1. The Assumptions a Causal DAG encodes Michael Rosenblum March 16, 2010

2. Overview I describe the set of assumptions encoded by a causal directed acyclic graph (DAG). I use an example from page 15 of the book Causality by Judea Pearl (2009). This presentation includes animations, so it’s best to watch it as a slideshow. (It may not make sense otherwise).

3. Causal Directed Acyclic Graphs (DAGs) Causal DAG encodes: 1. Assumptions about distribution generating observed data 2. How distribution under hypothetical intervention can be computed from distribution generating observed data T T W W S S R R A A Time of Year Accident Wet Sidewalk RainSprinkler Causal

4. Causal Directed Acyclic Graphs (DAGs) 1. Assumptions about distribution generating observed data (Markov assumption): Each node is conditionally independent of non-descendents given its parents. E.g.P(R|S,T) = P(R|T), P(W|R,S,T) = P(W|R,S), P(A|W,R,S,T) = P(A|W). Furthermore, these conditional independences hold under any interventions. T T W W S S R R A A Time of Year Accident Wet Sidewalk Rain Sprinkler

5. Causal Directed Acyclic Graphs (DAGs) 2. Assumptions about distribution under hypothetical interventions: Except for intervened-on nodes, probability of node given its parents is unchanged by interventions. E.g. Under intervention: do[Sprinkler = off] P(T|do[S=off]) = P(T), P(R|T,do[S=off]) = P(R|T), S=off w.p. 1, P(W|R,do[S=off]) = P(W|R,S=off), P(A|W,do[S=off]) = P(A|W). T T W W S S R R A A Time of Year Accident Wet Sidewalk Rain off Sprinkler

6. Causal Directed Acyclic Graphs (DAGs) 2. Assumptions about distribution under hypothetical interventions: Except for intervened-on nodes, probability of node given its parents is unchanged by interventions. E.g. Under intervention: “do[Wet Sidewalk = wet]” P(T|do[W=wet]) = P(T) P(R|T,do[W=wet]) = P(R|T), P(S|T,do[W=wet]) = P(S|T), W = wet w.p. 1. P(A|do[W=wet]) = P(A|W=wet). T T W W S S R R A A Time of Year Accident Wet Sidewalk Rain wet Sprinkler

7. Causal Directed Acyclic Graphs (DAGs) Structural Equation Model representation: For u 1,…,u 5 independent, unmeasured variables, and f T, f R, f S, f W, f A unknown functions, we have T=f T (u 1 ), R=f R (T,u 2 ), S=f S (T,u 3 ), W=f W (R,S,u 4 ), A=f A (W,u 5 ). T T W W S S R R A A Time of Year Accident Wet Sidewalk Rain Sprinkler

8. Causal Directed Acyclic Graphs (DAGs) Structural Equation Model representation: For u 1,…,u 5 independent, unmeasured variables, and f T, f R, f S, f W, f A unknown functions, we have T=f T (u 1 ), R=f R (T,u 2 ), S=f S (T,u 3 ), S=off, W=f W (R,S,u 4 ), W=f W (R,off,u 4 ) A=f A (W,u 5 ). Setting S=off gives mutilated set of equations. T T W W S S R R A A Time of Year Accident Wet Sidewalk Rain off Sprinkler

9. Causal Directed Acyclic Graphs (DAGs) Structural Equation Model representation: For u 1,…,u 5 independent, unmeasured variables, and f T, f R, f S, f W, f A unknown functions, we have T=f T (u 1 ), R=f R (T,u 2 ), S=f S (T,u 3 ), W=f W (R,S,u 4 ), A=f A (W,u 5 ). T T W W S S R R A A Time of Year Accident Wet Sidewalk Rain Sprinkler

10. Causal Directed Acyclic Graphs (DAGs) Structural Equation Model representation: For u 1,…,u 5 independent, unmeasured variables, and f T, f R, f S, f W, f A unknown functions, we have T=f T (u 1 ), R=f R (T,u 2 ), S=f S (T,u 3 ), W=f W (R,S,u 4 ), W=wet A=f A (W,u 5 ), A=f A (wet,u 5 ). Setting W=wet gives mutilated set of equations. T T W W S S R R A A Time of Year Accident Wet Sidewalk Rain wet Sprinkler

11. Counterfactuals Can Represent Counterfactuals using Structural Eqn. Models: T=f T (u 1 ), R=f R (T,u 2 ), S=f S (T,u 3 ), W=f W (R,S,u 4 ), A=f A (W,u 5 ). E.g. Counterfactual value of A setting W=wet is f A (wet,u 5 ); Counterfactual value of W setting S=off is f W (R,off,u 4 ). T T W W S S R R A A Time of Year Accident Wet Sidewalk Rain Sprinkler

12. MIRA Trial Example Randomized trial 2 study arms (diaphragm arm, control arm) Intensive condom counseling and provision to both arms We want to estimate effect of intervention assignment on HIV infection, holding condom use fixed. That is, we want: P(H=1|do[R=1,C=never])- P(H=1|do[R=0,C=never]). R R C C H H Study Arm Condom Use HIV Infection never

13. MIRA Trial Example We want to estimate effect of intervention assignment on HIV infection, holding condom use fixed. That is, we want: P(H=1|do[R=1,C=never])- P(H=1|do[R=0,C=never]). This causal DAG would imply: P(H=1|do[R=1,C=never]) =P(H=1|R=1,C=never). R R C C H H Study Arm Condom Use HIV Infection never

14. MIRA Trial Example Potential Confounders of effect of condom use on HIV infection: N = Number of Partners Then causal DAG implies: P(H=1|N,do[R=1,C=never]) =P(H=1|N,R=1,C=never). Can multiply each side by P(N) and sum over values of N to get P(H=1|do[R=1,C=never]). R R C C H H Study Arm Condom Use HIV Infection N N

15. MIRA Trial Example Potential Confounders of effect of condom use on HIV infection: N = Number of Partners P = Main Partner Seropositive Then P(H=1|N,P,do[R=1,C=never]) =P(H=1|N,P,R=1,C=never). But we don’t observe P!  R R C C H H Study Arm Condom Use HIV Infection N N P P

16. MIRA Trial Example Unmeasured (hidden) variables represented by dashed circle and dashed lines. In determining what assumptions a Causal DAG encodes, unmeasured variables treated just like measured variables. E.g. P(R|N) = P(R), P(H|N,P,R,do[C=never]) = P(H|N,P,R,C=never). R R C C H H Study Arm Condom Use HIV Infection N N P P