.of suchstudieswerethe sanlestatisticianswhofounded Theyaretheprimitivesofthetheoryandwillnotbe furtherthemodernera isticshasa greatdealSupposethatforeach unitu in U thereis emsandought value Y(u) of a variableY SupposefurtherthatY is ato playa moresignificantroleinphilosophicalanalysesof variableofscientificin thesensethatone wishesinterestcausationthanit has heretofore.In addition,I willtryto to understandwhythevaluesof Y o drawcausalinfer- U. Y is the responsevariablebecauseof its statusas aencesare distinctlydifferentfromthoseusedto drawas- "variableto be explained."In sone is organizedas follows.statisticalFirst,models Y e on theunitsof U. Let A be a eywillbe appliedtovari- U. DistinguishA fromY bycallingA an attributeof theous ideas about causationthathave been expressedby unitsin u. Logically,however,A and Y are on an equalseveralwriterson thissubject.One difficultythatarisesin footing,sincetheyarebothsimplyvariablesdefinedon U.aboutcausationis thevarietytalkingofquestionsthatareAll probabilities,and ing.Someauthorsfocuson the volvingvariablesare computedoverU. A rs meannothingmeaningfulnessmorenorlessthana proportionofunitsinare concernedwithdeducingthecausesofa giveneffect. U. The expectedvalueof a variableis merelyitsaverageStillothersare interestedin understandingthedetailsof valueoverall of U. Conditionalexpectedvaluesare avcausalmechanisms.The emphasisherewillbe on measur- ngtheeffectsof causesbecausethisseemsto be a place byconditioninginthevaluesofvariables.Itis models.Theroleoftimeneedstobementionedhere.* PaulW.Hollandis TestingService,Princeton,NJ08541.A preliminarydraftofthisarticlewasthebasisofan ngertothatdraftofthisarticle.? 96,Theoryand Methods945This content downloaded from 128.59.22.37 on Thu, 12 Mar 2015 13:41:49 UTCAll use subject to JSTOR Terms and Conditions
946Journal of the American Statistical Association, December 1986tionsofunitsexistwithina sticsofunitsthatvariablesrepresentmustalso be made at particulartimes.For associationalinference,however,theroleoftimeis simplyto affectthe definitionof the populationof unitsor totheoperationalmeaningof a particularspecifyvariable.As we willsee, in causalinferencetheroleoftimehas canhaveinthemodeljustdescribedis thevaluesof Y(u) andA (u) are all u inU. ThejointdistributionofY andA overU is specifiedbyrandomizedstudy.I do itto emphasizean idea thatI believereceivesinsufficientattentionin generaldiscussionsof causation.Thisis thefactthattheeffectof a causeisalwaysrelativeto anothercause.Forexample,thephrase"A causesB" almostalwaysmeansthatA causesB relativeto some othercause thatincludesthe condition"notA." The terminologybecomesrathertorturedifwe trytostickwiththeusualcausallanguage,butit is straightforwardifwe use thelanguageof experiments-treatment(i.e., onecause)versuscontrol(i.e., anothercause).InSection71 willdiscussthefundamentalquestionofwhatkindsPr( Y y, A a) proportionof u in U forwhichY(u) ofthingscan be causes.The keynotion,however,is the- y and A(u) a.ofwhetheritcanbe achievedinpracpotential(regardlessThe associationalparametersare determinedby this ticeor not)forexposingor notexposingeachunitto thejointdistribution.For example,theconditionaldistribu- actionofa f Y givenA is specifiedby Pr(Y y IA a) unitbe potentiallyexposableto anyone of the causes.Pr(Y y, A a)/Pr(A a). This conditionaldistribu- As an example,theschoolinga studentreceivescan be ationdescribeshow the distributionof Y valueschanges cause,inoursense,ofthestudent'sperformanceona test,overU as A varies.A typicalassociationalparameteristhe whereasthestudent'sraceor gendercannot.of Y on A, thatis, theconditionalregressionitshallbe assumedinthisarticleexpectation ForsimplicitythatthereE(Y I A a).are justtwocausesor levelsof treatment,denotedby in- (thetreatment)and c (thecontrol).Let S be a tions,etc.)about thatindicatesthecausetowhicheachunitin U is exposed;theassociationalparametersrelatingY andA on thebasis thatis, S t indicatesthattheunitis exposedto t and Sof