Causal Directed Acyclic Graphs - Harvard University · Elements of DAGs (Pearl. 2000. Causality. Cambridge UP) G= (E;V) 1 V: nodes or vertices variables (observed and onobserved)

Causal Directed Acyclic Graphs

Kosuke Imai

Harvard University

STAT186/GOV2002 CAUSAL INFERENCE

Fall 2019

Kosuke Imai (Harvard) Causal DAGs Stat186/Gov2002 Fall 2019 1 / 16

Elements of DAGs (Pearl. 2000. Causality. Cambridge UP)

G = (E ,V )1 V : nodes or vertices variables (observed and onobserved)2 E : directed arrows possibly non-zero direct causal effects

X

Z T Y

U

Acyclic: no simultaneity, the future does not cause the pastEncoded assumptions

Absence of variables: all common (observed and unobserved)causes of any pair of variablesAbsence of arrows: zero causal effect


DAG Terminology

X

Y Z

chain: X → Y → Zfork: Y ← X → Zinverted fork: X → Z ← Y

Parents (Children): directly causing (caused by) a vertex i → jAncestors (Descendents): directly or indirectly causing (causedby) a vertex i → · · · → j

Path: an acyclic sequence of adjacent nodesCausal path: all arrows pointing away from T and into YNon-causal path: some arrows going against causal order

Collider: a vertex on a path with two incoming arrows


Nonparametric Structural Equation Models (NPSEM)

Equivalence to the nonparametric structural equation models:

Y = f1(T ,U, ε1)T = f2(X ,Z , ε2)Z = f3(X , ε3)X = f4(U, ε4)

NPSEM allows:1 any functional form2 any form of heterogenous effects3 any form of interaction effects4 LSEM as a special case

Likelihood function:

P(X1,X2, . . . ,XJ) =J∏

j=1

P(Xj | pa(Xj))


D-separation

Does the conditional independence, A ⊥⊥ B | C, hold whereA,B,C are sets of vertices?

1 Identify all paths from any vertex in A to any vertex in B2 Check if each path is blocked3 If all paths are blocked, then A is d-separated from B by C

Path is blocked,1 if it includes a noncollider vertex that is in C, or2 if it includes a collider that is not in C and no descendant of any

collider is in C

If A and B are d-separated, A ⊥⊥ B | C holdsIf A and B are d-connected (i.e., not d-separated), A 6⊥⊥ B | C in atleast one distribution compatible with DAG


D-separation Example

U

Z X Y

TW

1 Are W and Y marginally independent of each other?2 What happens if we condition on Z , X , T , or any combination of

them?


Backdoor Criterion

Can we nonparametrically identify the average effect of T on Ygiven a set of variables X?

Backdoor criterion for X :1 No vertex in X is a decendent of T , and2 X d-separates every path between T and Y that has an incoming

arrow into T (backdoor path)

Need to block all non-causal pathsIn the previous example, does X satisfy the backdoor criterion?

Backdoor criterion implies the confounder selection criterion:(VanderWeele and Shpitser. 2011. Biometrics)

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 both.Estimation: P(Yi(t)) =

∑x P(Y | T = t ,X = x)P(X = x)


Example of Backdoor Criterion

U1

T

X1 X2

U2 Y

U4U3

Can we identify the causal effect of T on Y by conditioning on X1?What about conditioning on X1 and X2?


M-Structure and M-Bias

U1

T X

U2

Y

Should we condition on X or 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 measures


Frontdoor Criterion (Pearl. 1995. Biometrika)

T

M Y

U

U = unobserved confoundersM = mediator causal mechanismFrontdoor criterion for M:

1 M intercepts all directed paths from T to Y2 No backdoor path from T to M3 All backdoor paths from M to Y are blocked by T

P(Y (t)) =∑

m

{P(Mi = m | Ti = t)

∑t ′

P(Y | T = t ′,Mi = m)P(Ti = t ′)

}


Evaluating Backdoor and Frontdoor Criteria(Glynn and Kashin. 2018. J. Am. Stat. Assoc. )

National Job Training Partnership Act (JTPA) studyRandomized experiment: ATT on the wage after 18 months

adult 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 months

Comparison group for actual participantsbackdoor criterion: those assigned to the control groupfrontdoor criterion: those who chose not to participate


