OUTLINE CAUSES AND COUNTERFACTUALS IN THE EMPIRICAL SCIENCES. Judea Pearl University of California Los Angeles (

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1 CAUE AND COUNTERFACTUAL IN THE EMPIRICAL CIENCE Judea Pearl University of California Los Angeles ( OUTLINE Inference: tatistical vs. Causal, distinctions, and mental barriers Unified conceptualiation of counterfactuals, structural-equations, and graphs Inference to three types of claims:. Effect of potential interventions 2. Attribution (Causes of Effects) 3. Direct and indirect effects Frills 2 Data TRADITIONAL TATITICAL INFERENCE PARADIGM P Inference Q(P) (Aspects of P) FROM TATITICAL TO CAUAL ANALI:. THE DIFFERENCE Probability and statistics deal with static relations Data P change Inference P Q(P ) (Aspects of P ) e.g., Infer whether customers who bought product A would also buy product B. Q = B A) 3 What happens when P changes? e.g., Infer whether customers who bought product A would still buy A if we were to double the price. 4 FROM TATITICAL TO CAUAL ANALI:. THE DIFFERENCE Data What remains invariant when P changes say, to satisfy P (price=2)= P change P Q(P ) (Aspects of P ) FROM TATITICAL TO CAUAL ANALI:. THE DIFFERENCE (CONT). Causal and statistical concepts do not mix. CAUAL TATITICAL purious correlation Regression Randomiation / Intervention Association / Independence Confounding / Effect Controlling for / Conditioning Instrumental variable Odd and risk ratios trong Exogeneity Collapsibility / Granger causality Explanatory variables Propensity score 2. Inference Note: P (v) P (v price = 2) P does not tell us how it ought to change e.g. Curing symptoms vs. curing diseases e.g. Analogy: mechanical deformation

2 FROM TATITICAL TO CAUAL ANALI: 2. MENTAL BARRIER. Causal and statistical concepts do not mix. CAUAL TATITICAL purious correlation Regression Randomiation / Intervention Association / Independence Confounding / Effect Controlling for / Conditioning Instrumental variable Odd and risk ratios trong Exogeneity Collapsibility / Granger causality Explanatory variables Propensity score 2. No causes in no causes out (Cartwright, 989) statistical assumptions + data causal assumptions causal conclusions 3. Causal assumptions cannot be expressed in the mathematical language of standard statistics. 4. } 7 FROM TATITICAL TO CAUAL ANALI: 2. MENTAL BARRIER. Causal and statistical concepts do not mix. CAUAL TATITICAL purious correlation Regression Randomiation / Intervention Association / Independence Confounding / Effect Controlling for / Conditioning Instrumental variable Odd and risk ratios trong Exogeneity Collapsibility / Granger causality Explanatory variables Propensity score 2. No causes in no causes out (Cartwright, 989) 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) tructural equation models (Wright, 92; imon, 96) b) Counterfactuals (Neyman-Rubin ( x ), Lewis (x )) } 8 WH CAUALIT NEED PECIAL MATHEMATIC cientific Equations (e.g., Hooke s Law) are non-algebraic e.g., Length () equals a constant (2) times the weight () Correct notation: = := 2 = Process information Had been 3, would be 6. If we raise to 3, would be 6. Must wipe out =. = = 2 The solution WH CAUALIT NEED PECIAL MATHEMATIC cientific Equations (e.g., Hooke s Law) are non-algebraic e.g., Length () equals a constant (2) times the weight () Correct notation: (or) 2 = Process information Had been 3, would be 6. If we raise to 3, would be 6. Must wipe out =. = = 2 The solution 9 THE TRUCTURAL MODEL PARADIGM FAMILIAR CAUAL MODEL ORACLE FOR MANIPILATION Data Inference Data Generating Model M Q(M) (Aspects of M) M Invariant strategy (mechanism, recipe, law, protocol) by which Nature assigns values to variables in the analysis. INPUT OUTPUT Think Nature, not experiment! 2 2

