CAUSAL INFERENCE IN STATISTICS. A Gentle Introduction. Judea Pearl University of California Los Angeles (

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1 CAUSAL INFERENCE IN STATISTICS A Gentle Introduction Judea Pearl University of California Los Angeles (

2 OUTLINE Inference: Statistical vs. Causal, distinctions, and mental barriers Unified conceptualization of counterfactuals, structural-equations, and graphs Inference to three types of claims: 1. Causal effects and confounding 2. Attribution (Causes of Effects) 3. Direct and indirect effects Frills external validity and transportability

3 TRADITIONAL STATISTICAL INFERENCE PARADIGM Data P Joint Distribution Q(P) (Aspects of P) Inference e.g., Estimate the mean of Q( P) E( ) xp ( x) x

4 TRADITIONAL STATISTICAL INFERENCE PARADIGM Data P Joint Distribution Q(P) (Aspects of P) Inference e.g., Estimate the probability that a customer who bought product A would also buy product B. Q = P(B A)

5 FROM STATISTICAL TO CAUSAL ANALSIS: 1. THE DIFFERENCES Probability and statistics deal with static relations Data P Joint Distribution change Inference P Joint Distribution Q(P) (Aspects of P) What happens when P changes? e.g., Estimate the probability that a customer who bought A would buy B if we were to double the price.

6 FROM STATISTICAL TO CAUSAL ANALSIS: 1. THE DIFFERENCES What remains invariant when P changes say, to satisfy P (price=2)=1 Data P Joint Distribution change Inference P Joint Distribution Q(P) (Aspects of P) Note: P (B) P (B price = 2) e.g., Doubling price seeing the price doubled. P does not tell us how it ought to change.

7 FROM STATISTICAL TO CAUSAL ANALSIS: 1. THE DIFFERENCES (CONT) 1. Causal and statistical concepts do not mix. CAUSAL Spurious correlation Randomization / Intervention Confounding / Effect Instrumental variable Ignorability / Exogeneity Explanatory variables 2. STATISTICAL Regression Association / Independence Controlling for / Conditioning Odd and risk ratios Collapsibility / Granger causality Propensity score 3. 4.

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9 FROM STATISTICAL TO CAUSAL ANALSIS: 1. THE DIFFERENCES (CONT) 1. Causal and statistical concepts do not mix. CAUSAL Spurious correlation Randomization / Intervention Confounding / Effect Instrumental variable Ignorability / Exogeneity Explanatory variables 2. STATISTICAL Regression Association / Independence Controlling for / Conditioning Odd and risk ratios Collapsibility / Granger causality Propensity score 3. 4.

10 FROM STATISTICAL TO CAUSAL ANALSIS: 2. MENTAL BARRIERS 1. Causal and statistical concepts do not mix. CAUSAL STATISTICAL Spurious correlation Regression Randomization / Intervention Association / Independence Confounding / Effect Controlling for / Conditioning Instrumental variable Odds and risk ratios Ignorability / Exogeneity Collapsibility / Granger causality Explanatory variables Propensity score 2. No causes in no causes out (Cartwright, 1989) statistical assumptions + data causal assumptions causal conclusions 3. Causal assumptions cannot be expressed in the mathematical language of standard statistics. 4. Non-standard mathematics: a) Structural equation models (Wright, 1920; Simon, 1960) b) Counterfactuals (Neyman-Rubin ( x ), Lewis (x )) }

11 WH PHSICS IS COUNTERFACTUAL Scientific Equations (e.g., Hooke s Law) are non-algebraic e.g., Length () equals a constant (2) times the weight () = 2 = 1 Process information = 1 = 2 The solution

12 WH PHSICS IS COUNTERFACTUAL Scientific Equations (e.g., Hooke s Law) are non-algebraic e.g., Length () equals a constant (2) times the weight () Correct notation: := 2 = 3 = 1 Process information = 1 = 2 The solution Had been 3, would be 6. If we raise to 3, would be 6. Must wipe out = 1.

