Judea Pearl

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DIRECT AND INDIRECT EFFECTS

• Judea Pearl
• Computer Science Department
• UCLA
• www.cs.ucla.edu/judea

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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.

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WHY DECOMPOSE EFFECTS?

• Direct (or indirect) effect may be more
  transportable.
• Indirect effects may be prevented or controlled.
• Direct (or indirect) effect may be forbidden

?
Pill
Pregnancy


Thrombosis
Gender
Qualification
Hiring
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EFFECT-DECOMPOSITION IN LINEAR MODELS
Definition
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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 Xx.
Functional Bayes Net
Probability of Counterfactuals
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TOTAL, DIRECT, AND INDIRECT EFFECTS HAVE SIMPLE
SEMANTICS IN LINEAR MODELS
b
X
Z
z bx ?1 y ax cz ?2
a
c
Y
a bc
a
bc
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SEMANTICS BECOMES NONTRIVIAL IN NONLINEAR
MODELS (even when the model is completely
specified)
X
Z
z f (x, ?1) y g (x, z, ?2)
Y
Dependent on z?
Void of operational meaning?
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NEED OF FORMALIZATION
What is the direct effect of X on Y?
X
Z

AND
Y
Indirect Effect?
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TWO CONCEPTIONS OF DIRECT AND INDIRECT
EFFECTS Controlled vs. Natural
X
Z

AND
Y
Starting from X0, (and Z0 and Y0) Total
Effect Change X from 0 to 1, and test the
change in Y. Controlled DE Keep Z constant
at Z0, or Z1, and change X0 to X1. 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.
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LEGAL DEFINITIONS TAKE THE NATURAL
CONCEPTION (FORMALIZING DISCRIMINATION)
The central question in any employment-discrimi
nation 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? female y hire, y? not hire z
applicants qualifications
NO DIRECT EFFECT
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TWO CONCEPTIONS OF AVERAGE DIRECT AND INDIRECT
EFFECTS POPULATION-LEVEL DEFINITIONS
u3
u2
X
Z
(all other parents of Y)
Probabilistic causal model
y f (x,z,u)
ñ
á
u1
P(u)
M,
Y
Starting from Xx, (and ZZx(u) and Y
Yx(u)) Total Effect TE(x,xY) E(Yx)
E(Yx) Controlled DE CDEZ(x,xY) E(Yxz)
E(Yxz) Controlled IE None. Natural DE
NDE(x,xY) E(YxZx) E(Yx) Natural IE
NIE(x,xY) E(YxZx) E(Yx)
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THE OPERATIONAL MEANING OF AVERAGE DIRECT
EFFECTS
X
Z
z f (x, ?1) y g (x, z, ?2)
Y
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
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POLICY IMPLICATIONS (Who cares?)
indirect
What is the direct effect of X on Y?
The effect of Gender on Hiring if sex
discrimination is eliminated.
X
Z
IGNORE
f
Y
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THE OPERATIONAL MEANING OF INDIRECT EFFECTS
X
Z
z f (x, ?1) y g (x, z, ?2)
Y
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
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GRAPHICAL CONDITION FOR EXPERIMENTAL
IDENTIFICATION OF AVERAGE NATURAL DIRECT EFFECTS
Theorem If there exists a set W such that

Example
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HOW THE PROOF GOES?
Proof
Each factor is identifiable by experimentation.
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GRAPHICAL CRITERION FOR COUNTERFACTUAL
INDEPENDENCE
U3
U3
U2
U2
X
Z
Z
X
U1
Y
U1
Y
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GRAPHICAL CONDITION FOR NONEXPERIMENTAL
IDENTIFICATION OF AVERAGE NATURAL DIRECT EFFECTS

• Identification conditions
• There exists a W such that (Y Z W)GXZ
• There exist additional covariates that render all
  
• counterfactual terms identifiable.

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IDENTIFICATION IN MARKOVIAN MODELS
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).
Example S ?
X
Z
Y
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POLICY QUESTION ANSWERED BY NATURAL DIRECT EFFECT
Drug
X
W
Headache
Aspirin
Z
Outcome
Y
How effective would the drug be if we eliminate
its side-effect (Headache)?
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POLICY-BASED INTERPRETATION OF INDIRECT EFFECTS
X
(Advertisement Budget)
Z
(Competitors Budget)
Y
(Sales)

• NIE(x,xY) Expected increase in sales, if we
  bluff the competitor into believing that X is
  about to change from x to x.
• For Markovian models

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RELATIONS BETWEEN TOTAL, DIRECT, AND INDIRECT
EFFECTS
Theorem 5 The total, direct and indirect
effects obey The 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.
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GENERAL PATH-SPECIFIC EFFECTS (Def.)
X
X
Z
W
Z
W
Y
Y
Form a new model, , specific to active
subgraph g
Definition g-specific effect
Nonidentifiable even in Markovian models
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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.