data gatheredaboutY andA fromunitsin U. In this c indicatesexposureto c. In a controlledstudy,S issense,associationalinferenceis simplydescriptivestatis- constructedbytheexperimenter.In an uncontrolledstudy,tics.S is determinedto someextentbyfactorsbeyondtheexcontrol.In eithercase,thecriticalperimenter'sfeatureof3. RUBIN'SMODEL FOR CAUSAL INFERENCEthenotionofcausein thismodelis thatthevalueofS(u)Because experimentationis sucha t.and statisticaltool and one thatoftenintroducesclarity The variableS is analogousto thevariableA in Sectioninto discussionsof specificcases of causation,I una- 2, butwiththeessentialdiffkrencethatS(u) indicatesexbashedlydrawon thelanguageand frameworkofexperi- posureofu to a specificcause,whereasA (u) canindicatementsforthemodelforcausalinference.It is notthatI a propertyor characteristicofu. In thiscase thevalueofbelievean experimentis theonlypropersettingfordis- A(u) couldnothavebeendifferent.butI do feelthatan experimentcussingcausality,is hsetting.a model factthatwhena unitis exposedto a causethismustoccurThe purposeis to constructsimplestthatis complexenoughto allowus to formalizebasicin- atsomespecifictimeorwithina specifictimeperiod.Varituitionscause and effect.The pointof depar- ables now divideintotwo classes:pre-exposure-thoseconcerningtureis theanalysisofcausaleffectsgivenin Rubin(1974, whosevaluesaredeterminedtothecause;priorto exposure1977,1978,1980). It willbe sufficientforour purposes, post-exposure-thosewhosevaluesare determinedaftertodealwitha simplified,version exposureto issimplifiedmodelwasusedin HolTheroleofa responsevariableY istomeasuretheeffectinretro- landandRubin(1980)toanalyzecausalinferenceand rchclass.Thisgivesriseto anothercriticalelespective,in Hollandand Rubin(1983) to analyzeLord's"analysis mentofthemodel.The .I refertothisas "Rubin'smodel" are potentiallyaffectedbytheparticularcause,t or c, toeventhoughRubinwouldarguethattheideasbehindthe whichthe unitis exposed.This is nothingless thanthemodelhavebeen aroundsinceFisher.I thinkthatRubin statementwhichis theveryheartthatcauseshaveeffects,(1974)was theplacewheretheseideaswerefirstapplied of the notionof causation.For the modelto representto thestudyofcausation.thisstateofaffairswe neednota singlevariable,faithfullyThismodelalso beginswitha populationof units,U. Y, to representa responsebuttwovariables,Y, and Yc,Unitsin themodelforcausalinferenceare theobjectsof to representtwopotentialresponses.The interpretationstudyon whichcausesor treatmentsmayact. The terms ofthesetwovalues,Y,(u) and Yc(u)fora givenunitu, iscauseandtreatmentwillbe usedinterchangeably,andthe thatY,(u) is thevalueoftheresponsethatwouldbe obnotionthatthesetermsconveyis an importantpartofthe servediftheunitwereexposedto t andY (u)is thevaluemodel.It is importantto realizethatby usingtheterms thatwouldbe observedonthesameunitifitwereexposedcause and treatmentinterchangeablyI do notintendto to c.limitthe discussionto the activitieswithina controlled The notationYt(u) and Yc(u) is sometimesconfusingThis content downloaded from 128.59.22.37 on Thu, 12 Mar 2015 13:41:49 UTCAll use subject to JSTOR Terms and Conditions
Holland:Statisticsand Causal Inference947becausea variableusuallyrepresentsa measurementofsomesortand a measurementis usuallythoughtofas theresultofa processthatis appliedtoa unit.Thisisnotreallycorrect.For post-exposurevariablesthemeasurementisappliedto thepairing(u, t) (i.e., u afterexposureto t) orto(u, c) (i.e., u afterexposuretoc). A of Y on u and theexposed cause is Yt(u) Y(u, t) and Yc(u) Y(u, c). Ishalluse the Yt,Yc notation,however,sinceit leads tosimplerexpressions.The effectof the cause t on u as measuredby Y andrelativeto cause c is the differencebetweenYt(u) andYc(u).Inthemodelthiswillbe representedbythealgebraicdifferenceYt(u) - YC(u).(1)I shallcallthedifference(1) thecausaleffectoft (relativeto c) on u (as measuredby Y). Expression(1) is stbasicofall causalstatements.It saysthattreatmenttcausestheeffectY,(u) - Yc(u) on unitU (relativeto treatmentc)or moresimplythatt causes the effectY,(u) - YC(u).