1. From Propensity Scores And Mediation To External Validity
WHAT'S NEW IN
CAUSAL INFERENCE:
From Propensity Scores And Mediation To External Validity
Judea Pearl
University of California
Los Angeles (

2. OUTLINE
Unified conceptualization of counterfactuals, structural-equations, and graphs
Propensity scores demystified
Direct and indirect effects (Mediation)
External validity mathematized

3. TRADITIONAL STATISTICAL
INFERENCE PARADIGM
Data
Inference
Q(P)
(Aspects of P) P
Joint
Distribution
e.g.,
Infer whether customers who bought product A
would also buy product B.
Q = P(B | A)

4. THE STRUCTURAL MODEL PARADIGM Joint Distribution Data Generating Model
Q(M)
(Aspects of M)
Data M
Inference
M – Invariant strategy (mechanism, recipe, law, protocol) by which Nature assigns values to variables in the analysis.
“Think Nature, not experiment!”

5. ORACLE FOR MANIPILATION
FAMILIAR CAUSAL MODEL
ORACLE FOR MANIPILATION X Y Z
Here is a causal model we all remember from high-school -- a circuit diagram.
There are 4 interesting points to notice in this example:
(1) It qualifies as a causal model -- because it contains the information to confirm or refute all action, counterfactual and explanatory sentences concerned with the operation of the circuit.
For example, anyone can figure out what the output would be like if we set Y to zero, or if we change this OR gate to a NOR gate or if we perform any of the billions combinations of such actions.
(2) Logical functions (Boolean input-output relation) is insufficient for answering such queries
(3)These actions were not specified in advance, they do not have special names and they do not show up in the diagram.
In fact, the great majority of the action queries that this circuit can answer have never been considered by the designer of this circuit.
(4) So how does the circuit encode this extra information?
Through two encoding tricks:
4.1 The symbolic units correspond to stable physical mechanisms
(i.e., the logical gates)
4.2 Each variable has precisely one mechanism that determines its value.
INPUT
OUTPUT

6. STRUCTURAL CAUSAL MODELS
Definition: A structural causal model is a 4-tuple
V,U, F, P(u), where
V = {V1,...,Vn} are endogeneas variables
U = {U1,...,Um} are background variables
F = {f1,..., fn} are functions determining V,
vi = fi(v, u)
P(u) is a distribution over U
P(u) and F induce a distribution P(v) over observable variables
e.g.,

7. CAUSAL MODELS AND COUNTERFACTUALS
Definition:
The sentence: “Y would be y (in situation u), had X been x,” denoted Yx(u) = y, means:
The solution for Y in a mutilated model Mx, (i.e., the equations for X replaced by X = x) with input U=u, is equal to y.
The Fundamental Equation of Counterfactuals:

8. READING COUNTERFACTUALS
FROM SEM
Data shows: a = 0.7, b = 0.5, g = 0.4
A student named Joe, measured X=0.5, Z=1, Y=1.9
Q1: What would Joe’s score be had he doubled his study time?

9. READING COUNTERFACTUALS
Q1: What would Joe’s score be had he doubled his study time?
Answer: Joe’s score would be 1.9
Or,
In counterfactual notaion:

10. READING COUNTERFACTUALS
Q2: What would Joe’s score be, had the treatment been 0 and
had he studied at whatever level he would have studied had
the treatment been 1?

11. CAUSAL MODELS AND COUNTERFACTUALS
Definition:
The sentence: “Y would be y (in situation u), had X been x,” denoted Yx(u) = y, means:
The solution for Y in a mutilated model Mx, (i.e., the equations for X replaced by X = x) with input U=u, is equal to y.
Joint probabilities of counterfactuals:
In particular:

12. THE FIVE NECESSARY STEPS
OF CAUSAL ANALYSIS
Define:
Assume:
Identify:
Estimate:
Test:
Express the target quantity Q as a function
Q(M) that can be computed from any model M.
Formulate causal assumptions A using some formal language.
Determine if Q is identifiable given A.
Estimate Q if it is identifiable; approximate it,
if it is not.
Test the testable implications of A (if any).

13. THE LOGIC OF CAUSAL ANALYSIS
CAUSAL MODEL (MA)
CAUSAL MODEL (MA)
A - CAUSAL ASSUMPTIONS
A* - Logical
implications of A
Causal inference
Q Queries of interest
Q(P) - Identified estimands
T(MA) - Testable implications
Statistical inference
Data (D)
Q - Estimates of Q(P)
Goodness of fit
Provisional claims
Model testing

14. IDENTIFICATION IN SCM G Z1 Z2 Z3 Z4 Z5 X Z6 Y
Find the effect of X on Y, P(y|do(x)), given the
causal assumptions shown in G, where Z1,..., Zk
are auxiliary variables. G Z1 Z2 Z3 Z4 Z5 X Z6 Y
Can P(y|do(x)) be estimated if only a subset, Z,
can be measured?

