Causal Directed Acyclic Graphs - Harvard University

1 Causal Directed Acyclic GraphsKosuke ImaiHarvard UniversitySTAT186/GOV2002 CAUSALINFERENCEFall 2019 Kosuke Imai ( Harvard ) Causal DAGsStat186/Gov2002 Fall 20191 / 16 Elements of DAGs(Pearl. Cambridge UP)G= (E,V)1V: nodes or vertices variables (observed and onobserved)2E: Directed arrows possibly non-zerodirect Causal effectsXZTYUA cyclic: no simultaneity, the future does not cause the pastEncoded assumptionsAbsence of variables: all common (observed and unobserved)causes of any pair of variablesAbsence of arrows: zero Causal effectKosuke Imai ( Harvard ) Causal DAGsStat186/Gov2002 Fall 20192 / 16 DAG TerminologyXYZchain:X Y Zfork:Y X Zinverted fork:X Z YParents (Children): directly causing (caused by) a vertexi jAncestors (Descendents): directly or indirectly causing (causedby) a vertexi jPath: an Acyclic sequence of adjacent nodesCausal path : all arrows pointing away fromTand intoYNon- Causal path : some arrows going against Causal orderCollider.

2 A vertex on a path with two incoming arrowsKosuke Imai ( Harvard ) Causal DAGsStat186/Gov2002 Fall 20193 / 16 Nonparametric Structural Equation Models (NPSEM)Equivalence to the nonparametric structural equation models:Y=f1(T,U, 1)T=f2(X,Z, 2)Z=f3(X, 3)X=f4(U, 4)NPSEM allows:1any functional form2any form of heterogenous effects3any form of interaction effects4 LSEM as a special caseLikelihood function:P(X1,X2,..,XJ) =J j=1P(Xj|pa(Xj))Kosuke Imai ( Harvard ) Causal DAGsStat186/Gov2002 Fall 20194 / 16D-separationDoes the conditional independence,A B|C, hold whereA,B,Care sets of vertices?1 Identify all paths from any vertex inAto any vertex inB2 Check if each path is blocked3If all paths are blocked, thenAisd-separated fromBbyCPath is blocked,1if it includes a noncollider vertex that is inC, or2if it includes a collider that is not inCand no descendant of anycollider is inCIfAandBared-separated,A B|CholdsIfAandBared-connected ( , notd-separated),A6 B|Cin atleast one distribution compatible with DAGK osuke Imai ( Harvard ) Causal DAGsStat186/Gov2002 Fall 20195 / 16D-separation ExampleUZXYTW1 AreWandYmarginally independent of each other?

3 2 What happens if we condition onZ,X,T, or any combination ofthem?Kosuke Imai ( Harvard ) Causal DAGsStat186/Gov2002 Fall 20196 / 16 Backdoor CriterionCan we nonparametrically identify the average effect ofTonYgiven a set of variablesX?Backdoor criterion forX:1No vertex inXis a decendent ofT, and2X d-separates every path betweenTandYthat has an incomingarrow intoT(backdoor path )Need to block all non- Causal pathsIn the previous example, doesXsatisfy the backdoor criterion?Backdoor criterion implies the confounder selection criterion:(VanderWeele and Shpitser. )If there exist a set of observed covariates that meet the backdoorcriterion, it is sufficient to condition on all observed pretreatmentcovariates that either cause treatment, outcome, or :P(Yi(t)) = xP(Y|T=t,X=x)P(X=x)Kosuke Imai ( Harvard ) Causal DAGsStat186/Gov2002 Fall 20197 / 16 Example of Backdoor CriterionU1TX1X2U2YU4U3 Can we identify the Causal effect ofTonYby conditioning onX1?

4 What about conditioning onX1andX2?Kosuke Imai ( Harvard ) Causal DAGsStat186/Gov2002 Fall 20198 / 16M-Structure andM-BiasU1 TXU2 YShould we condition onXor not?Conditioning on too many variables can induce biasPearl s smoking and lung cancer example:X=wearing seatbeltU1=attitudes towards social normsU2=attitudes towards safety and health measuresKosuke Imai ( Harvard ) Causal DAGsStat186/Gov2002 Fall 20199 / 16 Frontdoor Criterion(Pearl. )TMYUU=unobserved confoundersM=mediator Causal mechanismFrontdoor criterion forM:1 Mintercepts all Directed paths fromTtoY2No backdoor path fromTtoM3 All backdoor paths fromMtoYare blocked byTP(Y(t)) = m{P(Mi=m|Ti=t) t P(Y|T=t ,Mi=m)P(Ti=t )}Kosuke Imai ( Harvard ) Causal DAGsStat186/Gov2002 Fall 201910 / 16 Evaluating Backdoor and Frontdoor Criteria(Glynn and Kashin. Am.)

5 Stat. Assoc.)National Job Training Partnership Act (JTPA) studyRandomized experiment: ATT on the wage after 18 monthsadult female: $702 (participation rate 55%)adult male: $700 (participation rate 57%)Non-experimental control groupT: encouragement to participate in the program,M: actual participationY: wage after 18 monthsComparison group for actual participantsbackdoor criterion: those assigned to the control groupfrontdoor criterion: those who chose not to participateKosuke Imai ( Harvard ) Causal DAGsStat186/Gov2002 Fall 201911 / 16 ResultsPG CFJOH FOSPMMFFT 8F FYQBOE PO UIJT SFTVMU IFSF 'JHVSF QSFTFOUT UIF DPNQBSBUJWF QFSGPSNBODFPG UIF GSPOU EPPS BOE CBDL EPPS FTUJNBUPST BDSPTT B WBSJFUZ PG TJNQMF DPOEJUJPOJOH TFUT GPS BEVMUNBMFT BOE BEVMU GFNBMFT SFTQFDUJWFMZ 8F EP OPU BTTVNF MJOFBSJUZ PS BEEJUJWJUZ JO UIF DPOEJUJPOBMFYQFDUBUJPO GVODUJPO&[:|B,N,Y]BOE UIVT VTF LFSOFM CBTFE SFHVMBSJ[FE MFBTU TRVBSFT ,3-4 UPPCUBJO UISFF DPOEJUJPOBM FYQFDUBUJPOT &[:|B.]]