(2)Causalinferenceis ultimatelyconcernedwiththeeffectsof causeson specificunits,thatis, withascertainingthevalue of the causal effectin (1). It is frustratedby aninherentfactof observationallifethatI call theFundamentalProblemof CausalInference.ingthatthesimultaneousof Y,(u) and Y,(u)observationis impossibleI do notmeanthatknowledgerelevanttothesevaluesis completelyabsent.It willdependon thesituationconsidered.Thereare rthesakeofconvenienceI willlabel thescientificsolutionand thestatisticalsolution.The scientificsolutionis toexploitvarioushomogeneityor invarianceassumptions.For example,bystudyingthebehaviorof a piece of laboratoryequipmentcarefullyascientistmaycometo believethatthevalueofYj(u) measuredat an earliertimeis equal to thevalueof Yj(u) forthecurrentexperiment.Allheneedstodo nowistoexposeu to t and measureY,(u) and he has overcometheFundamentalProblemof Causal sthasmadeanuntestablehomogeneityassumption.By carefulworkhe mayconvincehimselfandothersthatthisassumptionis right,buthe canneverbe absolutelycertain.Sciencehas solutionis a commonplaceaspectof oureverydaylifeas well.We all useit to makethe causal inferencesthatarisein our lives.Theseideas are amplifiedin Sections4.1 and 4.2.The statisticalsolutionis differentandmakesuse ofthepopulationU in a typicallystatisticalway. The averagecausaleffect,T, oft (relativeto c) overU is theexpectedvalueofthedifferenceYt(u) - Yj(u) overtheu's in U;thatis,E( Yt - YC) T.(3)FundamentalProblemof Causal Inference.It is imBy theusualpossibleto observethevalue of Y,(u) and Yc(u) on the T definedin (3) is 3)mayexpressedsameunitand, therefore,it is impossibleto observetheeffectoft on u.T E(Y) - E(Yc).(4)The emphasisis on thewordobserve.The impossibilityAlthoughthisdoes notlook like much,(4) revealsthatofobservingbothY,(u) and Yc(u) is self-evidentin some informationon differentunitsthatcan be le,iftheunit used to gainknowledgeabout T. For example,if someu is a specificfourthgrader,trepresentsa novelyear-long unitsareexposedtottheymaybe usedtogiveinformationprogramof studyof arithmetic,c representsa standard aboutE(Yt) (becausethisis themeanvalueofYtoverU),arithmeticand Y is a scoreon a testat theend and if otherunitsare exposedto c theymaybe used toprogram,oftheyear,thenitis evidentthatwe couldobserveeither giveinformationaboutE(YC). Formula(4) is thenusedtoY,(u) or Yc(u) butnotboth.We willneverobservewhat gainknowledgeaboutT. The exactwaythatunitswouldtheeffectoftwason u. On theotherhand,ifu is a room be selectedforexposureto t or c is veryimportantandin a house,t meansthatI flickon thelightswitchin that involvesall oftheusualconsiderationsofgoodstatisticalroom,c meansthatI do not,and Y indicateswhetherthe designof experiments.The importantpointis thatthelightis on or nota shorttimeafterapplyingeithert or c, ecausalthenI mightbe inclinedto believethatI can knowthe effectoft on a hY,(u)andYc(u)bysimplyflickingtheswitch. averagecausaleffectoftovera nlybecauseoftheplausibility ideaswillbe developedfurtherin Sections4.3 and eliefThe usefulnessof eitherthescientificor thestatisticalofminecanbe sharedbyanyoneelse.If,forexample,the solutionto theFundamentalProblemofCausalInferencelighthas beenflickingoffand on forno apparentreason dependson the truthof differentsetsof untestableaswhileI amcontemplatingbeginningthisexperiment,I might sumptions.In Section4 I willdiscusssomeofthetypicaldoubtthatI wouldknowthevaluesof Y,(u) and Yc(u) assumptionsthatare oftenusedto overcometheFundaafterflickingon the switch-atleast untilI was clever mentalProblemofCausalInference.enoughto figureouta newexperiment!It is usefulto have a notationto lProblemofCausal thecausalindicatorvariableS determineswhichvalue,YtInferenceis thatcausal inferenceis impossible.But we or Yc,is observedfora givenunit.IfS(u) t, thenYt(u)shouldnotjumpto thatconclusiontooquickly.By assert- is observed,andifS(u) c, thenYc(u)is observed.ThusThis content downloaded from 128.59.22.37 on Thu, 12 Mar 2015 13:41:49 UTCAll use subject to JSTOR Terms and Conditions