15. ELIMINATING CONFOUNDING BIAS THE BACK-DOOR CRITERION
P(y | do(x)) is estimable if there is a set Z of
variables such that Z d-separates X from Y in Gx. G Gx Z1 Z1 Z2 Z2 Z Z3 Z3 Z4 Z5 Z5 Z4 X Z6 Y X Z6 Y
Moreover,
(“adjusting” for Z)  Ignorability

16. EFFECT OF WARM-UP ON INJURY
(After Shrier & Platt, 2008)
Watch out!
???
Front Door
No, no!
Warm-up Exercises (X)
Injury (Y)

17. PROPENSITY SCORE ESTIMATOR
(Rosenbaum & Rubin, 1983) Z1 Z2
P(y | do(x)) = ? L Z4 Z3 Z5 X Z6 Y
Adjustment for L replaces Adjustment for Z
Theorem:

18. WHAT PROPENSITY SCORE (PS) PRACTITIONERS NEED TO KNOW
The asymptotic bias of PS is EQUAL to that of ordinary adjustment (for same Z).
Including an additional covariate in the analysis CAN SPOIL the bias-reduction potential of others. X Y Z Z
In particular, instrumental variables tend to amplify bias.
Choosing sufficient set for PS, requires knowledge of the model.

19. Instrumental variables are Bias-Amplifiers in linear
SURPRISING RESULT:
Instrumental variables are Bias-Amplifiers in linear
models (Bhattarcharya & Vogt 2007; Wooldridge 2009) Z U
(Unobserved) c3 c1 c2
“Naive” bias
Adjusted bias X c0 Y

20. When Z is allowed to vary, it absorbs (or explains)
INTUTION:
When Z is allowed to vary, it absorbs (or explains)
some of the changes in X. U c1 X Y Z c2 c3 c0 U c1 X Y Z c2 c3 c0
When Z is fixed the burden falls on U alone, and
transmitted to Y (resulting in a higher bias) Z U c3 c1 c2 X c0 Y

21. WHAT’S BETWEEN AN INSTRUMENT AND A CONFOUNDER? Should we adjust for Z?Y X
ANSWER:
CONCLUSION:
Yes, if
No, otherwise
Adjusting for a parent of Y is safer
than a parent of X

22. CONCLUSIONS
The prevailing practice of adjusting for all covariates, especially those that are good predictors of X (the “treatment assignment,” Rubin, 2009) is totally misguided.
The “outcome mechanism” is as important, and much safer
As X-rays are to the surgeon, graphs are for causation

23. REGRESSION VS. STRUCTURAL EQUATIONS (THE CONFUSION OF THE CENTURY)
Regression (claimless, nonfalsifiable):
Y = ax + Y
Structural (empirical, falsifiable):
Y = bx + uY
Claim: (regardless of distributions):
E(Y | do(x)) = E(Y | do(x), do(z)) = bx 
The mothers of all questions:
Q. When would b equal a?
A. When all back-door paths are blocked, (uY X)
Q. When is b estimable by regression methods?
A. Graphical criteria available

24. TWO PARADIGMS FOR CAUSAL INFERENCE Observed: P(X, Y, Z,...)
Conclusions needed: P(Yx=y), P(Xy=x | Z=z)...
How do we connect observables, X,Y,Z,…
to counterfactuals Yx, Xz, Zy,… ?
N-R model
Counterfactuals are
primitives, new variables
Super-distribution
Structural model
Counterfactuals are
derived quantities
Subscripts modify the
model and distribution

25. “SUPER” DISTRIBUTION IN N-R MODEL X Y Z Yx=0 Yx=1 Xz=0 Xz=1 Xy=0 U1 Y 1 Z 1
Yx=0 1
Yx=1 1
Xz=0 1
Xz=1 1
Xy=0
0
1 U u1 u2 u3 u4
inconsistency: x = 0  Yx=0 = Y
Y = xY1 + (1-x) Y0

26. ARE THE TWO PARADIGMS EQUIVALENT?
Yes (Galles and Pearl, 1998; Halpern 1998)
In the N-R paradigm, Yx is defined by consistency:
In SCM, consistency is a theorem.
Moreover, a theorem in one approach is a theorem in the other.
Difference: Clarity of assumptions and their implications

27. AXIOMS OF STRUCTURAL COUNTERFACTUALS
Yx(u)=y: Y would be y, had X been x (in state U = u)
(Galles, Pearl, Halpern, 1998):
Definiteness
Uniqueness
Effectiveness
Composition (generalized consistency)
Reversibility

28. FORMULATING ASSUMPTIONS
THREE LANGUAGES
1. English: Smoking (X), Cancer (Y), Tar (Z), Genotypes (U) X Y Z U
2. Counterfactuals: Z X Y
3. Structural:

29. COMPARISON BETWEEN THE
N-R AND SCM LANGUAGES
Expressing scientific knowledge
Recognizing the testable implications of one's assumptions
Locating instrumental variables in a system of equations
Deciding if two models are equivalent or nested
Deciding if two counterfactuals are independent given another
Algebraic derivations of identifiable estimands

30. GRAPHICAL – COUNTERFACTUALS SYMBIOSIS
Every causal graph expresses counterfactuals
assumptions, e.g., X  Y  Z
Missing arrows Y  Z
2. Missing arcs Y Z
consistent, and readable from the graph.
Express assumption in graphs
Derive estimands by graphical or algebraic methods

31. (direct vs. indirect effects)
EFFECT DECOMPOSITION
(direct vs. indirect effects)
Why decompose effects?
What is the definition of direct and indirect effects?
What are the policy implications of direct and indirect effects?
When can direct and indirect effect be estimated consistently from experimental and nonexperimental data?

