Otis Dudley Duncan Memorial Lecture THE RESURRECTION OF DUNCANISM

1. Otis Dudley Duncan Memorial Lecture THE RESURRECTION OF DUNCANISM Judea Pearl University of California Los Angeles (www.cs.ucla.edu/~judea/)

2. OUTLINE • Duncanism = Causally Assertive SEM • History: Oppression, Distortion, and Resurrection • The Old-New Logic of SEM • New Tools • 4.1 Local testing • 4.2 Non-parametric identification • 4.3 Logic of counterfactuals • 4.3 Non-linear mediation analysis

3. DUNCAN BOOK

4. DUNCANISM = ASSERTIVE SEM SEM – A tool for deriving causal conclusion from data and assumptions: • [y = bx + u] may be read as "a change in x or u produces a change in y” ... or “x and u are the causes of y” (Duncan, 1975, p. 1) • “[the disturbance]u stands for all other sources of variation in y” (ibid) • “doing the model consists largely in thinking about what kind of model one wants and can justify.” • (ibid, p. viii) • Assuming a model for sake of argument, we can express its properties in terms of correlations and (sometimes) find one or more conditions that must hold if the model is true“ (ibid, p. 20).

5. HISTORY: BIRTH, OPPRESSION, DISTORTION, RESURRECTION • Birth and Development (1920 - 1980) • Sewell Wright (1921), Haavelmo (1943), Simon (1950), • Marschak (1950), Koopmans (1953), Wold and Strotz (1963), • Goldberger (1973), Blalock (1964), and Duncan (1969, 1975) • The regressional assault (1970-1990) • Causality is meaningless, therefore, to be meaningful, SEM • must be a regression technique, not “causal modeling.” • Richard (1980) Cliff (1983), Dempster (1990), Wermuth (1992), • and Muthen (1987)     • The Potential-outcome assault (1985-present) • Causality is meaningful but, since SEM is a regression • technique, it could not possibly have causal interpretation. • Rubin (2004, 2010), Holland (1986) Imbens (2009), and • Sobel (1996, 2008)

6.   The mothers of all questions: Q. When would b equal a? A. When (uYX), read from the diagram REGRESSION VS. STRUCTURAL EQUATIONS (THE CONFUSION OF THE CENTURY) Regression (claimless, nonfalsifiable): Y = ax + Y Structural (empirical, falsifiable): Y = bx + uY ( := assignment) Claim: (regardless of distributions): E(Y | do(x)) = E(Y | do(x), do(z)) = bx Q. When is b a partial regression? b = YX • Z A. Shown in the diagram, Slide 36.

7. THE POTENTIAL-OUTCOME ASSAULT (1985-PRESENT) SEM = regressional strawman • “I am speaking, of course, about the equation: • What does it mean? The only meaning I have ever • determined for such an equation is that it is a shorthand way • of describing the conditional distribution of {y}given {x}.” • (Holland 1995) • “The use of complicated causal-modeling software • [read SEM] rarely yields any results that have any • interpretation as causal effects.” • (Wilkinson and Task Force 1999 on Statistical Methods in • Psychology Journals: Guidelines and Explanations)

8. THE POTENTIAL-OUTCOME ASSAULT (1985-PRESENT) (Cont) • In general (even in randomized studies), the structural • and causal parameters are not equal, implying that the • structural parameters should not be interpreted as effect.” • (Sobel 2008) • “Using the observed outcome notation entangles the • science...Bad! Yet this is exactly what regression • approaches, path analyses, directed acyclic graphs, • and so forth essentially compel one to do.” • (Rubin 1010)

9. SEM REACTION TO THE STRUCTURE-PHOBIC   ASSAULT NONE TO SPEAK OF! Galles, Pearl, Halpern (1998 - Logical equivalence) Heckman-Sobel Morgan Winship (1997) Gelman Blog

10. WHY SEM INTERPRETATION IS “SELF-CONTRADICTORY” D. Freedman JES (1987), p. 114, Fig. 3 "Now try the direct effect of Z on Y:  We intervene by fixing W and X but increasing Z by one unit; this should increase Y by d units.   However, this hypothetical intervention is self-contradictory, because fixing W and increasing Z causes an increase in X.   The oversight: Fixing XDISABLES equation (7.1);

11. SEM REACTION TO FREEDMAN CRITICS Total Surrender: • It would be very healthy if more researchers abandoned • thinking of and using terms such as cause and effect • (Muthen, 1987). • “Causal modeling” is an outdated misnomer (Kelloway, 1998). • Causality-free, politically-safe vocabulary: “covariance • structure” “regression analysis,” or “simultaneous equations.” • [Causal Modeling] may be somewhat dated, however, as it • seems to appear less often in the literature nowadays” • (Kline, 2004, p. 9)

12. THE RESURRECTION Why non-parametric perspective? What are the parameters all about and why we labor to identify them?Can we do without them?Consider: Only he who lost the parameters and needs to find substitutes can begin to ask:Do I really need them? What do they really mean? What role do the play?