13 WH PHSICS IS COUNTERFACTUAL Scientific Equations (e.g., Hooke s Law) are non-algebraic e.g., Length () equals a constant (2) times the weight () Correct notation: (or) 2 = 3 = 1 Process information Had been 3, would be 6. If we raise to 3, would be 6. Must wipe out = 1. = 1 = 2 The solution

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

15 FAMILIAR CAUSAL MODEL ORACLE FOR MANIPILATION INPUT OUTPUT

16 STRUCTURAL CAUSAL MODELS Definition: A structural causal model is a 4-tuple V,U, F, P(u), where V = {V 1,...,V n } are endogenous variables U = {U 1,...,U m } are background variables F = {f 1,..., f n } are functions determining V, v i = f i (v, u) e.g., y x u P(u) is a distribution over U P(u) and F induce a distribution P(v) over observable variables

17 STRUCTURAL MODELS AND CAUSAL DIAGRAMS The functions v i = f i (v,u) define a graph v i = f i (pa i,u i ) PA i V \ V i U i U Example: Price Quantity equations in economics U 1 I W U 2 q p b b 1 2 p q d d 1 i 2 w u 1 u 2 Q P PA Q

18 SIMULATING INTERVENTIONS IN STRUCTURAL MODELS do(x) Double the price Take a drug Raise taxes Make me laugh

19 SIMULATING INTERVENTIONS IN STRUCTURAL MODELS do(x) Let be a set of variables in V. The action do(x) sets to constants x regardless of the factors which previously determined. do(x) replaces all functions f i determining with the constant functions =x, to create a mutilated model M x q p b b 1 2 p q d d 1 i 2 w u 1 u 2 U 1 I W U 2 Q P

20 SIMULATING INTERVENTIONS IN STRUCTURAL MODELS Let be a set of variables in V. The action do(x) sets to constants x regardless of the factors which previously determined. do(x) replaces all functions f i determining with the constant functions =x, to create a mutilated model M x M p q p p b b 1 p 2 0 p q d d 1 i 2 w u 1 u 2 U 1 I W U 2 Q P P = p 0

21 CAUSAL MODELS AND COUNTERFACTUALS If I were a rich man Had we doubled the price

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

23 ), ( ), ' ( ) ( ) ( ) ( ' ) ' ( : ' ) ( : y x u P y x y PN u P y P y P y u u x x y u u x x In particular: do(x) CAUSAL MODELS AND COUNTERFACTUALS Definition: The sentence: would be y (in situation u), had been x, denoted x (u) = y, means: The solution for in a mutilated model M x, (i.e., the equations for replaced by = x) with input U=u, is equal to y. ) ( ), ( ) (, ) ( : u P z y P z u y u u w x w x Joint probabilities of counterfactuals:

24 READING COUNTERFACTUALS FROM SEM Score Treatment Study Time Data shows: = 0.7, = 0.5, = 0.4 A student named Joe, measured = 0.5, = 1.0, = 1.9 Q 1 : What would Joe s score be had he doubled his study time? x z x 2 y x z 3

25 READING COUNTERFACTUALS FROM SEM Score Treatment Study Time Data shows: = 0.7, = 0.5, = 0.4 A student named Joe, measured = 0.5, = 1.0, = 1.9 Q 1 : What would Joe s score be had he doubled his study time? Answer: =2 = = 2.30 x z x 2 y x z 3 z 2.0

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

27 THE FIVE NECESSAR STEPS OF CAUSAL ANALSIS Define: Assume: Identify: Estimate: Test: Express the target quantity Q as property of the model M. Express causal assumptions in structural or graphical form. Determine if Q is identifiable. Estimate Q if it is identifiable; approximate it, if it is not. If M has testable implications Repeat if necessary

28 THE LOGIC OF CAUSAL ANALSIS A - CAUSAL ASSUMPTIONS Q Queries of interest Q(P) - Identified estimands Data (D) CAUSAL MODEL (M A ) A* - Logical implications of A Causal inference T(M A ) - Testable implications Statistical inference Q - Estimates of Q(P) Q( D, A) Provisional claims g(t) Model testing Goodness of fit