3 TRUCTURAL CAUAL MODEL Definition: A structural causal model is a 4-tuple V,U, F,, where V = {V,...,V n } are endogeneas variables U ={U,...,U m } are background variables F = {f,..., f n } are functions determining V, v i = f i (v, e.g., y = α + βx + u is a distribution over U and F induce a distribution v) over observable variables TRUCTURAL MODEL AND CAUAL DIAGRAM The functions v i = f i (v, 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 q = b p + di + u p = b2q + d2w + u2 U I W U 2 Q P PA Q 3 4 TRUCTURAL MODEL AND INTERVENTION TRUCTURAL MODEL AND INTERVENTION 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 = to create a mutilated model M 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 = to create a mutilated model M x M p q = b p + di + u p = b2q + d2w + u2 U I W U 2 Q P q = b p + di + u p = b2q + d2w + u2 p = p U I W U 2 Q P P = p 5 6 CAUAL MODEL AND COUNTERFACTUAL Definition: The sentence: would be y (in situation, had been denoted x ( = 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 ( = M ( x 7 CAUAL MODEL AND COUNTERFACTUAL Definition: The sentence: would be y (in situation, had been denoted x ( = 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. probabilities of counterfactuals: x = y, w = = u: x ( = y, w( = In particular: x) ) Δ = x = = u: x ( = y x' = y' = u u: x ' ( = y' 8 3

4 THE FIVE NECEAR TEP OF CAUAL ANALI THE FIVE NECEAR TEP FOR EFFECT ETIMATION Define: Express the target quantity Q as a function Q(M) that can be computed from any model M. Assume: Formulate causal assumptions A using some formal language. Identify: Determine if Q is identifiable given A. Estimate: Estimate Q if it is identifiable; approximate it, if it is not. Test: Test the testable implications of A (if an. 9 Define: Express the target quantity Q as a function Q(M) that can be computed from any model M. ATE Δ = E( do( x )) E( do( Assume: Formulate causal assumptions A using some formal language. Identify: Determine if Q is identifiable given A. Estimate: Estimate Q if it is identifiable; approximate it, if it is not. Test: Test the testable implications of A (if an. 2 COUNTERFACTUAL AT WORK ETT EFFECT OF TREATMENT ON THE TREATED. 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? x = y x') 2 THE FIVE NECEAR TEP FOR EFFECT OF TREATMENT ON THE TREATED Define: Express the target quantity Q as a function Q(M) that can be computed from any model M. ETT Δ = x = y = x') Assume: Formulate causal assumptions A using some formal language. Identify: Determine if Q is identifiable given A. Estimate: Estimate Q if it is identifiable; approximate it, if it is not. Test: Test the testable implications of A (if an. 22 THE LOGIC OF CAUAL ANALI IDENTIFICATION IN CM A - CAUAL AUMPTION Q Queries of interest Q(P) - Identified estimands Data (D) CAUAL MODEL (M A ) A* - Logical implications of A Causal inference T(M A ) - Testable implications tatistical inference Find the effect of on, y do(, given the causal assumptions shown in G, where,..., k are auxiliary variables. 3 4 G 2 5 Q - Estimates of Q(P) Q ( Q D, A) g(t ) Conditional claims Model testing Goodness of fit 23 6 Can y do( be estimated if only a subset,, can be measured? 24 4

5 ELIMINATING CONFOUNDING BIA THE BACK-DOOR CRITERION is estimable if there is a set of variables such that d-separates from in G x. G 2 G x 2 EFFECT OF WARM-UP ON INJUR (After hrier & Platt, 28) Watch out! Front??? Door Moreover, = y, y = P ( x ) ( adjusting for ) Ignorability No, no! Warm-up Exercises () Injury () 26 FROM IDENTIFICATION TO ETIMATION PROPENIT CORE ETIMATOR (Rosenbaum & Rubin, 983) Define: Express the target quantity Q as a function Q(M) that can be computed from any model M. Q = Assume: Formulate causal assumptions using ordinary scientific language and represent their structural part in graphical form. Identify: Determine if Q is identifiable. Estimate: Estimate Q if it is identifiable; approximate it, if it is not. 27 =? L L (, 2, 3, 4, ) Δ 5 = =, 2, 3, 4, 5) Theorem: P ( y, x) = y L = l, x) L = l) l Adjustment for L replaces Adjustment for 28 WHAT PROPENIT CORE (P) PRACTITIONER NEED TO KNOW REGREION V. TRUCTURAL EQUATION (THE CONFUION OF THE CENTUR) L ( = = = P ( y, x) = y l, x) l) l. The asymptotic bias of P is EQUAL to that of ordinary adjustment (for same ). 2. Including an additional covariate in the analysis CAN POIL the bias-reduction potential of others. 3. In particular, instrumental variables tend to amplify bias. 4. Choosing sufficient set for P, requires knowledge of the model. 29 Regression (claimless, nonfalsifiable): = ax + ε tructural (empirical, falsifiable): = bx + u Claim: (regardless of distributions): E( do( = E( do( do() = 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 3 5

