Chen Avin

Ilya Shpitser

Judea Pearl

Computer Science Department

UCLA

IDENTIFIABILITY OF PATH-SPECIFIC

EFFECTS

QUESTIONS ASKED

• Why path-specific effects?• What are the semantics of path-specific effects

(in nonlinear and nonparametric models)?• What are the policy implications of path-specific

effects?• When can path-specific effects be estimated

consistently from experimental or nonexperimental data?

• Can these conditions be verified from accessible causal knowledge, i.e., graphs?

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: ))(),(|( zxYEx

a do do

CAUSAL MODELS AND COUNTERFACTUALS

Definition: A causal model is a 3-tupleM = V,U,F

(i) V = {V1…,Vn} endogenous variables,(ii) U = {U1,…,Um} background variables (unit)(iii) F = set of n functions,

The sentence: “Y would be y (in unit u), had X been x,”denoted Yx(u) = y, is the solution for Y in a mutilated model Mx, with the equations for X replaced by X = x. (“unit-based potential outcome”)

),( uvfv ii

COUNTERFACTUALS:STRUCTURAL SEMANTICS

Notation: Yx(u) = y 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

)(uM,P

tindependen- ))(),(|(

))(|(

DETEIE

ZzdoxdoYEx

DE

xdoYEx

TE

TOTAL, DIRECT, AND INDIRECT EFFECTS HAVE CONTROLLED-BASED

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

CONTROLLED-BASED SEMANTICS NONTRIVIAL IN NONLINEAR MODELS(even when the model is completely specified)

Dependent on z?

Void of operational meaning?

``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 OF DIRECT EFFECT

(FORMALIZING DISCRIMINATION)

NO DIRECT EFFECT

',' ' xxxx YYYYxZxZ

z = f (x, u)y = g (x, z, u)

X Z

Y

NATURAL SEMANTICS OFAVERAGE DIRECT EFFECTS

Average Direct Effect of X on Y:The expected change in Y, when we change X from x0 to x1 and, for each u, we keep Z constant at whatever value it attained before the change.

In linear models, DE = Controlled Direct Effect

][001 xZx YYE

x

);,( 10 YxxDE

Robins and Greenland (1992) – “Pure”

POLICY IMPLICATIONS(Who cares?)

f

GENDER QUALIFICATION

HIRING

What is the direct effect of X on Y?

Is employer guilty of sex-discrimination given data on (X,Y,Z)?

X Z

Y

CAN WE IGNORE THIS LINK?

tYYE xZxx

][0

01

z = f (x, u)y = g (x, z, u)

X Z

Y

NATURAL SEMANTICS OFINDIRECT EFFECTS

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 attained had X changed to x1.

In linear models, IE = TE - DE

][010 xZx YYE

x

);,( 10 YxxIE

POLICY IMPLICATIONS(Who cares?)

f

GENDER QUALIFICATION

HIRING

What is the indirect effect of X on Y?

The effect of Gender on Hiring if sex discriminationis eliminated.

X Z

Y

IGNORE

SEMANTICS AND IDENTIFICATION OF NESTED COUNTERFACTUALS

Consider the quantity

Given M, P(u), Q is well defined

Given u, Zx*(u) is the solution for Z in Mx*, call it z

is the solution for Y in Mxz

Can Q be estimated from data?

Experimental: nest-free expressionNonexperimental: subscript-free expression

)]([ )(*uYEQ uxZxu

entalnonexperim

alexperiment

)()(*uY uxZx

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

where S stands for any sufficient set of covariates.

IDENTIFICATION INMARKOVIAN MODELS

X ZExample:S =

Y

s z

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

z

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

Y

Z

X

W

x*

z* = Zx* (u)

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

EFFECT-INVARIANT

Rule 1 Rule 2

MAIN RESULT

Applying the two rules results in one of two cases:

Case 1: we obtain a ‘kite pattern.’ Then the path-specific effect is not identifiable.

R - Recanting witness

Z

Y

MAIN RESULT (Cont.)

X

Y

ZW Z’ Z”

Case 2: all blocked edges emanate from the root node. Then the effect is identifiable.

AZT EXAMPLE REVISITED

AZT

Pneumonia

Antibiotics

Headaches

Painkillers

Survival

Painkiller contribution to the total effect of AZT on survival

Antibiotics

AZT

PneumoniaHeadaches

Painkillers

Survival

Antibiotics contribution to the total effect of AZT on survival

RECANTING WITNESS

Antibiotics

AZT

PneumoniaHeadaches

Painkillers

Survival

Antibiotics contribution to the total effect of AZT on survival

R-Recanting Witness

R behaves as I

R behaves as II

P(RX,RX*) is not experimentally identifiable

SUMMARY OF RESULTS

1. Formal semantics of path-specific effects, based on signal blocking, instead of value fixing.

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

3. Meaningful (graphical) conditions for estimating effects from experimental and nonexperimental data.

4. Graphical techniques of inferring effects of policies involving signal blocking.