29 THE FIVE NECESSAR STEPS FOR EFFECT ESTIMATION Define: Assume: Identify: Estimate: Test: Express the target quantity Q as a property of the model M. P( x y) or P( y do( x)) Express causal assumptions in structural or graphical form. Determine if Q is identifiable. Estimate Q if it is identifiable; approximate it, if it is not. If M has testable implications

30 THE FIVE NECESSAR STEPS FOR AVERAGE TREATMENT EFFECT Define: Assume: Identify: Estimate: Test: Express the target quantity Q as a property of the model M. ATE E( do( x1 )) E( do( x0)) Express causal assumptions in structural or graphical form. Determine if Q is identifiable. Estimate Q if it is identifiable; approximate it, if it is not. If M has testable implications

31 THE FIVE NECESSAR STEPS FOR DNAMIC POLIC ANALSIS Define: Assume: Identify: Estimate: Test: Express the target quantity Q as a property of the model M. P( y do( x g( z)) Express causal assumptions in structural or graphical form. Determine if Q is identifiable. Estimate Q if it is identifiable; approximate it, if it is not. If M has testable implications

32 THE FIVE NECESSAR STEPS FOR TIME VARING POLIC ANALSIS Define: Assume: Identify: Estimate: Test: Express the target quantity Q as a property of the model M. P( y do( x, z, W w)) Express causal assumptions in structural or graphical form. Determine if Q is identifiable. Estimate Q if it is identifiable; approximate it, if it is not. If M has testable implications

33 THE FIVE NECESSAR STEPS FOR TREATMENT ON TREATED Define: Assume: Identify: Estimate: Test: Express the target quantity Q a property of the model M. ETT P( x y x') Express causal assumptions in structural or graphical form. Determine if Q is identifiable. Estimate Q if it is identifiable; approximate it, if it is not. If M has testable implications

34 THE FIVE NECESSAR STEPS FOR INDIRECT EFFECTS Define: Assume: Identify: Estimate: Test: Express the target quantity Q a property of the model M. IE E[ ) ] Express causal assumptions in structural or graphical form. Determine if Q is identifiable. Estimate Q if it is identifiable; approximate it, if it is not. If M has testable implications E[ x, ( x' x ]

35 THE FIVE NECESSAR STEPS FROM DEFINITION TO ASSUMPTIONS Define: Express the target quantity Q as a property of the model M. Assume: Identify: Estimate: Test: Express causal assumptions in structural or graphical form. Determine if Q is identifiable. Estimate Q if it is identifiable; approximate it, if it is not. If M has testable implications

36 testable? FORMULATING ASSUMPTIONS THREE LANGUAGES 1. English: Smoking (), Cancer (), Tar (), Genotypes (U) Not too friendly: 3. Structural: }, { ), ( ) ( ), ( ) ( ) ( ) ( ), ( ) ( u u u u u u u u z x zx z z zy y yx x 2. Counterfactuals: ), ( ),, ( ), ( x f z u z f y u f x U Consistent?, complete?, redundant?, plausible?,

37 FROM Q AND ASSUMPTIONS TO IDENTIFICATION Define: Assume: Identify: Estimate: Test: Express the target quantity Q as a function Q(M) that can be computed from any model M. Express causal assumptions in structural or graphical form. Determine if Q is identifiable. Estimate Q if it is identifiable; approximate it, if it is not. If M has testable implications SOLVED!!!

38 THE PROBLEM OF CONFOUNDING Find the effect of on, P(y do(x)), given measurements on auxiliary variables 1,..., k G Can P(y do(x)) be estimated if only a subset,, can be measured?