mber 1986theobservedresponseon unitu is Ys(u)(u). The observed 4.2 ce,eventhoughtheA secondwayof applyingthescientificsolutionto themodelcontainsthreevariables,S, Yt,and Yc,theprocess YI(u2)thatFundamentalProblemistoassumeY,(ul)of observationinvolvesonlytwo,thatis, S, Ys. The dis forandThistwounitsulu2.is thetinctionbetween(a) themeasurementprocess,Y, thatpro- and Y,(ul) plicableducestheresponsevariable;(b) thetwoversionsof thelaboratoriesandis also a hcause to workdonein scientificoftis tis exposed(and in termsof ryY,(ul) Y,(u2).aredefined);and(c) istsconvincethemselvesthehomogeveryimportantofand, often,is notmadein lytheycausation.These distinctionsneverarisein is,course,buttheyare crucialto isassumptionplausible.It is usefulto reviewthemodelforassociationalinference and Rubin'smodelside by side to emphasizetheirdifferences.Both involvea populationof units,U, and 4.3 Independencebothinvolvetwoobservablevariables:(A, Y) forassociIn mydiscussionof thestatisticalsolutionto theFunationand (S, Ys) forcausation.Thisis all, however,that damentalProblem,I didnotgiveanyspecificationto thetheyhavein common.WhereasA and Y are simplyvari- waythatunitsmightbe selectedforobservationof Y, orables definedon the unitsof U, S and Ys presupposeaOf course,thatitwasveryimportant.Y,. I onlyindicatedmorecomplicatedin orderforthemto applyto themostwell-knownstructurewaythatthisoccursin experimentalrealsituations.Two or morecauses(or treatments)must workis by randomization,and thissectionis concernedbe exposableto all oftheunits,andtheresponseY must withthattopic.be a post-exposurevariablein orderfortheobservedrein usingthestatisticalsolutionis thatThe suppositioninferenceinvolves thepopulationU does notconsistofone ortwounitsbutsponseYs to be defined.AssociationalofvaluesofY andA, is "large"in somesense.The butionsandcausalinferenceconcernsthevaluesY,(u) - Y,(u) on arevaluesofthepairofvariables(S, Ys).individualunits.Causal inferencesproceedfromtheobbetweenThe averagecausal effectT is thedifferencethatad- thetwoexpectedvaluesE(Yt) andservedvaluesofS and Ys andfromassumptionsE(Y,). The observedbut data(S, Ys), however,dresstheFundamentalProblemof Causal Inferenceaboutcanonlygiveus informationdo notnecthatare usuallyuntestable.CausalinferencesS t)(S)t) E(YtininvolvestatisticalbutassociationalE(YsS Sessarilyinferences,ferencesalmostalwaysdo.and4. SOME SPECIALCASES OF CAUSAL INFERENCEE(Ys S c)E(Yc S c).(6)It is importantto recognizethatE(Yt) and E(Yt IS eneral[similarlyforE(Yc) andE(YC IS c)]. To statethisdifferenceinwords,E(Yt) istheaveragevalueofYt(u)overall u in U, whereE(Yt IS t) is theaveragevalue4.1 Temporal Stabilityand Causal Transienceof Yt(u)overonlythosein u in U thatwereexposedto t.Thereis no reasonwhy,in general,thesetwoaveragesOne wayof applyingthescientificsolutionto theFunshouldbe equal. For example,ifS(u) tforall unitsfordamentalProblemof Causal Inferenceis to assumethatwhichY,(u) is small,thenE(Y, IS t) willbe smaller(a) thevalue of Y,(u) does notdependon whenthesethanE(Y,).quence"applyc to u thenmeasureY on u" occursand(b)makesThereis,however,an assumptionthat,ifplausible,thevalueof Y,(u) is aluesequal. It is theassumptionu to thesequencein (a). WhenthesetwoassumptionsareWhenunitsare assignedat randomeitherindependence.to measureY,(u) and Y,(u)plausibleitis a simplematterto cause t or to cause c, u to c thent,measuringY afterofwhicharecarriedoutso thatthedeterminationprocesseseach exposure.The firstassumptionis temporalstability,cause (t or c) u is exposedto is regardedas eovertime.Theof all othervariables,includingindependentYtand Yc.secondassumptionis causaltransience,becauseit assertsis carriedThismeansthatif thephysicalrandomizationthattheeffectofthecausec andthemeasurementprocessofthenit is plausiblethatS is independentoutcorrectly,thatresultsin Yc(u) is transientand does notchangeuand Y, and all othervariablesover U. Thisis theinY,enoughtoaffectY (u) y all of us in everydaylife-forexample,in the nsiderssomesimplespecialcasesofRubin's modelforcausalinference.The purposeis to showhowspecificaddedto estypes.This content downloaded from 128.59.22.37 on Thu, 12 Mar 2015 13:41:49 UTCAll use subject to JSTOR Terms and Conditions
