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Graphical methods for causal inference from observational data
Miguel A. Hernán
Department of EpidemiologyHarvard School of Public Healthwww.hsph.harvard.edu/causal
5/23/2002 Causal inference - Miguel Hernán 2
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
5/23/2002 Causal inference - Miguel Hernán 3
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?
5/23/2002 Causal inference - Miguel Hernán 4
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?
5/23/2002 Causal inference - Miguel Hernán 5
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
5/23/2002 Causal inference - Miguel Hernán 6
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
5/23/2002 Causal inference - Miguel Hernán 7
(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…
5/23/2002 Causal inference - Miguel Hernán 8
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
5/23/2002 Causal inference - Miguel Hernán 9
(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]
5/23/2002 Causal inference - Miguel Hernán 10
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
5/23/2002 Causal inference - Miguel Hernán 11
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])
5/23/2002 Causal inference - Miguel Hernán 12
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.
5/23/2002 Causal inference - Miguel Hernán 13
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
5/23/2002 Causal inference - Miguel Hernán 14
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
5/23/2002 Causal inference - Miguel Hernán 15
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)
5/23/2002 Causal inference - Miguel Hernán 16
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
5/23/2002 Causal inference - Miguel Hernán 17
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
5/23/2002 Causal inference - Miguel Hernán 18
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
5/23/2002 Causal inference - Miguel Hernán 19
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
5/23/2002 Causal inference - Miguel Hernán 20
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
5/23/2002 Causal inference - Miguel Hernán 21
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
5/23/2002 Causal inference - Miguel Hernán 22
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
5/23/2002 Causal inference - Miguel Hernán 23
Two variables are marginally associated if…
o They are cause and effect
o They share common causes
o (By chance)
5/23/2002 Causal inference - Miguel Hernán 24
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
5/23/2002 Causal inference - Miguel Hernán 25
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
5/23/2002 Causal inference - Miguel Hernán 26
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
5/23/2002 Causal inference - Miguel Hernán 27
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
5/23/2002 Causal inference - Miguel Hernán 28
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
5/23/2002 Causal inference - Miguel Hernán 29
Sources of association
o Cause and effect
o Common causes
o Conditioning on common effectsn In design or analysis
5/23/2002 Causal inference - Miguel Hernán 30
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
5/23/2002 Causal inference - Miguel Hernán 31
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
5/23/2002 Causal inference - Miguel Hernán 32
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
5/23/2002 Causal inference - Miguel Hernán 33
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?
5/23/2002 Causal inference - Miguel Hernán 34
Extreme examples
o No common causesn Association=causation
o No causal effectn Association=confounding
DA
A DL
5/23/2002 Causal inference - Miguel Hernán 35
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
5/23/2002 Causal inference - Miguel Hernán 36
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