1 what's new in causal inference: from propensity scores and mediation to external validity and...
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11
WHAT'S NEW IN CAUSAL INFERENCE:
From Propensity Scores And Mediation To External Validity
And Selection Bias
Judea PearlUCLA
(www.cs.ucla.edu/~judea/)
22
1. Unified conceptualization of counterfactuals,
structural-equations, and graphs
2. Propensity scores demystified
3. Direct and indirect effects (Mediation)
4. External validity mathematized
OUTLINE
33
TRADITIONAL STATISTICALINFERENCE PARADIGM
Data
Inference
Q(P)(Aspects of P)
PJoint
Distribution
e.g.,Infer whether customers who bought product Awould also buy product B.Q = P(B | A)
44
Data
Inference
Q(M)(Aspects of M)
Data Generating
Model
M – Invariant strategy (mechanism, recipe, law, protocol) by which Nature assigns values to variables in the analysis.
JointDistribution
THE STRUCTURAL MODELPARADIGM
M
“Think Nature, not experiment!”•
55
Z
YX
INPUT OUTPUT
FAMILIAR CAUSAL MODELORACLE FOR MANIPILATION
66
STRUCTURALCAUSAL MODELS
Definition: A structural causal model is a 4-tupleV,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 UP(u) and F induce a distribution P(v) over observable variables
Yuxy e.g.,
77
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.
)()( uYuY xMx
The Fundamental Equation of Counterfactuals:
88
READING COUNTERFACTUALSFROM SEM
Data shows: = 0.7, = 0.5, = 0.4A student named Joe, measured X=0.5, Z=1, Y=1.9Q1: What would Joe’s score be had he doubled his study time?
X Y
3
2
1
Z
X Y7.0
5.0 4.03
2
1
Z
Score
TimeStudy
Treatment
Y
Z
X
3
2
1
zxy
xz
x
99
0.2Z
Y 9.1
9.1)(2 uYz
Q1: What would Joe’s score be had he doubled his study time?Answer: Joe’s score would be 1.9Or,In counterfactual notation:
7.0
75.0
1 5.04.0
75.03
5.0X
2
READING COUNTERFACTUALS
X Y7.0
5.0 4.03
2
1
Z
Y7.0
5.0
75.0
1
0.1
5.0
Z4.0
75.03
5.0X 5.1
2
1010
25.1Z
1X 0 25.1
5.0
Y 5.1
0.1Z
5.0X Y 25.2
1 5.0
7.0
4.075.03
75.02
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?
READING COUNTERFACTUALS
X Y7.0
5.0 4.03
2
1
Z 0.2Z
Y 9.17.0
75.0
1 5.04.0
75.03
5.0X
2
Y7.0
5.0
75.0
1
0.1
5.0
Z4.0
75.03
5.0X 5.1
2
7.0
1 5.0
25.1Z4.0
75.03
X Y
75.02
1 25.2
5.0
1111
POTENTIAL AND OBSERVED OUTCOMES PREDICTED BY A STRUCTURAL MODEL
1212
In particular:
),|(),|'(
)()()|(
')(':'
)(:
yxuPyxyYP
uPyYPyP
yuxYux
yuxYux
)(xdo
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.
•
)(),()(,)(:
uPzZyYPzuZyuYu
wxwx
Joint probabilities of counterfactuals:
1313
Define:
Assume:
Identify:
Estimate:
Test:
THE FIVE NECESSARY STEPSOF CAUSAL ANALYSIS
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).
))(|())(|( 01 xdoYExdoYE ATEe.g., )'|( xXyYP x ETT e.g.,
1414
),(|),,( 010001 xxZxxxZx YYEIEYYEDE e.g., )'|( xXyYP x ETT e.g.,
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).