6 = ,Y] &[:|B ,.= ,Y] BOE&[:|B ,.= ,Y] )BJONVFMMFS BOE )B[MFUU 'JHVSF $PNQBSJTPO PG #BDL EPPS BOE 'SPOU EPPS "EKVTUNFOU PO +51" %BUBTFU CZ 5 BSHFU (SPVQ VTJOH,3-4 F DPOEJUJPOJOH TFUT JODMVEF QFSNVUBUJPOT PG UIF GPMMPXJOH WBSJBCMFT BHF SBDF EVNNJFT GPS XIJUF CMBDL BOE PUIFS TJUF EVNNJFT BOE UPUBM FBSOJOHT JO NPOUI PG SBOEPN BTTJHONFOU FMJHJCJMJUZ TDSFFOJOH 3" &4 F FYQFSJNFOUBM FTUJNBUF JT EFOPUFE BT B EPUUFE EBSL HSFZ MJOF XJUI UIF TIBEFE HSFZ SFHJPO SFQSF TFOUJOH UIF DPO EFODF JOUFSWBM CPPUTUSBQQFE QFSDFOUJMF DPO EFODF JOUFSWBMT GPS CPUI BEKVTUNFOUNFUIPET BOE UIF FYQFSJNFOUBM CFODINBSL BSF CBTFE PO SFQMJDBUFT 500005000 NoneAgeRaceSiteAge,RaceAge,SiteRace,Site Earn at t=0 Earn at t=0,AgeEarn at t=0,RaceEarn at t=0,SiteAllConditioning SetEffect on 18 Month EarningsAdult Males 010002000300040005000 NoneAgeRaceSiteAge,RaceAge,SiteRace,Site Earn at t=0 Earn at t=0,AgeEarn at t=0,RaceEarn at t=0,SiteAllConditioning SetAdult FemalesEstimation Method Front doorBack door F SFTVMU JT TUSJLJOH JO UIBU GPS BEVMU NBMFT UIF GSPOU EPPS FTUJNBUFT FYIJCJU VOJGPSNMZ MFTTFTUJNBUJPO FSSPS UIBO UIF CBDL EPPS FTUJNBUFT BDSPTT BMM UIF TQFDJ DBUJPOT XF FYBNJOF F FS SPS VTJOH UIF OVMM DPOEJUJPOJOH TFU GSPN UIF CBDL EPPS FTUJNBUF JT JT OFHBUJWF FSSPS 8F SFQPSU SFTVMUT GSPN ,3-4 IFSF EVF UP PVS SFMVDUBODF UP NBLF TUSPOH QBSBNFUSJD BTTVNQUJPOT CVU XF PCUBJOTJNJMBS SFTVMUT XIFO VTJOH PUIFS NFUIPET TVDI BT 0-4 GPS FTUJNBUJPO Kosuke Imai ( Harvard )

7 Causal DAGsStat186/Gov2002 Fall 201912 / 16 Instrumental Variables(Brito and Pearl. 2002. UAI. )U1 ZTU2YU3 WZis a valid instrumental variable conditional onWif1 Wcontains only non-descendants ofY2W d-separatesZfromYin the subgraphGsobtained by removingedgeT Y3 Wdoes notd-separateZfromTinGsKosuke Imai ( Harvard ) Causal DAGsStat186/Gov2002 Fall 201913 / 16 DAGitty ( )Kosuke Imai ( Harvard ) Causal DAGsStat186/Gov2002 Fall 201914 / 16 Potential Outcomes vs. DAGs ControversyImbens and Rubin (2015):Pearl s work is interesting, and many researchers find his arguments thatpath diagrams are a natural and convenient way to express assumptionsabout Causal structures appealing. In our own work, perhaps influencedby the type of examples arising in social and medical sciences, we havenot found this approach to aid drawing of Causal s blog post:So, what is it about epidemiologists that drives them to seek the light ofnew tools, while economists seek comfort in partial blindness, whilemissing out on the Causal revolution?

8 Can economists do in their headswhat epidemiologists observe in their Graphs ? Can they, for instance,identify the testable implications of their own assumptions? Can theydecide whether the IV assumptions are satisfied in their own models ofreality? Of course they can t; such decisions are intractable to thegraph-less Imai ( Harvard ) Causal DAGsStat186/Gov2002 Fall 201915 / 16 Concluding RemarksPotential outcomes are useful when thinking about treatmentassignment mechanism experiments, quasi-experimentsDAGs are useful when thinking about the Causal structure complex Causal relationships, Causal mechanismsGrowing literature on Causal discoveryReadings:Pearl. (2009).Causality. Cambridge UPElwert. (2013). Chapter 13: Graphical Causal Models inHandbook ofCausal Analysis for Social ResearchPeters, Janzing, and Sch lkopf.

9 (2018).Elements of Causal Inference:Foundations and Learning Algorithms. MIT Imai ( Harvard ) Causal DAGsStat186/Gov2002 Fall 201916 / 16