PANEL ON EDIAION AND OHER IRACLES OF HE 21 S CENUR Judea Pearl UCLA With Kosuke Imai and many others HE QUESION OF EDIAION (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? 2 WH DECOPOSE EFFECS? 1. o understand how Nature works (Gender) LEGAL IPLICAIONS OF DIREC EFFEC Can data prove an employer guilty of hiring discrimination? (Qualifications) 2. o comply with legal requirements 3. o predict the effects of new type of interventions: Signal re-routing and mechanism deactivating, rather than variable fixing 3 (Hiring) What is the direct effect of on? CDE = E( do(x 1 ),do(m)) E( do(x 0 ),do(m)) (m-dependent) Adjust for? No! No! CDE identification is completely solved (ian et al. 2002) 4 NAURAL INERPREAION OF AVERAGE DIREC EFFECS DEFINIION OF INDIREC EFFECS Robins and Greenland (1992) Pearl (2001) m = f (x, u) y = g (x, m, u) Natural Direct Effect of on : DE(x 0, x 1 ; ) he 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[ x1 x x 0 0 ] Note the 3-way symbiosis 5 m = f (x, u) y = g (x, m, u) Indirect Effect of on : IE( x0, x1; ) he 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[ x0 x 1 x 0 ] In linear models, IE = E - DE No controlled indirect effect 6 1
POLIC IPLICAIONS OF INDIREC EFFECS What is the indirect effect of on? he effect of Gender on Hiring if sex discrimination is eliminated. GENDER IGNORE f HIRING QUALIFICAION Deactivating a link a new type of intervention 7 HE EDIAION FORULAS IN UNCONFOUNDED ODELS m = f (x, u 1 ) y = g (x, m, u 2 ) u 1 independent of u 2 DE = [E( x 1,m) E( x 0,m)]P(m x 0 ) m IE = [E( x 0,m)[P(m x 1 ) P(m x 0 )] m E = E( x 1 ) E( x 0 ) E DE + IE IE = Fraction of responses explained by mediation (sufficient) E DE = Fraction of responses owed to mediation (necessary) NONPARAERIC IDENIFICAION (Natural mediation is a solved problem) WHEN CAN WE IDENIF EDIAED EFFECS? he nonparametric estimability of natural (and controlled) direct and indirect effects can be determined in polynomial time given any causal graph G with both measured and unmeasured variables. If NDE (or NIE) is estimable, then its estimand can be derived in polynomial time. he algorithm is complete and was extended to any path-specific effects (Shpitser, 2013). (d) (e) (c) (f) W 1 WHEN CAN WE IDENIF EDIAED EFFECS? WHA CAN EDIAION FORULA DO FOR PARAERIC ANALSS? (d) (e) (c) (f) W 1 α W γ 2 β 4 γ 1 β 1 β 3 β 2 ulti-mediators non-linear model y = β 1 m + β 2 x + β 3 xm + β 4 w + u 1 m = γ 1 x + γ 2 w + u 2 w = αx + u 3 What combination of parameters gives the effect mediated by? IE( ) = β 1 (γ 1 + αγ 2 ) What combination of parameters gives the effect owed to? E DE( ) = (β 1 + β 3 )(γ 1 + αγ 2 ) 2
IGNORABILI CONDIIONS FOR NDE IDENIFICAION (Sequential Ignorability) (Imai et al (2010)) No confounding W (confounder) here exists a set W of measured covariates such that: SI-1 x xm (,W ) SI-2 ( xm, x ) W WEAKER AND RANSPAREN CONDIIONS FOR NDE IDENIFICAION No confounding here exists a set W such that: A-1 No member of W is a descendant of. A-2 W blocks all back-door paths from to, disregarding the one through. A-3 he W -specific effect of on is identifiable. P(m do(x),w) A-4 he W specific effect of {, } on is identifiable. P(y do(x,m),w) W (confounder) FRIENDL ECHANGE CONCERNING IGNORABILI VS. GRAPHICAL ASSUPIONS Psychological ethods (2014) Imai et al. proved that graphical and ignorability assumptions are identical for randomized treatments. Concensus achieved regarding transparency of graphical assumptions. Semi-concensus regarding other aspects of the graphical vs. ignorability languages. DAGS VS. POENIAL COUCOES AN UNBIASED PERSPECIVE 1. Semantic Equivalence 2. Both are abstractions of Structural Equation odels (SE). x (u) = x (u) y = f (x,z,u) x (u) = All factors that affect when is held constant at =x. FORULAING A PROBLE IN HREE LANGUAGES 1. English: Smoking (), Cancer (), ar (Z), Genotypes (U) U FORULAING A PROBLE IN HREE LANGUAGES 1. English: Smoking (), Cancer (), ar (Z), Genotypes (U) U Z 2. Counterfactuals: Z x (u) = Z yx (u), y (u) = zy (u) = z (u) = (u), z (u) = zx (u), Z x { z, } Not too friendly: Consistent?, complete?, redundant?, plausible?, testable? Z 2. Counterfactuals: Z x (u) = Z yx (u), y (u) = zy (u) = z (u) = (u), z (u) = zx (u), Z x { z, } 3. Structural: U x = f 1 (u,ε 1 ) y = f 3 (z,u,ε 3 ) Z z = f 2 (x,ε 2 ) ε 1 ε 2 ε 3 3
HE SRUCURAL-COUNERFACUAL SBIOSIS 1. Express theoretical assumptions in structural language. 2. Express queries in counterfactual language. 3. ranslate (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). hank you JON S QUESIONS O PANEL 1. Do you think an experiment has any value without mediational analysis? 2. Is a separate study directly manipulating the mediator useful? How is the second study any different from the first one? 3. Imai's correlated residuals test seems valuable for distinguishing fake from genuine mediation. Is that so? 4. Why isn't it easy to test whether participants who show the largest increases in the posited mediator show the largest changes in the outcome? 1 0 = f m (Z 0 Z 1 ) f m monotonic 5. Why is mediational analysis any worse than any other method of investigation? CHRISIAN S QUESIONS O PANEL 1. Can we go beyond if assumptions, then conclusions? es, testable implications, experimental evidence, other studies. 2. How your framework would use the results of one mediation analysis to inform the setup of a second, new mediation analysis? It is not a question of framework but of information HE EDIAION FALLAC OR HOW RADIIONALISS CONFUSED AN ENIRE CENUR HE EDIAION FALAC OR HOW RADIIONALISS CONFUSED AN ENIRE CENUR L L A A Standard model DE = Whatever changes we see in when we vary and hold constant Holding constant controlling for Statistics has no operator for holding constant Controlling for leads to fallacies in the presence of unobserved confounders A B B Conditioning on creates dependency between and though DE = 0 C Fixing correctly shows DE = 0 No dependence C 4
DIVIDE AND CONQUER AKES A DIFFERENCE DIVIDE AND CONQUER AKES A DIFFERENCE deconfounds deconfounds {, } confounds wo separate sets can accomplish what their union can t. wo separate sets can accomplish what their union can t. o identify NDE, we must condition on and separately. SEQUENIAL IGNORABILI IS NO NEEDED Z W = 0 No set can deconfound. easuring Z permits the identification of P ( = m do(x)) through the front-door formula. NDE is identified. 5