Judea Pearl Computer Science Department UCLA cs.ucla/~judea

1. Judea Pearl Computer Science Department UCLA www.cs.ucla.edu/~judea DIRECT AND INDIRECT EFFECTS 2005

2. QUESTIONS ASKED • Why decompose effects? • What are the semantics of direct and indirect effects (in nonlinear and nonparametric models)? • What are the policy implications of direct and indirect effects? • When can direct and indirect effects be estimatedconsistently from experimental or nonexperimental data? • Can these conditions be verified from accessible causal knowledge, i.e., graphs?

3. WHY DECOMPOSE EFFECTS? • Direct (or indirect) effect may be more transportable. • Indirect effects may be prevented or controlled. • Direct (or indirect) effect may be forbidden  Pill Pregnancy + + Thrombosis Gender Qualification Hiring

4. a bc EFFECT-DECOMPOSITION IN LINEAR MODELS b X Z a c Y Definition:

5. Definition: A causal model is a 3-tuple M = V,U,F (i) V = {V1…,Vn} endogenous variables, (ii) U = {U1,…,Um} background variables (unit) • F = set of n functions, • The sentence: “Y would be y (in unit u), had X been x,” • denoted Yx(u) = y, is the solution for Y in a mutilated model • Mx, with the equations for X replaced by X = x. • (“unit-based potential outcome”) CAUSAL MODELS AND COUNTERFACTUALS

6. COUNTERFACTUALS: STRUCTURAL SEMANTICS u u W W Z X X=x Z Y Yx(u)=y Notation: Yx(u) = y Y has the value y in the solution to a mutilated system of equations, where the equation for X is replaced by a constant X=x. Functional Bayes Net Probability of Counterfactuals:

7. TOTAL, DIRECT, AND INDIRECT EFFECTS HAVE CONTROLLED-BASED SEMANTICS IN LINEAR MODELS b X Z z = bx + 1 y = ax + cz + 2 a c Y a+bc a bc

8. CONTROLLED-BASED SEMANTICS NONTRIVIAL IN NONLINEAR MODELS (even when the model is completely specified) X Z z = f (x, 1) y = g (x, z, 2) Y Dependent on z? Void of operational meaning?

9. LEGAL DEFINITION OF DIRECT EFFECT (FORMALIZING DISCRIMINATION) ``The central question in any employment-discrimination case is whether the employer would have taken the same action had the employee been of different race (age, sex, religion, national origin etc.) and everything else had been the same’’ [Carson versus Bethlehem Steel Corp. (70 FEP Cases 921, 7th Cir. (1996))] x = male, x = female y = hire, y = not hire z = applicant’s qualifications NO DIRECT EFFECT

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

11. GENDER QUALIFICATION HIRING POLICY IMPLICATIONS (Who cares?) What is the direct effect of X on Y? Is employer guilty of sex-discrimination given data on (X,Y,Z)? X Z CAN WE IGNORE THIS LINK? f Y

12. SEMANTICS 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

13. NATURAL SEMANTICS OF INDIRECT EFFECTS X Z z = f (x, u) y = g (x, z, u) Y 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

14. GENDER QUALIFICATION HIRING POLICY IMPLICATIONS (Who cares?) What is the indirect effect of X on Y? The effect of Gender on Hiring if sex discrimination is eliminated. X Z IGNORE f Y

15. RELATIONS BETWEEN TOTAL, DIRECT, AND INDIRECT EFFECTS Theorem 5: The total, direct and indirect effects obey The following equality In words, the total effect (on Y) associated with the transition from x* to x is equal to the difference between the direct effect associated with this transition and the indirect effect associated with the reverse transition, from x to x*.

16. EXPERIMENTAL IDENTIFICATION OF AVERAGE DIRECT EFFECTS Theorem: If there exists a set W such that Then the average direct effect Is identifiable from experimental data and is given by

17. HOW THE PROOF GOES? Proof: Each factor is identifiable by experimentation.

18. GRAPHICAL CONDITION FOR EXPERIMENTAL IDENTIFICATION OF DIRECT EFFECTS Theorem: If there exists a set W such that then, Example:

19. GRAPHICAL CONDITION FOR NONEXPERIMENTAL IDENTIFICATION OF AVERAGE NATURAL DIRECT EFFECTS • Identification conditions • There exists a W such that (YZ | W)GXZ • There exist additional covariates that render all • counterfactual terms identifiable.

20. IDENTIFICATION IN MARKOVIAN MODELS Corollary 3: The average direct effect in Markovian models is identifiable from nonexperimental data, and it is given by where S stands for any sufficient set of covariates. Example: S =  X Z Y

21. x* z* = Zx*(u) GENERAL PATH-SPECIFIC EFFECTS (Def.) X X W Z W Z Y Y Form a new model, , specific to active subgraph g Definition: g-specific effect Nonidentifiable even in Markovian models

22. SUMMARY OF RESULTS • Formal semantics of path-specific effects, based on signal blocking, instead of value fixing. • Path-analytic techniques extended to nonlinear and nonparametric models. • Meaningful (graphical) conditions for estimating direct and indirect effects from experimental and nonexperimental data. • Estimability conditions hold in Markovian models. • Graphical techniques of inferring effects of policies involving signal blocking.