Holland: Statisticsand CausalInference949andThe assumptionof trelevanttoeveryunitand,there(8)E(Yc) E(Yc I S c).at thefore,allowsT to be usedto drawcausalinferencesHence underthe independenceassumptionthe average unitlevel.causaleffectT effectcanbe partiallyis usuallyin thesamewaythattheadditivityassumptionT E(YSIS t) - E(YsIS c).(9)investigated.Forexample,U canbe dividedintosubpopThe data(S, Ys) cannowbe usedto estimateT bytaking ulations U1, U2, . . , and on each U, the average rved effectcan be estimated,Ti, T2,. If theT1'svary,theresponseYs fortheunitswithS tandfortheunitswith constanteffectassumptioncannothold.If theTi'sdo notS c. Hence,ifrandomizationis possible,theaverage vary,thentheconstanteffectmaybe plausible.assumptioncausaleffectT can alwaysbe estimated.If U is large,TThe constanteffectassumptionis impliedby theunitcan be estimatedwithhighaccuracy.thatis, if Y,(ul) Y,(u2)andhomogeneityassumption;It is usefulto havea namefortherightsideofEquation Y,(ul) Y,(u2), thenclearlyY,(ul) - Y,(ul) Y,(u2) (9) evenwhentheassumptionof independencedoes not Y,(u2). Hencewe mayviewtheconstanteffectassumptionhold.I willcallittheprimafaciecausaleffectoft(relative as a weakeningoftheassumptionofunithomogeneity.to c) and denoteitbyIfwe makeonlytheconstantassumptionwe saleffect,TPF E(Yt IS t)-E(YC I S c),(10) equals the averagecausal effect,T, in (3). To see thiswhichis algebraicallyfunctionofthe observethatunderconstanteffectwe haveequaltothefollowingonofS:regression Ys T TPF E(Ys I S t) - E(Ys I S c).(11)Y,(u)Y,(u)(13)forall units,u. HenceThe termprimafaciecausaleffectis adaptedfromSuppesE(Yt IS t) T E(Yc I S t),(14)(see Sec. 5) andusedheretodistinguish(11) fromthetrueinEquation(3). Theprima soaveragecausaleffect,T, ntparameterTPF T {E(YC I S t) - E(YC I S c)}. (15)distributionoftheobservablepair(Ys, S). In general,theaveragecausal effectT does not equal the primafacie The termin bracesin (15) is not0 in general,thatis, ifhow- theindependenceofindependence,causaleffectTPF.The assumptionis nottrue.assumptionofunitIt is easyto showthatthestrongerassumptionever, does allow the conclusion that T TPF,that is,does implyequalitybetweenT and TPF.Equation(9).homogeneity4.4 ConstantEffectThe valueoftheaveragecausaleffectT is udies.Itwouldbe ofinterestto a ogramwouldbe thebestto ctgradersof the best programwouldbe reflectedin increasesinstatewideaveragereadingscores.The averagecausaleffectT is an averageand as ages.For example,if the variabilityin the causal effectsYt(u) - Yc(u) is largeoverU, thenT maynotrepresentthecausaleffectofa specificunit,uo,verywell.Ifuois theunitofinterest,thenT maybe irrelevant,no matterhowwe estimatecarefullyit!The assumptionofconstantis thattheeffectofteffecton everyunitis thesame,and underthisassumptionwehavetheequation4.5 Causal Inferencein NonrandomizedObservational StudiesIt is beyondthescopeofthisarticleto applythemodelforcausalinferenceto dthereaderis c),Rosenbaumand Rubin(1983a,b,1984a,b,1985a,b),and temphasiscanis on thewaysthatpre-exposurevariables be usedtowithless stringentreplacethe independenceassumptionthatare ationalstudies.Rosenbaumand Rubinreferredone suchassumptionas "strongignorability."5. COMMENTSON SELECTEDPHILOSOPHERSaboutcausalityby philosoSo muchhas been writtento givean adequatecoverageofphersthatitis impossibletheideasthattheyhaveexpressedin a deasinthecontextT Yt(u) - Yc(u),forall u in U.(12) modelforcausalinferencegivenin Sections3 and 4. Itorevenrepresentative.Hence underthe assumptiontobe exhaustiveof constanteffectT is the makesno attemptAristotledistinguishedfour";causes"'of a thingin hiscausaleffectforeveryunitin U. Thisassumptionis in statisticalmodelsforexperimentsbe- Physics:The ntowhichcause the treatmentt adds a constantamountT to thetheefficientcause (thatwhichmakesthething),and thecontrolresponseforeachunit.This content downloaded from 128.59.22.37 on Thu, 12 Mar 2015 13:41:49 UTCAll use subject to JSTOR Terms and Conditions