6 TWO PARADIGM FOR CAUAL INFERENCE Observed:,,,...) Conclusions needed: x =, y =x =... UPER DITRIBUTION IN N-R MODEL x= x= = = y= U How do we connect observables,,,, to counterfactuals x,, y,? N-R model Counterfactuals are primitives, new variables uper-distribution P *(,,...,,...),, constrain y,... tructural model Counterfactuals are derived quantities ubscripts modify the model and distribution x = = PM ( = x 3 inconsistency: x = x= = = x + (-x) Defines : P *(,,,... y... xy,......) P *( x = y, ) x y u u 2 u 3 u 4 32 ARE THE TWO PARADIGM EQUIVALENT? AIOM OF TRUCTURAL COUNTERFACTUAL es (Galles and Pearl, 998; Halpern 998) In the N-R paradigm, x is defined by consistency: = x + ( x) In CM, consistency is a theorem. Moreover, a theorem in one approach is a theorem in the other. Difference: Clarity of assumptions and their implications 33 x (=y: would be y, had been x (in state U = (Galles, Pearl, Halpern, 998):. Definiteness x s. t. y ( = x 2. Uniqueness ( y ( = x) & ( y( = x') x = x' 3. Effectiveness xw ( = x 4. Composition (generalied consistenc w ( = x wx( = w( 5. Reversibility ( xw ( = & ( Wxy( = w) x( = y 34 FORMULATING AUMPTION THREE LANGUAGE. English: moking (), Cancer (), Tar (), Genotypes (U) U 2. Counterfactuals: x( = yx(, 3. tructural: y( = y( = ( = (, ( = x(, x {, } x = f( u, ε) = f2( ε2) y = f3(, u, ε3) 35 GRAPHICAL COUNTERFACTUAL MBIOI Every causal graph expresses counterfactuals assumptions, e.g.,. Missing arrows x, ( = x( 2. Missing arcs x y consistent, and readable from the graph. Express assumption in graphs Derive estimands by graphical or algebraic methods 36 6

7 DETERMINING THE CAUE OF EFFECT (The Attribution Problem) our Honor! My client (Mr. A) died BECAUE he used that drug. THE ATTRIBUTION PROBLEM Definition:. What is the meaning of PN(: Probability that event y would not have occurred if it were not for event given that x and y did in fact occur. Court to decide if it is MORE PROBABLE THAN NOT that A would be alive BUT FOR the drug!? A is dead, took the drug) >.5 Answer: Computable from M PN( = x ' = y' PN = THE ATTRIBUTION PROBLEM Definition:. What is the meaning of PN(: Probability that event y would not have occurred if it were not for event given that x and y did in fact occur. Identification: 2. Under what condition can PN( be learned from statistical data, i.e., observational, experimental and combined. 39 TPICAL THEOREM (Tian and Pearl, 2) Bounds given combined nonexperimental and experimental data y ) ( ) max x' P y' PN min x' Identifiability under monotonicity (Combined data) y x) y x' ) y x' ) y ) PN = + x' y x) corrected Excess-Risk-Ratio 4 CAN FREQUENC DATA DECIDE LEGAL REPONIBILIT? Experimental Nonexperimental do(x) do(x ) x x Deaths ( urvivals (y ) ,,,, Nonexperimental data: drug usage predicts longer life Experimental data: drug has negligible effect on survival Plaintiff: Mr. A is special.. 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 Δ = x' = y' >.5 4 OLUTION TO THE ATTRIBUTION PROBLEM WITH PROBABILIT ONE y x Combined data tell more that each study alone 42 7

