Counterfactual models Time dependent confounding Based on Gran and Røysland lectures

Counterfactuals May-19 HS

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

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

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

Time dependent confounding May-19 HS

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

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

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

Process graphs May-19 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 May-19 HS

DAGs and process graphs X2 X1 X3 Y2 Y1 Y3 Z2 Z1 Z3 … Y Z X May-19 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 May-19 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 May-19 HS

Analysis under TimeDependentConfounding May-19 HS

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

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

InverseProbability ofTreatmentWeighting C 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

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

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

Courtesy of JM Gran May-19 HS

May-19 HS

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

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

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