),|( 1100yYxXyYPPC x e.g.,
Define:
Assume:
Identify:
Estimate:
Test:
THE FIVE NECESSARY STEPSOF CAUSAL ANALYSIS
1515
CAUSAL MODEL
(MA)
A - CAUSAL ASSUMPTIONS
Q Queries of interest
Q(P) - Identified estimands
Data (D)
Q - Estimates of Q(P)
Causal inference
T(MA) - Testable implications
Statistical inference
Goodness of fit
Model testingProvisional claims
),|( ADQQ )(Tg
A* - Logicalimplications of A
CAUSAL MODEL
(MA)
THE LOGIC OF CAUSAL ANALYSIS
1616
IDENTIFICATION IN SCM
Find the effect of X on Y, P(y|do(x)), given the
causal assumptions shown in G, where Z1,..., Zk are auxiliary variables.
Z6
Z3
Z2
Z5
Z1
X Y
Z4
G
Can P(y|do(x)) be estimated if only a subset, Z, can be measured?
1717
ELIMINATING CONFOUNDING BIASTHE BACK-DOOR CRITERION
P(y | do(x)) is estimable if there is a set Z ofvariables such that Z d-separates X from Y in Gx.
Z6
Z3
Z2
Z5
Z1
X Y
Z4
Z6
Z3
Z2
Z5
Z1
X Y
Z4
Z
Gx G
• Moreover,
(“adjusting” for Z) Ignorability
z z zxP
zyxPzPzxyPxdoyP
)|(),,(
)(),|())(|(
1818
Front Door
EFFECT OF WARM-UP ON INJURY (After Shrier & Platt, 2008)
No, no!
Watch out!
Warm-up Exercises (X) Injury (Y)
???
1919
PROPENSITY SCORE ESTIMATOR(Rosenbaum & Rubin, 1983)
Z6
Z3
Z2
Z5
Z1
X Y
Z4
),,,,|1(),,,,( 5432154321 zzzzzXPzzzzzL
Adjustment for L replaces Adjustment for Z
z l
lLPxlLyPzPxzyP )(),|()(),|(Theorem:
P(y | do(x)) = ?
L
Can L replace {Z1, Z2, Z3, Z4, Z5} ?
2020
WHAT PROPENSITY SCORE (PS)PRACTITIONERS NEED TO KNOW
1. The asymptotic bias of PS is EQUAL to that of ordinary adjustment (for same Z).
2. Including an additional covariate in the analysis CAN SPOIL the bias-reduction potential of others.
)|1()( zZXPzL
z l
lPxlyPzPxzyP )(),|()(),|(
3. In particular, instrumental variables tend to amplify bias.4. Choosing sufficient set for PS, requires knowledge of the
model.
ZX YX Y X Y
Z
X Y
ZZ
2121
U
c1
X Y
Z
c2c3
c0
SURPRISING RESULT:Instrumental variables are Bias-Amplifiers in linear models (Bhattarcharya & Vogt 2007; Wooldridge 2009)
“Naive” bias
Adjusted bias
2123
2102
3
210
11))(|(),|( cc
c
ccc
c
cccxdoYE
xzxYE
xBz
2102100 ))(|()|( ccccccxdoYEx
xYEx
B
(Unobserved)
2222
INTUTION:When Z is allowed to vary, it absorbs (or explains) some of the changes in X.
When Z is fixed the burden falls on U alone, and transmitted to Y (resulting in a higher bias)
U
c1
X Y
Z
c2c3
c0
U
c1
X Y
Z
c2c3
c0
U
c1
X Y
Z
c2c3
c0
2323
c0
c2
Z
c3
U
YX
c4
T1
c1
WHAT’S BETWEEN AN INSTRUMENT AND A CONFOUNDER?Should we adjust for Z?
T2
ANSWER:
CONCLUSION:
23
12
3
4
1 c
cccc
Yes, if
No, otherwise
Adjusting for a parent of Y is safer than a parent of X
2424
WHICH SET TO ADJUST FOR
Should we adjust for {T}, {Z}, or {T, Z}?