8 EFFECT DECOMPOITION (direct vs. indirect effects) WH DECOMPOE EFFECT?. 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? 43. To understand how Nature works 2. To comply with legal requirements 3. To predict the effects of new type of interventions: ignal routing, rather than variable fixing 44 (Gender) LEGAL IMPLICATION OF DIRECT EFFECT Can data prove an employer guilty of hiring discrimination? (Hiring) What is the direct effect of on? (Qualifications) E ( do( x ), do( ) E( do( do( ) (averaged over Adjust for? No! No! 45 NATURAL INTERPRETATION OF AVERAGE DIRECT EFFECT Robins and Greenland (992) Pure = f ( y = g (, Natural Direct Effect of on : DE( x; ) The expected change in, when we change from x to x and, for each u, we keep constant at whatever value it attained before the change. E[ x ] x x In linear models, DE = Controlled Direct Effect = β ( x x ) 46 DEFINITION AND IDENTIFICATION OF NETED COUNTERFACTUAL Consider the quantity Q Δ = Eu [ x ( ] x*( Given M,, Q is well defined Given u, x* ( is the solution for in M x*, call it x x ( ) is the solution for in M * ( u x experimental Can Q be estimated from data? nonexperimental Experimental: nest-free expression Nonexperimental: subscript-free expression 47 DEFINITION OF INDIRECT EFFECT = f ( y = g (, Indirect Effect of on : IE( x; ) The expected change in when we keep constant, say at x, and let change to whatever value it would have attained had changed to x. E[ x ] x x In linear models, IE = TE - DE 48 8

9 POLIC IMPLICATION OF INDIRECT EFFECT What is the indirect effect of on? The effect of Gender on Hiring if sex discrimination is eliminated. GENDER IGNORE f QUALIFICATION MEDIATION FORMULA. The natural direct and indirect effects are identifiable in Markovian models (no confounding), 2. And are given by: DE = [ E( do( ) E( do( )] do(. IE = E( do( )[ do( x )) do( ] TE = DE EI( rev) HIRING Blocking a link a new type of intervention Applicable to linear and non-linear models, continuous and discrete variables, regardless of distributional form. 5 WH m TE DE + m 2 IE TE = DE IE(rev) MEDIATION FORMULA IN UNCONFOUNDED MODEL β In linear systems IE( rev) = IE TE = β + m m 2 TE DE = β TE - DE IE = m m2 = TE DE DE IE = Effect sustained by mediation alone Is NOT equal to: Disabling TE DE = Effect prevented by disabling mediation mediation IE Disabling direct path 5 DE = [ E( E( ] x) IE = [ E( [ x) x)] TE = E( x) E( x) IE = Fraction of responses explained by mediation TE DE = Fraction of responses owed to mediation 52 TRANPORTABILIT -- WHEN CAN WE ETRPOLATE EPERIMENTAL FINDING TO DIFFERENT POPULATION? = age Experimental study in LA Measured: y, = age Observational study in NC Measured: P* ( y, Problem: We find (LA population is younger) What can we say about P* ( Intuition: P *( = 53 TRANPORT FORMULA DEPEND ON THE TOR (a) (b) a) represents age P *( = b) represents language skill P* ( =??? c) represents a bio-marker P* ( =??? (c) 54 9

10 TRANPORTABILIT (Pearl and Bareinboim, 2) Definition (Transportabilit Given two populations, denoted Π and Π*, characteried by probability distributions P and P*, and causal diagrams G and G*, respectively, a causal relation R is said to be transportable from Π to Π* if. R(Π) is estimable from the set I of interventional studies on Π, and 2. R(Π*) is identified from I, P, P*, G, and G*. TRANPORT FORMULA DEPEND ON THE TOR (a) (b) a) represents age P *( = b) represents language skill P *( = c) represents a bio-marker P *( = x ) (c) WHICH MODEL LICENE THE TRANPORT OF THE CAUAL EFFECT (a) (b) (c) DETERMINE IF THE CAUAL EFFECT I TRANPORTABLE U T W V What measurements need to be taken in the study and in the target population? W W (c) (d) (e) (f) 57 The transport formula = w) w do( t) t) w t 58 CONCLUION CONCLUION I TOLD OU CAUALIT I IMPLE 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) He is wise who bases causal inference on an explicit causal structure that is defensible on scientific grounds. (Aristotle B.C.) 59 6

11 QUETION??? They will be answered 6