The Assumptions a Causal DAG encodes

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# The Assumptions a Causal DAG encodes - PowerPoint PPT Presentation

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

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Michael Rosenblum

March 16, 2010

The Assumptions a Causal DAG encodes

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).

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

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

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

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

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

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

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

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

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

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

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

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

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

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