39 ELIMINATING CONFOUNDING BIAS THE BACK-DOOR CRITERION P(y do(x)) is estimable if there is a set of variables that d-separates from in G x. G G x Moreover, P(y do(x)) = P(y x,z) P(z) z ( adjusting for )

40 EFFECT OF INTERVENTION BEOND ADJUSTMENT Theorem (Tian-Pearl 2002) We can identify P(y do(x)) if there is no child of connected to by a confounding path. G

41 EFFECT OF WARM-UP ON INJUR (After Shrier & Platt, 2008) No, no!

42 COUNTERFACTUALS AT WORK ETT EFFECT OF TREATMENT ON THE TREATED 1. Regret: I took a pill to fall asleep. Perhaps I should not have? 2. Program evaluation: What would terminating a program do to those enrolled? P( x y x')

43 IDENTIFICATION OF COUNTERFACTUALS ETT P( x y x') ETT is identifiable in G iff P(y do(x),w) is identifiable in G G' Moreover, W ETT P( y x') x P( y do( x), w) Complete graphical criterion (Shpitser-Pearl, 2009) G' wx'

44 ETT - THE BACK-DOOR CRITERION ETT is identifiable in G if there is a set of variables that d-separates from in G x. G G x Moreover, ETT z P ( y x, z) P( z x' ) 6 Standardized morbidity

45 FROM IDENTIFICATION TO ESTIMATION Define: Assume: Identify: Estimate: Express the target quantity Q as a function Q(M) that can be computed from any model M. e.g., Q P( y do( x)) Formulate causal assumptions using ordinary scientific language and represent their structural part in graphical form. Determine if Q is identifiable. Estimate Q if it is identifiable; approximate it, if it is not.

46 PROPENSIT SCORE ESTIMATOR (Rosenbaum & Rubin, 1983) L ),,,, 1 ( ),,,, ( z z z z z P z z z z z L Adjustment for L replaces Adjustment for z l l L P x l L y P z P x z y P ) ( ), ( ) ( ), ( Theorem: P(y do(x)) =?

47 WHAT PROPENSIT SCORE (PS) PRACTITIONERS NEED TO KNOW z L( z) P( 1 z) P ( y z, x) P( z) P( y l, x) P( l) 1. The assymptotic bias of PS is EQUAL to that of ordinary adjustment (for same ). 2. Including an additional covariate in the analysis CAN SPOIL the bias-reduction potential of others. 3. Choosing sufficient set for PS, requires knowledge about the model. l

48 WHICH COVARIATES MA / SHOULD BE ADJUSTED FOR? Assignment Treatment B 1 Age M Hygiene Outcome B 2 Cost Follow-up Question: Which of these eight covariates may be included in the propensity score function (for matching) and which should be excluded. Answer: Must include: Must exclude: May include: Age B 1, M, B 2, Follow-up, Assignment without Age Cost, Hygiene, {Assignment + Age}, {Hygiene + Age + B 1 }, more...

49 WHICH COVARIATES MA / SHOULD BE ADJUSTED FOR? Assignment Treatment B 1 Age M Hygiene Outcome B 2 Cost Follow-up Question: Which of these eight covariates may be included in the propensity score function (for matching) and which should be excluded. Answer: Must include: Must exclude: May include: Age B 1, M, B 2, Follow-up, Assignment without Age Cost, Hygiene, {Assignment + Age}, {Hygiene + Age + B 1 }, more...

50 WHAT PROPENSIT SCORE (PS) PRACTITIONERS NEED TO KNOW z L( z) P( 1 z) P( y z, x) P( z) 1. The assymptotic bias of PS is EQUAL to that of ordinary adjustment (for same ). 2. Including an additional covariate in the analysis CAN SPOIL the bias-reduction potential of others. 3. Choosing sufficient set for PS, requires knowledge about the model. 4. That any empirical test of the bias-reduction potential of PS, can only be generalized to cases where the causal relationships among covariates, observed and unobserved is the same. 1 P( y l, x) P( l)

51 THE STRUCTURAL-COUNTERFACTUAL SMBIOSIS 1. Express assumptions in structural or graphical language. 2. Express queries in counterfactual language. 3. Translate (1) into (2) for algebraic analysis, Or (2) into (1) for graphical analysis. 4. Use either graphical or algebraic machinery to answer the query in (2).

52 GRAPHICAL COUNTERFACTUALS TRANSLATION Every causal graph expresses counterfactuals assumptions, e.g., 1. Missing arrows x, z ( u) x ( u) 2. Missing arcs x y consistent, and readable from the graph. Every theorem in SCM is a theorem in Potential-Outcome Model, and conversely.

