Graphical methods for causal inference from observational data

Miguel A. Hernán

Department of EpidemiologyHarvard School of Public Healthwww.hsph.harvard.edu/causal

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Overview

I. Definition of causal effectn Counterfactuals

II. Representation of causal effectsn Directed acyclic graphs

III. Causation and Associationn D-separation

IV. Identifiability of causal effectsn Back-door criterion/Unmeasured confounding

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An intuitive definition of cause

o Ian took the pill on April 1, 2002n Five days later, he died

o Had Ian not taken the pill on April 1, 2002 (all others things being equal)n Five days later, he would have died

o Did the pill cause Ian’s death?

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An intuitive definition of cause

o Jim didn’t take the pill on April 1, 2002n Five days later, he was alive

o Had Jim taken the pill on April 1, 2002 (all others things being equal)n Five days later, he would have been alive

o Did the pill cause Jim’s survival?

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Notation for actual data

o D=1 if patient died, 0 otherwisen Di=1, Dj=0

o A=1 if patient treated, 0 otherwisen Ai=1, Aj=0

00Jim

11Ian

DAID

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Notation for ideal datao Da=0=1 if patient would have died, had he

not taken the pilln Di, a=0=1, Dj, a=0=0

o Da=1=1 if patient would have died, had he taken the pilln Di, a=1=1, Dj, a=1=0

0000Jim

1111Ian

Da=1Da=0DAID

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(Individual) Causal effect

o For Ian: n Pill has a causal effect if Di, a=0 ? Di, a=1

o For Jim: n Pill has a causal effect if Dj, a=0 ? Dj, a=1

o Unfortunately, individual causal effects cannot be determined because…

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Available data set

o Da=0 and Da=1 are counterfactual outcomeso Unobserved but linked to observed

outcomes: If Ai=1, then Di, a=1 =Di

?110Leo…

0?01Ken?000Jim1?11IanDa=1Da=0DAID

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(Average) Causal effect

o In the population, the pill has a causal effect if E[Da=0] ? E[Da=1]

o E[Da=0] and E[Da=1] can be computed under certain conditions

o Without loss of generality, we will use dichotomous outcomes: E[Da] = Pr[Da=1]

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Ideal randomized experiment

o Large sample size, full compliance, no loss to follow-up

o Pr[Da=0=1], Pr[Da=1=1] can be estimated

o Treatment assignment is independent of counterfactual outcome: Da î An Pr[Da=1=1] = Pr[Da=1=1|A=1] =

Pr[D=1|A=1]o Intention to treat analysis has causal

interpretation

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Point exposures

o Uncommon in epidemiologyn Surgery, one-dose vaccine, …

o Only one possible causal questionn Pr[Da=1=1] ? Pr[Da=0=1] ?

o Effect measured in different scalesn Pr[Da=1=1] - Pr[Da=0=1]n Pr[Da=1=1] / Pr[Da=0=1]n (Pr[Da=1=1]/Pr[Da=1=0])/(Pr[Da=0=1]/Pr[Da=0=0])

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Time-varying exposures

o Common in epidemiology: drugs, diet, smoking…

o More than two counterfactual outcomes

o Many possible causal questionsn Pr[Da=1,b=1=1] ? Pr[Da=0, b=0=1] ?n Pr[Da=1,b=0=1] ? Pr[Da=0, b=1=1] ?n etc.

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Counterfactual theories of increasing generalityo Neyman (1923)n Effects of point exposures

in randomized experiments

o Rubin (1974)n Effects of point exposures

in randomized andobservational studies

o Robins (1986)n Total and direct effects of

time-varying exposures in longitudinal studies

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Overview

I. Definition of causal effectn Counterfactuals

II. Representation of causal effectsn Directed acyclic graphs

III. Causation and Associationn D-separation

IV. Identifiability of causal effectsn Back-door criterion/Unmeasured confounding

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Diagrams for causal structures

A DL

o DIRECTED edges (arrows) linking nodes (variables) o ACYCLIC links because no arrows from descendants

(effects) to ancestors (causes)o GRAPHSo Pearl (1995); Spirtes, Glymour and Scheines

(1993)

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Causal DAGs

A Dn means Pr[Da=1=1] = Pr[Da=0=1]

