1. Counterfactual models Time dependent confounding
Based on Gran and Røysland lectures

2. Counterfactuals
May-19 HS

3. Notation Disease (outcome) D or Y
Exposure (treatment, action) E or X, A
Confounder (liability) C or L
May-19 HS

4. Causal effect Two possible outcomes Causal effect
Outcome if treated: D1
Outcome if untreated: D0
Counterfactuals
Potential outcomes
Causal effect
Individual: D1i-D0i
Average: E(D1-D0)
Fundamental problem: either D1 or D0 is missing
May-19 HS

5. Association vs. causation
unexposed
exposed
vs.
P(D|E=0)
P(D|E=1)
Association
vs.
P(D0)
P(D1)
Causation
P(D|E=1)  P(D1)
conditional  marginal
May-19 HS

6. Time dependent confounding
May-19 HS

7. Time dependence Individuals followed over time Censoring
Time varying exposure: E1, E2, …
Time varying covariates: C1, C2,…
Outcome: D
May-19 HS

8. Time dependent confounding
“Normal” confounding (point exposure)
Time dependent confounding C
C is a common cause of E and D E D C1 C2
Time points t1 and t2
C is a common cause of E and D
C is in the causal path from E to D E1 E2 D
Conditioning on C will remove confounding
but will also remove part of the effect
May-19 HS

9. TimeDependentConfounding, Exercise
E is treatment, D is disease
C is a prognostic factor E1 E2 D
Initial treatment (at t1) will influence C
which will determine later treatment (at t2)
Verify that C is a time dependent confounder
May-19 HS

10. TimeDependentConfounding, Workshop
E is treatment, D is disease
C is a prognostic factor E1 E2 D
Time points t1 and t2
Find examples of time dependent confounding
in our data
May-19 HS

11. Process graphs
May-19 HS

12. Process graphs Notation Variables over time replaced by process
One process may drive another
Feedback loops
P P P3 P P S P S
May-19 HS

13. DAGs and process graphsX2 X1 X3 Y2 Y1 Y3 Z2 Z1 Z3 … Y Z X
May-19 HS

14. TimeDependentConfounding as process
DAG
Process C C1 C2 E D E1 E2 D
Conditions for TimeDependentConfounding
1) C is a confounder for E on D
2) C is a mediator for E on D
May-19 HS

15. 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-19 HS

16. Analysis under TimeDependentConfounding
May-19 HS

17. Methods of adjusting E D C Conditioning, Stratification Close path
Action
Effect
Conditioning, Stratification
Close path
Matching in cohort
Remove arrow
Stratification: non-parametric adjustment
Regression: parametric adjustment E D C
Matching in Case-Control
Remove arrow
Matching: smaller matched sample
InverseProbabilityWeighting: lager “randomized” sample
May-19
H.S.
MSM, NSM
←parametric→
regression

18. Handling TimeDependentConfounding
Conditioning
Matching, IPTW C C V E D E D
May-19 HS

19. InverseProbability ofTreatmentWeightingC
Simple point treatment (exposure) E D
Subjects E 1
sum C
300
100
400
200
600
1000
Probabilities E 1
sum C
0.75
0.25
0.33
0.67
Weights E 1 C
1.3
4.0
3.0
1.5
𝑃(𝐸)
𝑃( 𝐸 )
1/𝑃(𝐸)
1/𝑃( 𝐸 )
Propensity
scores
"Subjects" E 1
sum C
400
800
600
1200
1001
1000
2000
N*w
N*w  C E D 
May-19 HS

20. InverseProbability ofTreatmentWeighting
Time varying treatment (exposure) C1 C2
E is treatment, D is disease
C is a prognostic factor E1 E2 D
Time points t1 and t2
Weights: w1 w2
Weight at E2:
Weight for the entire exposure and covariate history
up to time 2
Weight by w1*w2
observed, factual
counterfactuals
May-19 HS

21. IPTW for time varying exposures
Courtesy of JM Gran
May-19 HS

22. Courtesy of JM Gran
May-19 HS

23. May-19 HS

24. Counterfactual modeling
Aim
Effect of intervention (treatment, action, exposure)
What if? Treated vs not treated
Mimic randomized trial
Courtesy of JM Gran
May-19 HS

25. 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-19 HS

26. 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-19 HS