Answer 1: (From bias-amplification considerations){T} is better than {T, Z} which is the same as {Z}
Answer 2: (From variance considerations){T} is better than {T, Z} which is better than {Z}
Z T
X Y
2525
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, from both bias and variance viewpoints
• As X-rays are to the surgeon, graphs are for causation
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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
Q. When is b estimable by regression methods?A. Graphical criteria available
The mothers of all questions:Q. When would b equal a?A. When all back-door paths are blocked, (uY X)
2727
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 modelCounterfactuals areprimitives, new variables
Super-distribution
Structural modelCounterfactuals are derived quantities
Subscripts modify the model and distribution )()( yYPyYP xMx ,...,,,
,...),,...,,(*
yx
zxZYZYX
XYYXP
constrain
2828
“SUPER” DISTRIBUTIONIN N-R MODEL
X
0
0
0
1
Y 0
1
0
0
Yx=0
0
1
1
1
Z
0
1
0
0
Yx=1
1
0
0
0
Xz=0
0
1
0
0
Xz=1
0
0
1
1
Xy=0 0
1
1
0
U
u1
u2
u3
u4
inconsistency: x = 0 Yx=0 = Y Y = xY1 + (1-x) Y0
yx
zx
xyxzyx
ZXY
XZyYP
ZYZYZYXP
|
),|(*
...),...,...,,...,,(*
:Defines
2929
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
01 )1( YxxYY
3030
AXIOMS OF STRUCTURAL COUNTERFACTUALS
1. Definiteness
2. Uniqueness
3. Effectiveness
4. Composition (generalized consistency)
5. Reversibility
xuXtsXx y )( ..
')')((&))(( xxxuXxuX yy
xuX xw )(
)()()( uYuYxuX wwxw
yuYwuWyuY xxyxw )())((&))((
Yx(u)=y: Y would be y, had X been x (in state U = u)
(Galles, Pearl, Halpern, 1998):
3131
FORMULATING ASSUMPTIONSTHREE LANGUAGES
},{),()(
),()()()(
),()(
XYZuYuY
uXuXuXuX
uZuZ
zxzxz
zzyy
yxx
2. Counterfactuals:
1. English: Smoking (X), Cancer (Y), Tar (Z), Genotypes (U)
X YZ
U
ZX Y
3. Structural:
)3,,(
),(
),(
3
22
11
uzfy
xfz
ufx
3232
1. Expressing scientific knowledge2. Recognizing the testable implications of one's
assumptions 3. Locating instrumental variables in a system of
equations4. Deciding if two models are equivalent or nested5. Deciding if two counterfactuals are independent
given another6. Algebraic derivations of identifiable estimands
COMPARISON BETWEEN THEN-R AND SCM LANGUAGES
3333
GRAPHICAL – COUNTERFACTUALS SYMBIOSIS
Every causal graph expresses counterfactuals assumptions, e.g., X Y Z
consistent, and readable from the graph.
• Express assumption in graphs• Derive estimands by graphical or algebraic
methods
)()(, uYuY xzx
2. Missing arcs Y Z yx ZY
1. Missing arrows Y Z
3434
EFFECT DECOMPOSITION(direct vs. indirect effects)
1. Why decompose effects?
2. What is the definition of direct and indirect effects?
3. What are the policy implications of direct and indirect effects?
4. When can direct and indirect effect be estimated consistently from experimental and nonexperimental data?
3535
WHY DECOMPOSE EFFECTS?
1. To understand how Nature works
2. To comply with legal requirements
3. To predict the effects of new type of interventions:
Signal routing, rather than variable fixing
3636
X Z
Y
LEGAL IMPLICATIONSOF DIRECT EFFECT
What is the direct effect of X on Y ?
(averaged over z)
))(),(())(),( 01 zdoxdoYEzdoxdoYECDE ||(
(Qualifications)
(Hiring)
(Gender)
Can data prove an employer guilty of hiring discrimination?
Adjust for Z? No! No!
