The Assumptions a Causal DAG encodes

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

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) Time of Year T Causal DAG encodes: 1. Assumptions about distribution generating observed data 2. How distribution under hypothetical intervention can be computed from distribution generating observed data Causal Sprinkler Rain R S Wet Sidewalk W A Accident

4. Causal Directed Acyclic Graphs (DAGs) Time of Year T 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. Sprinkler Rain R S Wet Sidewalk W A Accident

5. Causal Directed Acyclic Graphs (DAGs) Time of Year T 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). Sprinkler Rain R S Wet Sidewalk off W A Accident

6. Causal Directed Acyclic Graphs (DAGs) Time of Year T 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). Sprinkler Rain R S Wet Sidewalk W wet A Accident

7. Causal Directed Acyclic Graphs (DAGs) Time of Year T Structural Equation Model representation: For u1,…,u5 independent, unmeasured variables, and fT, fR, fS, fW, fA unknown functions, we have T=fT(u1), R=fR(T,u2), S=fS(T,u3), W=fW(R,S,u4), A=fA(W,u5). Sprinkler Rain R S Wet Sidewalk W A Accident

8. Causal Directed Acyclic Graphs (DAGs) Time of Year T Structural Equation Model representation: For u1,…,u5 independent, unmeasured variables, and fT, fR, fS, fW, fA unknown functions, we have T=fT(u1), R=fR(T,u2), S=fS(T,u3),S=off, W=fW(R,S,u4), W=fW(R,off,u4) A=fA(W,u5). Setting S=off gives mutilated set of equations. Sprinkler Rain R S Wet Sidewalk off W A Accident

9. Causal Directed Acyclic Graphs (DAGs) Time of Year T Structural Equation Model representation: For u1,…,u5 independent, unmeasured variables, and fT, fR, fS, fW, fA unknown functions, we have T=fT(u1), R=fR(T,u2), S=fS(T,u3), W=fW(R,S,u4), A=fA(W,u5). Sprinkler Rain R S Wet Sidewalk W A Accident

10. Causal Directed Acyclic Graphs (DAGs) Time of Year T Structural Equation Model representation: For u1,…,u5 independent, unmeasured variables, and fT, fR, fS, fW, fA unknown functions, we have T=fT(u1), R=fR(T,u2), S=fS(T,u3), W=fW(R,S,u4), W=wet A=fA(W,u5), A=fA(wet,u5). Setting W=wet gives mutilated set of equations. Sprinkler Rain R S Wet Sidewalk W wet A Accident

11. Counterfactuals Time of Year T Can Represent Counterfactuals using Structural Eqn. Models: T=fT(u1), R=fR(T,u2), S=fS(T,u3), W=fW(R,S,u4), A=fA(W,u5). E.g. Counterfactual value of A setting W=wet is fA(wet,u5); Counterfactual value of W setting S=off is fW(R,off,u4). Sprinkler Rain R S Wet Sidewalk W A Accident

12. MIRA Trial Example Study Arm 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]). Condom Use R C never H HIV Infection

13. MIRA Trial Example Study Arm 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). Condom Use R C never H HIV Infection

14. MIRA Trial Example Study Arm 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]). Condom Use R C H N HIV Infection

15. MIRA Trial Example Study Arm 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!  Condom Use R C P H N HIV Infection

16. MIRA Trial Example Study Arm 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). Condom Use R C P H N HIV Infection