13. THE LOGIC OF SEM CAUSAL MODEL (MA) CAUSAL MODEL (MA) A - CAUSAL ASSUMPTIONS A* - Logical implications of A Causal inference Q Queries of interest Q(P) - Identified estimands T(MA) - Testable implications Statistical inference Data (D) Q - Estimates of Q(P) Goodness of fit Conditional claims Model testing

14. P Joint Distribution Q(P) (Aspects of P) Data Inference TRADITIONAL STATISTICAL INFERENCE PARADIGM e.g., Infer whether customers who bought product A would also buy product B. Q = P(B | A)

15. FROM STATISTICAL TO CAUSAL ANALYSIS: 1. THE DIFFERENCES Probability and statistics deal with static relations P Joint Distribution P Joint Distribution Q(P) (Aspects of P) Data change Inference What happens when P changes? e.g., Infer whether customers who bought product A would still buy Aif we were to double the price.

16. FROM STATISTICAL TO CAUSAL ANALYSIS: 1. THE DIFFERENCES What remains invariant when Pchanges say, to satisfy P(price=2)=1 P Joint Distribution P Joint Distribution Q(P) (Aspects of P) Data change Inference Note: P(v)P (v | price = 2) P does not tell us how it ought to change Causal knowledge: what remains invariant

17. Causal and statistical concepts do not mix. CAUSAL Spurious correlation Randomization / Intervention Confounding / Effect Instrumental variable Exogeneity / Ignorability Mediation STATISTICAL Regression Association / Independence “Controlling for” / Conditioning Odd and risk ratios Collapsibility / Granger causality Propensity score FROM STATISTICAL TO CAUSAL ANALYSIS: 1. THE DIFFERENCES (CONT)

18. Causal and statistical concepts do not mix. CAUSAL Spurious correlation Randomization / Intervention Confounding / Effect Instrumental variable Exogeneity / Ignorability Mediation STATISTICAL Regression Association / Independence “Controlling for” / Conditioning Odd and risk ratios Collapsibility / Granger causality Propensity score • No causes in – no causes out (Cartwright, 1989) } statistical assumptions + data causal assumptions  causal conclusions FROM STATISTICAL TO CAUSAL ANALYSIS: 2. MENTAL BARRIERS • Causal assumptions cannot be expressed in the mathematical language of standard statistics.

19. Causal and statistical concepts do not mix. CAUSAL Spurious correlation Randomization / Intervention Confounding / Effect Instrumental variable Exogeneity / Ignorability Mediation STATISTICAL Regression Association / Independence “Controlling for” / Conditioning Odd and risk ratios Collapsibility / Granger causality Propensity score • No causes in – no causes out (Cartwright, 1989) } statistical assumptions + data causal assumptions  causal conclusions • Non-standard mathematics: • Structural equation models (Wright, 1920; Simon, 1960) • Counterfactuals (Neyman-Rubin (Yx), Lewis (xY)) FROM STATISTICAL TO CAUSAL ANALYSIS: 2. MENTAL BARRIERS • Causal assumptions cannot be expressed in the mathematical language of standard statistics.

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

21. e.g., STRUCTURAL CAUSAL MODELS • Definition: A structural causal model is a 4-tuple • V,U, F, P(u), where • V = {V1,...,Vn} are endogenous variables • U={U1,...,Um} are background variables • F = {f1,...,fn} are functions determining V, • vi = fi(v, u) • P(u) is a distribution over U • P(u) and F induce a distribution P(v) over observable variables

22. U U X (u) Y (u) X = x YX (u) M Mx CAUSAL MODELS AND COUNTERFACTUALS Definition: The sentence: “Y would be y (in unit u), had X beenx,” 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.

23. The Fundamental Equation of Counterfactuals: CAUSAL MODELS AND COUNTERFACTUALS Definition: The sentence: “Y would be y (in unit u), had X beenx,” 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.

