Judea Pearl

Computer Science Department

UCLA

www.cs.ucla.edu/~judea

DIRECT AND INDIRECT EFFECTS

QUESTIONS ASKED

• Why decompose effects?• What is the semantics of direct and indirect effects?• What are the policy implications of direct and

indirect effects?• Can path-analytic techniques be extended to

nonlinear and nonparametric models?• When can direct and indirect effect be estimated

consistently from experimental and nonexperimental data.

1. Direct (or indirect) effect may be more transportable.2. Indirect effects may be prevented or controlled.

3. Direct (or indirect) effect may be forbidden

WHY DECOMPOSEEFFECTS?

Pill

Thrombosis

Pregnancy

+

+

Gender

Hiring

Qualification

EFFECT-DECOMPOSITIONIN LINEAR MODELS

X Z

Y

ca

b

effect Indirect effect Direct effect Total

a bc

Definition: ))(),(|( zdoxdoYEx

a

COUNTERFACTUALS:STRUCTURAL SEMANTICS

Notation: Yx(u) = y Abbreviation: yx

Formal: Y has the value y in the solution to a mutilated system of equations, where the equation for X is replaced by a constant X=x.

u

Yx(u)=y

Z

W

X=x

u

Y

Z

W

X

Probability of Counterfactuals:

FunctionalBayes Net

))(|()())(()( xdoyPu

uPyuxYPyxYP

tindependen- ))(),(|(

))(|(

DETEIE

ZzdoxdoYEx

DE

xdoYEx

TE

TOTAL, DIRECT, AND INDIRECT EFFECTS HAVE SIMPLE SEMANTICS

IN LINEAR MODELS

X Z

Y

ca

b z = bx + 1

y = ax + cz + 2

a + bc

bc

a

z = f (x, 1)y = g (x, z, 2)

????

))(),(|(

))(|(

IE

zdoxdoYEx

DE

xdoYEx

TE

X Z

Y

SEMANTICS BECOMES NONTRIVIALIN NONLINEAR MODELS

(even when the model is completely specified)

Dependent on z?

Void of operational meaning?

Indirect Effect?

NEED OF FORMALIZATION

X Z

AND

Y

=

What is the direct effect of X on Y?

Starting from X=0, (and Z=0 and Y=0)

Total Effect: Change X from 0 to 1, and test the change in Y.

Controlled DE: Keep Z constant at Z=0, or Z=1, and change X=0 to X=1.

Controlled IE: None.

Natural DE: Keep Z constant at its current value, and change X to 1.Natural IE: Keep X at 0, but set Z to what it would be if X were 1.

TWO CONCEPTIONS OF DIRECTAND INDIRECT EFFECTS:

Controlled vs. Natural

X Z

AND

Y

=

``The central question in any employment-discrimination case is whether the employer would have taken the same action had the employee been of different race (age, sex, religion, national origin etc.) and everything else had been the same’’

[Carson versus Bethlehem Steel Corp. (70 FEP Cases 921, 7th Cir. (1996))]

x = male, x = femaley = hire, y = not hirez = applicant’s qualifications

LEGAL DEFINITIONS TAKE THE NATURAL CONCEPTION

(FORMALIZING DISCRIMINATION)

NO DIRECT EFFECT

',' ' xxxx YYYYxZxZ

Starting from X=x*, (and Z=Zx*(u) and Y= Yx*(u))Total Effect: TE(x,x*;Y) = E(Yx) – E(Yx*)

Controlled DE: CDEZ(x,x*;Y) = E(Yxz) – E(Yx*z)Controlled IE: None.

Natural DE: NDE(x,x*;Y) = E(YxZx*) – E(Yx*)

Natural IE: NIE(x,x*;Y) = E(Yx*Zx) – E(Yx*)

TWO CONCEPTIONS OF AVERAGE DIRECT AND INDIRECT EFFECTS:

POPULATION-LEVEL DEFINITIONS

X Z

y = f (x,z,u)

Y

u2u3

u1

Probabilisticcausal model:

P(u)M,

(all other parents of Y)

z = f (x, 1)y = g (x, z, 2)

X Z

Y

THE OPERATIONAL MEANING OFAVERAGE DIRECT EFFECTS

“Natural” Direct Effect of X on Y:The expected change in Y per unit change of X, when we keep Z constant at whatever value it attains before the change.