950Journal of the American Statistical Association, December hichthethingismade).Itishisnotion ndto ofconstantWecausethatis relevantto ionsof causationthatgrowout of inquiries maythinkwe havea case of"A andnotB" butwe reallyintothemethodsofscience.Locke (1690)proposedthese havea caseof"A' andnotB" forsomeA' thatwemistookforexamplesof "notA and B"). In thedefinitions:"Thatwhichproducesanysimpleor complex forA notebythegeneralname'cause',andthatwhich s produced,'effect'."Althoughit is evidentthatthese ern The other,morefundamentaldefinitionsreferto thesamekindsofthingsthemodelin Section3, theydo littlemorethansuggest tioncan failin the modelis forthe constanteffectasthatthe modelis not out of linewithan ancientphilo- sumptionto failto hold,thatis, forthe causal effectssophicaltradition.It shouldbe noted,however,that Y,(u) - Y,(u) to varywiththeunitu. Hence,ifwe esofa thingratherthanthe regardthosecasesofnonconstanterror,we see a littlemoreeven-handed. to odel.ofthehis- requiresBunge(1959)gavea veryaccessiblediscussionof - othemodel.Wewillhavetobe satisfiedthatat5.1 HumeleastHume's analysisfitsintothe modeland let e judgetheutilit
Statistics and Causal Inference PAUL W. HOLLAND* Problems involving causal inference have dogged at the heels of statistics since its earliest days. Correlation does not imply causation, and yet causal conclusions drawn from a carefully designed experiment are often valid. What can a statistical model say about causation? This question is ad-
Chapter 1 (pp. 1 -7 & 24-33) of J. Pearl, M. Glymour, and N.P. Jewell, Causal Inference in Statistics: A Primer, Wiley, 2016. Correlation Is Not Causation The gold rule of causal analysis: no causal claim can be established purely by a statistical method. . Every causal inf
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causal inference across the sciences. The authors of any Causal Inference book will have to choose which aspects of causal inference methodology they want to emphasize. The title of this introduction reflects our own choices: a book that helps scientists–especial
For Causal Inference We Need: 6/7/2018 dechter, class 8, 276-18 1. A working definition of “causation” 2. A method by which to formally articulate causal assumptions—that is, to create causal models 3. A method by which to link the structure of a causal model to features of data 4.
So a causal effect of X on Y was established, but we want more! X M Y The directed acyclic graph (DAG) above encodes assumptions. Nodes are variables, directed arrows depict causal pathways Here M is caused by X, and Y is caused by both M and X. DAGs can be useful for causal inference: clarify the assumptions taken and facilitate the discussion.
Causal inference with graphical models – in small and big data 1 Outline Association is not causation How adjustment can help or harm Counterfactuals - individual-level causal effect - average causal effect Causal graphs - Graph structure, joint distribution, conditional independencies - how to esti
Causal inference primer 2. Causality from non-experimental data 3. Text as a control 4. Double machine learning . Text and Causal Inference: A Review of Using Text to Remove Confounding from Causal Estimates (ACL 2020) . chern.handout.pdf. Double ML Text Examples. What We’v
polypeptide, or protein. Chapter 8 – From DNA to Proteins Translation converts mRNA messages into polypeptides. A codon is a sequence of three nucleotides that codes for an amino acid. codon for methionine (Met) codon for leucine (Leu) Chapter 8 – From DNA to Proteins The genetic code matches each codon to its amino acid or function. –three stop codons –one start codon .