3737
X Z
Y
FISHER’S GRAVE MISTAKE(after Rubin, 2005)
Compare treated and untreated lots of same density
No! No!
))(),(())(),( 01 zdoxdoYEzdoxdoYE ||(
(Plant density)
(Yield)
(Soil treatment)
What is the direct effect of treatment on yield?
zZ zZ
(Latent factor)
Proposed solution (?): “Principal strata”
3838
z = f (x, u)y = g (x, z, u)
X Z
Y
NATURAL INTERPRETATION OFAVERAGE DIRECT EFFECTS
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
][001 xZx YYE
x
);,( 10 YxxDE
Robins and Greenland (1992) – “Pure”
)( 01 xx
3939
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 expressionNonexperimental: subscript-free expression
)]([ )(*uYEQ uxZxu
entalnonexperim
alexperiment
)()(*uY uxZx
4040
z = f (x, u)y = g (x, z, u)
X Z
Y
DEFINITION 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
No Controlled Indirect Effect
4141
POLICY IMPLICATIONS OF INDIRECT EFFECTS
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
Deactivating a link – a new type of intervention
4242
1. The natural direct and indirect effects are identifiable in Markovian models (no confounding),
2. And are given by:
3. Applicable to linear and non-linear models, continuous and discrete variables, regardless of distributional form.
MEDIATION FORMULAS
)(
))](|())(|())[,(|(
)).(|())],(|()),(|([
010
001
revIEDETE
xdozPxdozPzxdoYEIE
xdozPzxdoYEzxdoYEDE
z
z
4343
Z
m2
X Y
m1
IEDETE WHY
IEDETE
mmIE
DE
mmTE
21
21
In linear systems
22
11
xzzmxy
xmz
xz
1
21
121
mIEDETE
mmIE
DE
mmmTELinear + interaction
4444
MEDIATION FORMULASIN UNCONFOUNDED MODELS
X
Z
Y
)|()|(
])|()|()[,|([
)|()],|(),|([
01
010
001
xYExYETE
xzPxzPzxYEIE
xzPzxYEzxYEDE
z
z
mediation to owed responses of Fraction
mediationby explained responses of Fraction
DETE
IE
4545
Z
m2
X Y
m1
)(revIEDETE
DETEmmIE
DE
mmTE
21
21
mediation
disablingby prevented Effect
alone mediationby sustained Effect
DETE
IE
IEDETE WHY
IErevIE )(
Disabling mediation
Disabling direct path
DE
TE - DE
TE
IE
In linear systems
Is NOT equal to:
4646
MEDIATION FORMULAFOR BINARY VARIABLES
X
Z
Y
))((
)()1)((
000101
0011100010gghhIE
hgghggDE
XX ZZ YY E(Y|x,z)=gxz E(Z|x)=hx
nn11 00 00 00
nn22 00 00 11
nn33 00 11 00
nn44 00 11 11
nn55 11 00 00
nn66 11 00 11
nn77 11 11 00
nn88 11 11 11
0021
2 gnn
n
0143
4 gnn
n
1065
6 gnn
n
1187
8 gnn
n
04321
43 hnnnn
nn
18765
87 hnnnn
nn
4747
RAMIFICATION OF THEMEDIATION FORMULA
• DE should be averaged over mediator levels,
IE should NOT be averaged over exposure levels.
• TE-DE need not equal IETE-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.
4848
TRANSPORTABILITY -- WHEN CANWE EXTRAPOLATE EXPERIMENTAL FINDINGS TO
DIFFERENT POPULATIONS?