In linear models, NDE = Controlled Direct Effect

][001 xZx YYE

x

POLICY IMPLICATIONS(Who cares?)

f

GENDER QUALIFICATION

HIRING

What is the direct effect of X on Y?

The effect of Gender on Hiring if sex discrimination is eliminated.

indirect

X Z

Y

IGNORE

z = f (x, 1)y = g (x, z, 2)

X Z

Y

THE OPERATIONAL MEANING OFINDIRECT EFFECTS

“Natural” Indirect Effect of X on Y:The expected change in Y when we keep X constant, say at x0, and let Z change to whatever value it would have under a unit change in X.

In linear models, NIE = TE - DE

][010 xZx YYE

x

Example:

Theorem: If there exists a set W such that

GRAPHICAL CONDITION FOR EXPERIMENTAL IDENTIFICATION

OF AVERAGE NATURAL DIRECT EFFECTS

zw

xzxxz wPwzZPwYEwYEYxxNDE,

** )()|()|()|()*;,(

)()|( ZXNDWWZYXZG and

HOW THE PROOF GOES?

)(,*;, ** xZx YEYEYxxNDEx

xzWZYW xxz and all for If |*

Proof:

)()|(

),|()(

*

*, *

wWPwWzZP

wWzZYEYE

x

w zxxzZx x

)()|(

)|()(

*

, *

wWPwWzZP

wWYYEYE

x

w zxzZx x

Each factor is identifiable by experimentation.

GRAPHICAL CRITERION FORCOUNTERFACTUAL INDEPENDENCE

xzWZY xxz and all for |*

XZGWZY )|(

U3

U1

X

Y

Z

U2

U3

U1

X Z

Y

U2

U3

U1

X

Y

U2

ZXZG

21 UU

21 UU

GRAPHICAL CONDITION FOR NONEXPERIMENTAL IDENTIFICATION

OF AVERAGE NATURAL DIRECT EFFECTS

Identification conditions1. There exists a W such that (Y Z | W)GXZ

2. There exist additional covariates that render all counterfactual terms identifiable.

zw

xzxxz wPwzZPwYEwYEYxxNDE

,** )()|()|()|(

)*;,(

Corollary 3:The average natural direct effect in Markovian models is identifiable from nonexperimental data, and it is given by

where S stands for all parents of X (or another sufficient set).

IDENTIFICATION INMARKOVIAN MODELS

X ZExample:S =

Y

s z

sPsxzPzxYEzxYEYxxNDE )()*,|()*,|(),|()*;,(

z

xzPzxyEzxYEYxxNDE *)|()*,|(),|()*;,(

How effective would the drug be if we eliminate its side-effect (Headache)?

POLICY QUESTION ANSWERED BY NATURAL DIRECT EFFECT

Outcome

Drug

Aspirin

X

Y

Z

W Headache

z

xYExzPzxYEYxxNDE *)|(*)|(),|()*;,(

• NIE(x,x*;Y) = Expected increase in sales, if we bluff the competitor into believing that X is about to change from x* to x.

• For Markovian models:

POLICY-BASED INTERPRETATION OF INDIRECT EFFECTS

X

Y

Z

(Sales)

(Advertisement Budget) (Competitor’s Budget)

z

xzPxzPzxYEYxxNDE *)|()|()*,|()*;,(

Theorem 5: The total, direct and indirect effects obeyThe following equality

In words, the total effect (on Y) associated with the transition from x* to x is equal to the difference between the direct effect associated with this transition and the indirect effect associated with the reverse transition, from x to x*.

RELATIONS BETWEEN TOTAL, DIRECT, AND INDIRECT EFFECTS

);*,()*;,();,( * YxxNIEYxxNDEYxxTE

Y

Z

X

W

x*

z* = Zx* (u)

Nonidentifiable even in Markovian models

GENERAL PATH-SPECIFICEFFECTS (Def.)

)),(*),(();,(* ugpagpafgupaf iiiii

*);,();,( **gMMg YxxTEYxxE

Y

Z

X

W

Form a new model, , specific to active subgraph g*gM

Definition: g-specific effect

SUMMARY OF RESULTS

1. New formulation of path-specific effects, based on signal blocking, instead of value fixing.

2. Path-analytic techniques extended to nonlinear and nonparametric models.

3. Conditions for estimating direct and indirect effects from experimental and nonexperimental data.

4. Estimability conditions hold in Markovian models.5. Graphical techniques of inferring effects of

nonstandard policies, involving signal blocking.