53 DEMSTIFING CONDITIONAL IGNORABILIT { (0), (1)} (Ignorability) ( ) G (Back-door) Where in the graph are { (0), (1)}? W plays the role of { (0), (1)} in the graph W { } (0), (1) S (a) S (b)

54 DETERMINING THE CAUSES OF EFFECTS (The Attribution Problem) our Honor! My client (Mr. A) died BECAUSE he used that drug.

55 DETERMINING THE CAUSES OF EFFECTS (The Attribution Problem) our Honor! My client (Mr. A) died BECAUSE he used that drug. Court to decide if it is MORE PROBABLE THAN NOT that A would be alive BUT FOR the drug! PN = P(? A is dead, took the drug) > 0.50

56 THE ATTRIBUTION PROBLEM Definition: 1. What is the meaning of PN(x,y): Probability that event y would not have occurred if it were not for event x, given that x and y did in fact occur. Answer: Computable from M PN( x, y) P( ' y' x, y) x

57 THE ATTRIBUTION PROBLEM Definition: 1. What is the meaning of PN(x,y): Probability that event y would not have occurred if it were not for event x, given that x and y did in fact occur. Identification: 2. Under what condition can PN(x,y) be learned from statistical data, i.e., observational, experimental and combined.

58 PARTIAL IDENTIFICATION (Tian and Pearl, 2000) Bounds given combined nonexperimental and experimental data 0 P( y) P( y max P( x,y) x' ) PN 1 P( y' ) min x' P( x,y) Identifiability under monotonicity (Combined data) PN P( y x) P( y x' ) P( y x) P( y x' ) P( y P( x,y) x' ) corrected Excess-Risk-Ratio

59 CAN FREQUENC DATA DECIDE LEGAL RESPONSIBILIT? Experimental Nonexperimental do(x) do(x) x x Deaths (y) Survivals (y) ,000 1,000 1,000 1,000 Nonexperimental data: drug usage predicts longer life Experimental data: drug has negligible effect on survival Plaintiff: Mr. A is special. 1. He actually died 2. He used the drug by choice Court to decide (given both data): Is it more probable than not that A would be alive but for the drug? PN P( ' y' x, y) x 0.50

60 SOLUTION TO THE ATTRIBUTION PROBLEM WITH PROBABILIT ONE 1 P(y x x,y) 1 Combined data tell more that each study alone

61 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?

62 WH 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

63 LEGAL IMPLICATIONS OF DIRECT EFFECT Can data prove an employer guilty of hiring discrimination? (Gender) (Qualifications) What is the direct effect of on? (averaged over z) (Hiring) E ( do( x1 ), do( z)) E( do( x0), do( z)) Adjust for? No! No!

64 NATURAL INTERPRETATION OF AVERAGE DIRECT EFFECTS Robins and Greenland (1992) Pure z = f (x, u) y = g (x, z, u) Natural Direct Effect of on : The expected change in, when we change from x 0 to x 1 and, for each u, we keep constant at whatever value it attained before the change. E[ x DE( x0, x1; ) 1 x x 0 0 In linear models, DE = Controlled Direct Effect ] ( x1 x0 )

65 DEFINITION OF INDIRECT EFFECTS z = f (x, u) y = g (x, z, u) IE( x0, x1; ) Indirect Effect of on : The expected change in when we keep constant, say at x 0, and let change to whatever value it would have attained had changed to x 1. E[ x 0x x 1 0 In linear models, IE = TE - DE ]

66 POLIC IMPLICATIONS OF INDIRECT EFFECTS What is the indirect effect of on? The effect of Gender on Hiring if sex discrimination is eliminated. GENDER QUALIFICATION IGNORE f HIRING Blocking a link a new type of intervention

67 MEDIATION FORMULAS IN UNCONFOUNDED MODELS DE IE TE IE TE DE [ E( x1, z) E( x0, z)] P( z z [ E( z E( x0, z)[ P( z x1 ) E( x0) Fraction of effect Fraction of effect x1 ) P( z explained owed z f ( x, u1) y g( x, z, u x0) x0)] x u 1 by mediation to mediation 2 u ) 2

68 SUMMAR OF MEDIATION RESULTS 1. Formal semantics of path-specific effects, based on disabling mechanisms, instead of value fixing. 2. Path-analytic techniques extended to nonlinear and nonparametric models. 3. Meaningful (graphical) conditions for estimating direct and indirect effects from experimental and nonexperimental data.