Experimental study in LAMeasured:
Problem: We find
(LA population is younger)
What can we say about Intuition:
)),(|(),,(
zxdoyPzyxP
))(|(* xdoyP
Observational study in NYCMeasured: ),,(* zyxP
)(*)( zPzP
z
zPzxdoyPxdoyP )(*)),(|())(|(*
X Y
Z = age
X Y
Z = age
4949
TRANSPORT FORMULAS DEPEND ON THE STORY
X Y
Z
(a)X Y
Z
(b)
a) Z represents age
b) Z represents language skill
c) Z represents a bio-marker
z
zPzxdoyPxdoyP )(*)),(|())(|(*
???))(|(* xdoyP
???))(|(* xdoyP
X Y(c)
Z
5050
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*
5151
TRANSPORT FORMULAS DEPEND ON THE STORY
a) Z represents age
b) Z represents language skill
c) Z represents a bio-marker
z
zPzxdoyPxdoyP )(*)),(|())(|(*
))(|(* xdoyP
X YZ
(b)
S
X Y
Z
(a)
S
X Y(c)
Z
S
))(|( xdoyP
z
xzPzxdoyP )|(*)),(|())(|(* xdoyP
?
?
5252
X Y(f)
Z
S
X Y(d)
Z
S
W
WHICH MODEL LICENSES THE TRANSPORT OF THE CAUSAL EFFECT
X Y(e)
Z
S
W
(c)X YZ
S
X YZ
S
WX YZ
S
W
(b)YX
S
(a)YX
S
5353
U
W
DETERMINE IF THE CAUSAL EFFECT IS TRANSPORTABLE
X YZ
V
ST
S
The transport formula
What measurements need to be taken in the study and in the target population?
tz w
tPtxdowPwzPzxdoyP
xdoyP
)(*)),(|()|(*)),(|(
))(|(*
5454
SURROGATE ENDPOINTS – A CAUSAL PERSPECTIVE
The problem:Infer effects of (randomized) treatment (X) on outcome (Y) from
measurements taken on a surrogate variable (Z), which is more readily measurable, and is sufficiently well correlated with the first (Ellenberg and Hamilton 1989).
Prentice 1989: "strong correlation is not sufficient," there should be "no pathways that bypass the surrogate" (2005).
1989-2011 - Everyone agrees that correlation is notsufficient, and no one explains why.
5555
WHY STRONG CORRELATION IS NOT SUFFICIENT FOR SURROGACY
Joffe and Green (2009):“A surrogate outcome is an outcome for which knowing the effect of treatment on the surrogate allows prediction of the effect of treatment on the more clinically relevant outcome.”
Two effects = Two experiments conducted under two different conditions.
Surrogacy = “Strong correlation,” + robustness to the new conditions.
New condition = Interventions to change the surrogate Z.
5656
WHO IS A GOOD SURROGATE?
S
S
S
X SX Y
(a)
Z
S
X Y
(b)Z
Y
(c)Z
X
(d)
Z Y X Y
(e)
Z
S
X Y
(f)
Z
U W
5757
Definition (Pearl and Bareinboim, 2011):A variable Z is said to be a surrogate endpoint relative the effect of X on Y if and only if:
1. P(y|do(x), z) is highly sensitive to Z in the experimental study, and
2. P(y|do(x), z, s) = P(y|do(x), z, s ) where S is a selection
variable added to G and directed towards Z.
In words, the causal effect of X on Y can be reliably predicted
from measurements of Z, regardless of the mechanism
responsible for variations in Z.
SURROGACY: CORRELATIONS AND ROBUSTNESS
5858
Z and X d-separate S from Y.