Results

PG CFJOH FOSPMMFFT� 8F FYQBOE PO UIJT SFTVMU IFSF� 'JHVSF � QSFTFOUT UIF DPNQBSBUJWF QFSGPSNBODF

PG UIF GSPOU�EPPS BOE CBDL�EPPS FTUJNBUPST BDSPTT B WBSJFUZ PG TJNQMF DPOEJUJPOJOH TFUT GPS BEVMU

NBMFT BOE BEVMU GFNBMFT SFTQFDUJWFMZ� 8F EP OPU BTTVNF MJOFBSJUZ PS BEEJUJWJUZ JO UIF DPOEJUJPOBM

FYQFDUBUJPO GVODUJPO &[:|B,N, Y] BOE UIVT VTF LFSOFM�CBTFE SFHVMBSJ[FE MFBTU TRVBSFT ,3-4 UP

PCUBJO UISFF DPOEJUJPOBM FYQFDUBUJPOT� &[:|B�,. = �, Y] &[:|B�,. = �, Y] BOE &[:|B�,. = �, Y]

)BJONVFMMFS BOE )B[MFUU ��ƬƬ

'JHVSF �� $PNQBSJTPO PG #BDL�EPPS BOE 'SPOU�EPPS "EKVTUNFOU PO +51" %BUBTFU CZ 5BSHFU (SPVQ VTJOH,3-4� ćF 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 TDSFFOJOH3"�&4� ćF FYQFSJNFOUBM FTUJNBUF JT EFOPUFE BT B EPUUFE EBSL HSFZ MJOF XJUI UIF TIBEFE HSFZ SFHJPO SFQSF�TFOUJOH UIF �� DPOĕEFODF JOUFSWBM� �� CPPUTUSBQQFE QFSDFOUJMF DPOĕEFODF JOUFSWBMT GPS CPUI BEKVTUNFOUNFUIPET BOE UIF FYQFSJNFOUBM CFODINBSL BSF CBTFE PO �� SFQMJDBUFT�

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Age Race

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ite

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Conditioning Set

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Adult Males

● ● ● ● ● ● ● ● ● ● ● ●

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1000

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Age Race

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Conditioning Set

Adult FemalesEstimation Method● Front−door

Back−door

ćF SFTVMU JT TUSJLJOH JO UIBU GPS BEVMU NBMFT UIF GSPOU�EPPS FTUJNBUFT FYIJCJU VOJGPSNMZ MFTT

FTUJNBUJPO FSSPS UIBO UIF CBDL�EPPS FTUJNBUFT BDSPTT BMM UIF TQFDJĕDBUJPOT XF FYBNJOF� ćF FS�

SPS VTJOH UIF OVMM DPOEJUJPOJOH TFU GSPN UIF CBDL�EPPS FTUJNBUF JT �� ćJT OFHBUJWF FSSPS

ƬƬ8F 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�

��


Instrumental Variables (Brito and Pearl. 2002. UAI. )

U1

Z T

U2

Y

U3 W

Z is a valid instrumental variable conditional on W if1 W contains only non-descendants of Y2 W d-separates Z from Y in the subgraph Gs obtained by removing

edge T → Y3 W does not d-separate Z from T in Gs


DAGitty (http://dagitty.net/)


http://dagitty.net/

Potential Outcomes vs. DAGs Controversy

Imbens 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 inferences.

Pearl’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? 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 mind.


Concluding Remarks

Potential outcomes are useful when thinking about treatmentassignment mechanism experiments, quasi-experiments

DAGs are useful when thinking about the causal structure complex causal relationships, causal mechanisms

Growing literature on causal discovery

Readings:Pearl. (2009). Causality. Cambridge UPElwert. (2013). Chapter 13: Graphical Causal Models in Handbook ofCausal Analysis for Social ResearchPeters, Janzing, and Schölkopf. (2018). Elements of Causal Inference:Foundations and Learning Algorithms. MIT Press.


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Causal Directed Acyclic Graphs - Harvard University · Elements of DAGs (Pearl. 2000. Causality. Cambridge UP) G= (E;V) 1 V: nodes or vertices variables (observed and onobserved)