69 ETERNAL VALIDIT From Threats to Licenses `External validity asks the question of generalizability: To what population, settings, treatment variables, and measurement variables can this effect be generalized? (Shadish, Cook and Campbell 2002) An experiment is said to have `external validity if the distribution of outcomes realized by a treatment group is the same as the distribution of outcome that would be realized in an actual program. (Manski, 2007) "A threat to external validity is an explanation of how you might be wrong in making a generalization." (Wikipedia 2011, after Trochin) A license of validity is a set of theoretical assumptions that neutralizes all conceivable threats. (Anon, 2011)

70 TRANSPORTABILIT ACROSS DOMAINS 1. A Theory of causal transportability When can causal relations learned from experiments be transferred to a different environment in which no experiment can be conducted? 2. A Theory of statistical transportability When can statistical information learned in one domain be transferred to a different domain in which a. only a subset of variables can be observed? Or, b. only a few samples are available? 3. Applications to Meta Analysis Combining results from many diverse studies

71 MOTIVATION WHAT CAN EPERIMENTS IN LA TELL ABOUT NC? (Age) (Age) * (Intervention) (Outcome) (Observation) (Outcome) Experimental study in LA Measured: P( x, y, z) P( y do( x), z) Observational study in NC Measured: P* ( x, y, z) P* ( z) P( z) Needed: P* ( y do( x))? z P ( y do( x), z) P*( z) Transport Formula (calibration): F( P, P, P*) do

72 TRANSPORT FORMULAS DEPEND ON THE STOR S (a) a) represents age P*( y do( x)) z P( y do( x), z) P*( z)

73 TRANSPORT FORMULAS DEPEND ON THE STOR S S (a) (b) a) represents age P*( y do( x)) z b) represents language skill P( y do( x), z) P*( z) P *( y do( x)) P( y do( x))

74 TRANSPORT FORMULAS DEPEND ON THE STOR (a) S a) represents age P*( y b) represents language skill do( x)) P* ( y do( x)) z? (b) P( y S do( x), z) P*( z) S Factors producing differences

75 TRANSPORT FORMULAS DEPEND ON THE STOR (a) S a) represents age P*( y do( x)) z P( y b) represents language skill P* ( y do( x)) c) represents a bio-marker P* ( y do( x)) (b) S do( x), z) P*( z) P( y do( x)) z P ( y do( x), z) P*( z x) S (c)

76 GOAL: ALGORITHM TO DETERMINE IF AN EFFECT IS TRANSPORTABLE Back to Transportability U T S W V S INPUT: Annotated Causal Graph S Factors creating differences OUTPUT: 1. Transportable or not? 2. Measurements to be taken in the experimental study 3. Measurements to be taken in the target population 4. A transport formula P* ( y do( x)) f [ P( y, v, z, w, t, u do( x)); P* ( y, v, z, w, t, u)]

77 TRANSPORTABILIT REDUCED TO CALCULUS Theorem 1 A causal relation R is transportable from to * if and only if R(*) is reducible, using the rules of do-calculus, to an expression in which S appears only as a conditioning variable in do-free terms. R * P*( y do( x)) P( y do( x), s) w w w P ( y do( x), s, w) P( w do( x), s) P ( y do( x), w) P( w s) P ( y do( x), w) P*( w) S W

78 RESULT: ALGORITHM TO DETERMINE IF AN EFFECT IS TRANSPORTABLE U T S W V S INPUT: Annotated Causal Graph S Factors creating differences OUTPUT: 1. Transportable or not? 2. Measurements to be taken in the experimental study 3. Measurements to be taken in the target population 4. A transport formula P* ( y do( x)) z P ( y do( x), z) P* ( z w) P( w do( x), t) w t P* ( t)