SURROGACY: A GRAPHICAL CRITERION
S
S
S
X SX Y
(a)
Z
S
X Y
(b)Z
Y
(c)Z
X
(d)
Z Y X Y
(e)
Z
S
X Y
(f)
Z
U W
5959
X
S
UY
Yc0
1 2
No selection biasSelection bias activated by a virtual colliderSelection bias activated by both a virtual collider and real collider
THE ORIGIN OFSELECTION BIAS:
UX
6060
Can be eliminated by randomization or adjustment
X
UY
Y
S2
CONTROLLING SELECTION BIAS BY ADJUSTMENT
U1 U2
S3 S1
6161
Cannot be eliminated by randomization, requires adjustment for U2
X
UY
Y
S2
CONTROLLING SELECTION BIAS BY ADJUSTMENT
U1 U2
S3 S1
6262
Cannot be eliminated by randomization or adjustment
X
UY
Y
S2
CONTROLLING SELECTION BIAS BY ADJUSTMENT
U1 U2
S3 S1
6363
Cannot be eliminated by adjustment or by randomization
CONTROLLING SELECTION BIAS BY ADJUSTMENT
X
S
UY
Yc0
1 2
UX
6464
Adjustment for U2 gives
If all we have is P(u2 | S2 = 1), not P(u2), then only the U2-specific effect is recoverable
X
UY
Y
S2
CONTROLLING BY ADJUSTMENT MAY REQUIRE EXTERNAL
INFORMATION U1 U2
S3 S1
2
222 )(),1,|())(|(u
uPuSxyPxdoyP
6565
WHEN WOULD THE ODDSRATIO BE RECOVERABLE?
(a) OR is recoverable, despite the virtual collider at Y. whenever (X Y | Y,Z)G or (Y S | X , Z)G, giving:
OR (X,Y | S = 1) = OR(X,Y) (Cornfield 1951, Wittemore 1978; Geng 1992)
(b) the Z-specific OR is recoverable, but is meaningless. OR (X ,Y | Z) = OR (X,Y | Z, S = 1)
(c) the C-specific OR is recoverable, which is meaningful. OR (X ,Y | C) = OR (X,Y | W,C, S = 1)
YX S
(a) (b) (c)YX
Z
S
YX
W
C
S),(
)|()/(
/)|()|(
),(00
01
10
11
XYOR
xyPxyP
xyPxyP
YXOR
6666
Theorem (Bareinboim and Pearl, 2011)Let graph G contain the arrow X Y and a selection node S. A necessary condition for G to permit the G-recoverability of OR(Y,X | C) for some set C of pre-treatment covariates is that every ancestor Ai
of S that is also a descendant of X have a separating set Ti that either d-separates Ai from X given Y, or d-separates Ai from Y given X.
Moreover, For every C s.t.where ii TTTXNdT and ,)('
GRAPHICAL CONDITION FORODDS-RATIO RECOVERABILITY
)1,,|,()|,( STCXYORCXYORXCTY ,|'
6767
W2
EXAMPLES OF ODDSRATIO RECOVERABILITY
(a) (b)
W3W4
S
YX
W1
W3W4
S
YX
W2W1
Recoverable Non-recoverable
)1,,,|,(),( 421 SWWWXYORXYOR (a)
{W1, W2, W4}{W4, W3, W1}
Separator
6868
W1
EXAMPLES OF ODDSRATIO RECOVERABILITY
(a) (b)
W3W4
S
YX
W2W1
W3W4
S
YX
W2W1
Recoverable Non-recoverable
)1,,,|,(),( 421 SWWWXYORXYOR (a)
Separator
6969
0
EXAMPLES OF ODDSRATIO RECOVERABILITY
(a) (b)
W3W4
S
YX
W2W1
W3W4
S
YX
W2W1
Recoverable Non-recoverable
)1,,,|,(),( 421 SWWWXYORXYOR (a)
W2 W1
Separator
7070
W4
EXAMPLES OF ODDSRATIO RECOVERABILITY
(a) (b)
W3W4
S
YX
W2W1
W3W4
S
YX
W2W1
Recoverable Non-recoverable
)1,,,|,(),( 421 SWWWXYORXYOR (a)
Separator
7171
W1
EXAMPLES OF ODDSRATIO RECOVERABILITY
(a) (b)
W3W4
S
YX
W2
W3W4
S
YX
W2W1
Recoverable Non-recoverable
)1,,,|,(),( 421 SWWWXYORXYOR (a)
W2
0
Separator
7272
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)
CONCLUSIONS
7373
Thank you for agreeing with everything I said.