Counterfactual models
Time dependent confounding
Based on Gran and Røysland
lectures
Counterfactuals
May-16
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Notation
• Disease (outcome)
D
or
Y
• Exposure (treatment, action)
E
or
X, A
• Confounder (liability)
C
or
L
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Causal effect
• Two possible outcomes
– Outcome if treated:
– Outcome if untreated:
D1
D0
Counterfactuals
Potential outcomes
• Causal effect
– Individual:
– Average:
D1i-D0i
E(D1-D0)
Fundamental problem: either D1 or D0 is missing
May-16
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Association vs. causation
unexposed
exposed
Causation
Association
vs.
vs.
P(D|E=0)
P(D0)
P(D|E=1)
P(D1)
P(D|E=1)  P(D1)
conditional  marginal
May-16
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Time dependent confounding
May-16
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Time dependence
• Individuals followed over time
• Censoring
• Time varying exposure: E1, E2, …
• Time varying covariates: C1, C2,…
• Outcome:
May-16
D
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Time dependent confounding
• “Normal” confounding (point exposure)
C
E
C is a common cause of E and D
D
• Time dependent confounding
C1
E1
C2
E2
D
Time points t1 and t2
C is a common cause of E and D
C is in the causal path from E to D
Conditioning on C will remove confounding
but will also remove part of the effect
May-16
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TimeDependentConfounding, Exercise
C2
E1
E2
E is treatment, D is disease
C is a prognostic factor
D
Initial treatment (at t1) will influence C
which will determine later treatment (at t2)
Verify that C is a time dependent confounder
May-16
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TimeDependentConfounding, Workshop
C1
C2
E1
E2
E is treatment, D is disease
C is a prognostic factor
D
Time points t1 and t2
Find examples of time dependent confounding
in our data
May-16
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Process graphs
May-16
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Process graphs
• Notation
– Variables over time
– replaced by process
P1
P2
P3
P
– One process may drive another
P
S
– Feedback loops
P
S
May-16
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DAGs and process graphs
DAG
Process
X1
X2
X3
…
Y1
Y2
Y3
…
Z1
Z2
Z3
…
May-16
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Y
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TimeDependentConfounding as process
DAG
C1
Process
C
C2
E
E1
E2
D
D
Conditions for TimeDependentConfounding
1) C is a confounder for
2) C is a mediator for
May-16
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E on D
E on D
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Exercise: HIV treatment
Follow HIV patients over time
Treat is CD4 count is low, treatment will increase CD4 count
Estimate the effect of treatment on death
CD4
Censoring
Death
Treatment
Assuming that DAG rules carry over:
Show that censoring gives bias
Show that CD4 count is a TimeDependentConfounder
May-16
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Analysis under TimeDependentConfounding
May-16
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Methods of adjusting
Method
Action
Effect
C
Conditioning, Stratification
Close path
E
D
C
Matching in cohort
Remove arrow
E
Matching in Case-Control
Matching:
InverseProbabilityWeighting:
May-16
C
Remove arrow
smaller matched sample
lager “randomized” sample
H.S.
D
E
D
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Handling TimeDependentConfounding
Conditioning
Matching, IPTW
C
C
V
E
May-16
D
E
HS
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InverseProbability ofTreatmentWeighting
C
Simple point treatment (exposure)
E
C
1
0
D
Subjects
Probabilities
Weights
E
E
E
1
0
300
200
100
400
sum
400
600
1000
"Subjects"
N*w
C
1
0
E
N*w
1
0
400
600
1001
400
600
1000
May-16
sum
800
1200
2000
𝑃(𝐸)
1
C
1
0
0.75
0.33
𝑃(𝐸)
sum
0
0.25 1
0.67 1
1/𝑃(𝐸) 1/𝑃(𝐸)
C
1
0
1
0
1.3
3.0
4.0
1.5
Propensity
scores

C

E
HS
D
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InverseProbability ofTreatmentWeighting
Time varying treatment (exposure)
Weights:
C1
C2
E1
E2
D
w1
w2
Weight at E2:
Weight for the entire exposure and covariate history
up to time 2
Weight by w1*w2
E1
1
0
1
0
May-16
E2
1
1
0
0
E is treatment, D is disease
C is a prognostic factor
Time points t1 and t2
observed, factual
counterfactuals
HS
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IPTW for time varying exposures
Courtesy of JM Gran
May-16
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May-16
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Courtesy of JM Gran
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May-16
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Counterfactual modeling
• Aim
– Effect of intervention (treatment, action,
exposure)
– What if?
Treated vs not treated
– Mimic randomized trial
Courtesy of JM Gran
May-16
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• Counterfactual and graphical models
• Counterfactual models and graphical
models can be seen as the two main
frameworks for causal inference
• Has been shown that many fundamental
concepts are equivalent in both
frameworks
Courtesy of JM Gran
May-16
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History of counterfactual modeling
• Goes back to Neyman (1923), Fisher
(1935) and Cochran and Cox (1950)
• Formalized by Rubin (1974 and later) typically referred to as the potential
outcome framework
• Roots in economic literature through Roy
(1951), Quandt (1972) and Heckman
(1974 and later)
• Extended by Robins (1986 and later)
Courtesy of JM Gran
May-16
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