79 WHICH MODEL LICENSES THE TRANSPORT OF THE CAUSAL EFFECT S External factors creating disparities es No es S S S (a) (b) (c) S es S es S No W (d) W (e) (f)

80 STATISTICAL TRANSPORTABILIT Why should we transport statistical information? i.e., Why not re-learn things from scratch? 1. Measurements are costly. Limit measurements to a subset V * of variables called scope. 2. Samples are scarce. Pooling samples from diverse populations will improve precision, if differences can be filtered out.

81 STATISTICAL TRANSPORTABILIT Definition: (Statistical Transportability) A statistical relation R(P) is said to be transportable from to * over V * if R(P*) is identified from P, P*(V *), where V* is a subset of variables. R=P* (y x) is transportable over S V* = {,}, i.e., R is estimable without re-measuring R P*( z x) P( z y) z Transfer Learning S If few samples (N 2 ) are available from * and many samples (N 1 ) from, then estimating R = P*(y x) by R P*( y x, z) P( z x) z achieves a much higher precision

82 META-ANALSIS OR MULTI-SOURCE LEARNING Target population * R = P*(y do(x)) (a) (b) (c) S W W W (d) (e) S (f) S W W W (g) (h) S (i) S W W W

83 CAN WE GET A BIAS-FREE ESTIMATE OF THE TARGET QUANTIT? Target population * R = P*(y do(x)) (a) (d) W W S Is R identifiable from (d) and (h)? R P*( y do( x), w) P*( w do( x)) w P w ( h) ( y do( x), w) P( d) ( w do( x)) P( h) ( y do( x), w) P( d) ( w x) w R(*) is identifiable from studies (d) and (h). R(*) is not identifiable from studies (d) and (i). (h) S (i) S W W

84 FROM META-ANALSIS TO META-SNTHESIS The problem How to combine results of several experimental and observational studies, each conducted on a different population and under a different set of conditions, so as to construct an aggregate measure of effect size that is "better" than any one study in isolation. Definition (Meta-Estimability) A relation R is said to be "meta estimable" from a set of populations { 1, 2,..., K } to a target population * iff it is identifiable from the information set I = {I( 1 ), I( 2 ),..., I( K ), I(P*)}.

85 FROM META-ANALSIS TO META-SNTHESIS (Cont.) Theorem { 1, 2,..., K } a set of studies. {D 1, D 2,..., D k } selection diagrams (relative to *). A relation R(*) is "meta estimable" if it can be decomposed into terms of the form: Q k P( Vk do( Wk ), k ) such that each Q k is transportable from D k. Open-problem: Systematic decomposition

86 BIAS VS. PRECISION IN META-SNTHESIS Principle 1: Calibrate estimands before pooling (to minimize bias) Principle 2: Decompose to sub-relations before calibrating (to improve precision) R ( *) P *( y do ( x )) (a) (g) (h) S (i) (d) S W W W P ( * g) ( y do( x)) Pooling W W Calibration P ( * ) ( y do( x)) d

87 BIAS VS. PRECISION IN META-SNTHESIS (Cont.) (a) (g) (h) S (i) (d) S W W W W W R( *) P *( y do( x)) P ( * g) ( y do( x)) P ( * ) ( y w, do( x)) P * ( ( )) h ( * ) ( w do( x)) P ( ) w do x i d Pooling P ( w do( )) Composition ( * i, d ) x P* ( i, d, k) ( y do( x)) w P* ( h) ( y w, do( x)) P* ( i, d) ( w do( x)) P* ( all) x Pooling ( y do( ))

88 CONCLUSIONS I TOLD OU CAUSALIT IS SIMPLE Principled methodology 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.

89 CONCLUSIONS He is wise who bases causal inference on an explicit causal structure that is defensible on scientific grounds. (Aristotle B.C.) From Charlie Poole

90 QUESTIONS??? Now is the time!