32. WHY DECOMPOSE EFFECTS? To understand how Nature works
To comply with legal requirements
To predict the effects of new type of interventions:
Signal routing, rather than variable fixing

33. LEGAL IMPLICATIONS OF DIRECT EFFECT X Z Y
Can data prove an employer guilty of hiring discrimination?
(Gender) X Z
(Qualifications) Y
(Hiring)
What is the direct effect of X on Y ?
(averaged over z)
Adjust for Z? No! No!

34. FISHER’S GRAVE MISTAKE
(after Rubin, 2005)
What is the direct effect of treatment on yield?
(Soil treatment) X Z
(Plant density)
(Latent factor) Y
(Yield)
Compare treated and untreated lots of same density
No! No!
Proposed solution (?): “Principal strata”

35. NATURAL INTERPRETATION OF AVERAGE DIRECT EFFECTS
Robins and Greenland (1992) – “Pure” X Z
z = f (x, u)
y = g (x, z, u) Y
Natural 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 = Natural Direct Effect

36. DEFINITION 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 expression
Nonexperimental: subscript-free expression

37. DEFINITION OF INDIRECT EFFECTS X Z z = f (x, u) y = g (x, z, u) Y
No Controlled Indirect Effect
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

38. POLICY IMPLICATIONS OF INDIRECT EFFECTS
What is the indirect effect of X on Y?
The effect of Gender on Hiring if sex discrimination
is eliminated.
GENDER
QUALIFICATION
HIRING X Z
IGNORE f Y
Deactivating a link – a new type of intervention

39. MEDIATION FORMULAS
The natural direct and indirect effects are identifiable in Markovian models (no confounding),
And are given by:
Applicable to linear and non-linear models, continuous and discrete variables, regardless of distributional form.

40. Z
g xz
Linear + interaction m1 m2 X Y
In linear systems

41. IN UNCONFOUNDED MODELS
MEDIATION FORMULAS
IN UNCONFOUNDED MODELS X Z Y

42. Z m1 m2 X Y In linear systems Disabling mediation direct path DE
TE - DE TE IE
Is NOT equal to:

43. MEDIATION FORMULA FOR BINARY VARIABLES Z X Y X Z Y E(Y|x,z)=gxz
E(Z|x)=hx n1 n2 1 n3 n4 n5 n6 n7 n8

44. RAMIFICATION OF THE MEDIATION FORMULA
DE should be averaged over mediator levels,
IE should NOT be averaged over exposure levels.
TE-DE need not equal IE
TE-DE = proportion for whom mediation is necessary
IE = proportion for whom mediation is sufficient
TE-DE informs interventions on indirect pathways
IE informs intervention on direct pathways.

45. TRANSPORTABILITY -- WHEN CAN
WE EXTRPOLATE EXPERIMENTAL FINDINGS TO DIFFERENT POPULATIONS? X Y
Z = age X Y
Z = age
Experimental study in LA
Measured:
Problem: We find
(LA population is younger)
What can we say about
Intuition:
Observational study in NYC
Measured:

46. TRANSPORT FORMULAS DEPEND
ON THE STORY X Y Z
(a) X Y Z
(b) X Y
(c) Z
a) Z represents age
b) Z represents language skill
c) Z represents a bio-marker

47. (Pearl and Bareinboim, 2010)
TRANSPORTABILITY
(Pearl and Bareinboim, 2010)
Definition 1 (Transportability)
Given two populations, denoted  and *,
characterized by models M = <F,V,U> and
M* = <F,V,U+S>, respectively, a causal relation
R is said to be transportable from  to * if
1. R() is estimable from the set I of interventional studies on , and
2. R(*) is identified from I, P*, G, and G + S.
S = external factors responsible for M  M*

48. TRANSPORT FORMULAS DEPEND
ON THE STORY X Y Z
(a) S X Y Z
(b) S X Y
(c) Z S
a) Z represents age
b) Z represents language skill
c) Z represents a bio-marker ? ?

49. WHICH MODEL LICENSES THE TRANSPORT OF THE CAUSAL EFFECTY X S
(b) Y X S
(c) X Y Z S X Y
(d) Z S W X Y ( Z S
(c) W X Y
(f) Z S
(c) W X Y (f Z S S S S X X X W W W Z Z Z Y Y Y
(e)

50. DETERMINE IF THE CAUSAL EFFECT IS TRANSPORTABLEV
What measurements need to be taken in the study and
in the target population? T S X W Z Y
The transport formula

51. CONCLUSIONS I TOLD YOU CAUSALITY IS SIMPLE
Formal basis for causal and counterfactual inference (complete)
Unification of the graphical, potential-outcome and structural equation approaches
Friendly and formal solutions to
century-old problems and confusions.
No other method can do better (theorem)

52. Thank you for agreeing
with everything I said.