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## Counterfactual models Time dependent confounding

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**Counterfactual modelsTime dependent confounding**Based on Gran and Røysland lectures**Notation**• Disease (outcome)D or Y • Exposure (treatment, action)E or X, A • Confounder (liability)C or L HS**Causal effect**• Two possible outcomes • 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 HS**Association vs. causation**unexposed exposed Causation Association vs. vs. P(D|E=0) P(D|E=1) P(D0) P(D1) P(D|E=1) P(D1) conditional marginal HS**Time dependence**• Individuals followed over time • Censoring • Time varying exposure: E1, E2, … • Time varying covariates: C1, C2,… • Outcome: D HS**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 HS**TimeDependentConfounding, Exercise**C2 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 HS**TimeDependentConfounding, Workshop**C1 C2 E is treatment, D is disease C is a prognostic factor Time points t1 and t2 E1 E2 D Find examples of time dependent confounding in our data HS**Process graphs**• Notation • Variables over time • replaced by process • One process may drive another • Feedback loops P1 P2 P3 P P S P S HS**DAGs and process graphs**DAG Process X1 X2 X3 … X Y1 Y2 Y3 … Y Z Z1 Z2 Z3 … HS**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 HS**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 HS**Methods of adjusting**Method Action Effect Conditioning, Stratification Close path Matching in cohort Remove arrow C Matching in Case-Control Remove arrow E D Matching: smaller matched sample InverseProbabilityWeighting: lager “randomized” sample H.S. MSM, NSM ←parametric→ regression**Handling TimeDependentConfounding**Conditioning Matching, IPTW C C V E D E D HS**InverseProbabilityofTreatmentWeighting**C Simple point treatment (exposure) E D Propensity scores N*w N*w C E D HS**InverseProbabilityofTreatmentWeighting**Time varying treatment (exposure) C1 C2 E is treatment, D is disease C is a prognostic factor Time points t1 and t2 E1 E2 D w1 w2 Weights: Weight at E2: Weight for the entire exposure and covariate history up to time 2 Weight by w1*w2 observed, factual counterfactuals HS**IPTW for time varying exposures**Courtesy of JM Gran HS**Counterfactual modeling**• Aim • Effect of intervention (treatment, action, exposure) • What if? Treated vs not treated • Mimic randomized trial Courtesy of JM Gran HS**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 HS**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 HS