A Dn means Pr[Da=1=1] = Pr[Da=0=1] or

Pr[Da=1=1] ? Pr[Da=0=1]

o Information is in the missing arrows

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Expert knowledge and causal DAGs

o Complete DAGs do not exclude any possible causal effect

o Incomplete DAGsencode expert knowledge in the form of missing arrows

A DL

A DL

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DAGs and causal DAGs

o A DAG is a causal DAG if the common causes of any pair of variables in the graph are also in the DAG

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Overview

I. Definition of causal effectn Counterfactuals

II. Representation of causal effectsn Directed acyclic graphs

III. Causation and Associationn D-separation

IV. Identifiability of causal effectsn Back-door criterion/Unmeasured confounding

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Causal effect implies association

o Pr[Da=1=1] ? Pr[Da=0=1]

o Pr[D=1|A=1] ? Pr[D=1|A=0]

BA B DA

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Common causes imply association

o Pr[Da=1=1] = Pr[Da=0=1]

o Pr[D=1|A=1] ? Pr[D=1|A=0] in general

o Confounding

A DL

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What do common effects imply?

o Pr[Da=1=1] = Pr[Da=0=1]

o Pr[D=1|A=1] = Pr[D=1|A=0]D î A

D LA

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Two variables are marginally associated if…

o They are cause and effect

o They share common causes

o (By chance)

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Conditional independence

o Pr[Da=1=1] ? Pr[Da=0=1]o Pr[D=1|A=1,B=b] =

Pr[D=1|A=0,B=b] D î A |B for all values b

o Pr[Da=1=1] = Pr[Da=0=1]o Pr[D=1|A=1,L= l] =

Pr[D=1|A=0,L= l] D î A |L for all values l

B DA

A DL

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Conditioning on common effects

o Pr[Da=1=1] = Pr[Da=0=1]

o Pr[D=1|A=1,L= l] ? Pr[D=1|A=0,L= l] for some value l

o Selection bias

D LA

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Similarly…

o Pr[Da=1=1] = Pr[Da=0=1]

o Pr[D=1|A=1,S=s] ? Pr[D=1|A=0,S=s] for some value s

o Selection bias

D LA S

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Examples of selection bias

o Cohort studiesA: HAART, D: deathC: censoringU: immunologic status

o Case-control studiesA: PM hormonesD: ThromboembolismB: Hip fractureS: Selection

C D

U

A

D SA

B

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Examples of ascertainment bias

o A: exogenous estrogens

o E: endometrial cancero C: vaginal bleedingo D: ascertained

endometrial cancer

o A: oral contraceptives

o E: thromboembolismo C: medical careo D: ascertained

thromboembolism

DA E C

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Sources of association

o Cause and effect

o Common causes

o Conditioning on common effectsn In design or analysis

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D-separation / Moralizationo Graphical rules to decide whether two variables are

(conditionally) independento Pearl (1995); Spirtes, Glymour, and Scheines (1993)n Appendix of Hernán et al, AJE 2002

Judea PearlProfessor of Computer Science, UCLA

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Faithfulness

o A may have a causal effect on D and yetn Pr[Da=1=1] = Pr[Da=0=1]n Pr[D=1|A=1] = Pr[D=1|A=0]

o For example, if A causes D in half of the population, and prevents D in the other half

o DAG not faithful to joint distributiono Probably rare

DA

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Overview

I. Definition of causal effectn Counterfactuals

II. Representation of causal effectsn Directed acyclic graphs

III. Causation and Associationn D-separation

IV. Identifiability of causal effectsn Back-door criterion/Unmeasured confounding

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Can we estimate the causal effect of interest?

o The association between exposure and outcome has 2 components:n Cause and effectn Common causes (confounding)

o Can we eliminate the latter?

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Extreme examples

o No common causesn Association=causation

o No causal effectn Association=confounding

DA

A DL

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Causal effect can be estimated if

o No common causes n No back-door pathn No confoundingn Da î A

o Common causes butn Enough data to block

the back-door pathsn No unmeasured

confoundingn Da î A |L

A DL

DA

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Standard definition of confounder

o L is a confounder if it isn associated to An associated to D conditional on An Not in the causal pathway

o No reference to common causes

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Does it work?

A DU

L

A DL A DLU

A D

U1

L

U2

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Message:

o We need expert knowledge to determine if we should adjust for a variable