24. Joint probabilities of counterfactuals: In particular: CAUSAL MODELS AND COUNTERFACTUALS Definition: The sentence: “Y would be y (in situation u), had X beenx,” 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.

25. 3-LEVEL HIERARCHY OF CAUSAL MODELS • Probabilistic Knowledge P(y | x) • Bayesian networks, covariance structure • Interventional Knowledge P(y | do(x)) • Causal Bayesian Networks (CBN) • (Agnostic graphs, manipulation graphs) • Counterfactual Knowledge P(Yx = y,Yx=y) • Structural equation models, physics, • functional graphs

26. 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 model Counterfactuals are primitives, new variables Super-distribution Structural model Counterfactuals are derived quantities Subscripts modify the model and distribution

27. ARE THE TWO PARADIGMS EQUIVALENT? • Yes (Galles and Pearl, 1998; Halpern 1998) • In the N-R paradigm, Yx is defined by consistency: • In SEM, consistency is a theorem. • Moreover, a theorem in one approach is a theorem in the other.

28. THE FIVE NECESSARY STEPS OF CAUSAL ANALYSIS Define: Assume: Identify: Estimate: Test: Express the target quantity Q as a function Q(M) that can be computed from any model M. Formulate causal assumptions Ausing 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).

29. THE FIVE NECESSARY STEPS FOR EFFECT ESTIMATION Define: Assume: Identify: Estimate: Test: Express the target quantity Q as a function Q(M) that can be computed from any model M. Formulate causal assumptions Ausing 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).

30. 2. Counterfactuals: 3. Structural: X Z Y FORMULATING ASSUMPTIONS THREE LANGUAGES 1. English: Smoking (X), Cancer (Y), Tar (Z), Genotypes (U) Not too friendly: consistent? complete? redundant? arguable?

31. IDENTIFYING CAUSAL EFFECTS IN POTENTIAL-OUTCOME FRAMEWORK Define: Assume: Identify: Estimate: Express the target quantity Q as a counterfactual formula, e.g., E(Y(1) – Y(0)) Formulate causal assumptions using the distribution: Determine if Q is identifiable using P* and Y=x Y (1) + (1 – x) Y (0). Estimate Q if it is identifiable; approximate it, if it is not.

32. GRAPHICAL – COUNTERFACTUALS SYMBIOSIS Every causal graph expresses counterfactuals assumptions, e.g., X  Y Z • Missing arrows Y Z 2. Missing arcs Y Z • consistent, and readable from the graph. • Express assumption in graphs • Derive estimands by graphical or algebraic methods

33. NON-PARAMETRIC IDENTIFIABILITY Definition: Let Q(M) be any quantity defined on a causal model M, andlet Abe a set of assumption. Qis identifiable relative to A iff P(M1) = P(M2) ÞQ(M1) = Q(M2) for all M1, M2, that satisfy A. In other words, Qcan be determined uniquely from the probability distribution P(v) of the endogenous variables, V, and assumptions A. A is displayed in graph G.

34. G IDENTIFICATION IN SEM Find the effect ofXonY, P(y|do(x)), given the causal assumptions shown inG, whereZ1,..., Zk are auxiliary variables. Z1 Z2 Z3 Z4 Z5 X Z6 Y CanP(y|do(x)) be estimated if only a subset,Z, can be measured?

35. G Gx Moreover, (“adjusting” for Z) Ignorability NON-PARAMETRIC IDENTIFICATION USING THE BACK-DOOR CRITERION P(y | do(x)) is estimable if there is a set Z of variables such thatZd-separates X from YinGx. Z1 Z1 Z2 Z2 Z Z3 Z3 Z4 Z5 Z5 Z4 X X Z6 Y Y Z6

36. S THE MOTHER OF ALL QUESTIONS Question: Under what conditions can a path coefficient  be estimated as a regression coefficient, and what variables should serve as the regressors? Answer: It is all in the diagram (Pearl 2009, p. 180)  X Y The single door criterion • No member of S is a descendant of Y, and • S blocks all paths from X to Y, except the direct path X Y.

37. SEARCHING FOR THE TESTABLE

38. FINDING THE TESTABLES BY INSPECTION Z1 Z2 W1 W2 Z3 X W3 Y Missing edges imply conditional independencies and vanishing regression coefficients. • Software for automatic detection of all tests (Kyono, 2010)

39. FINDING INSTRUMENTAL VARIABLES Find an instrument for identifying b34 (Duncan, 1975) Find a variable Z that, conditioned on S, is independent of the disturbance v. By inspection: Xis d-separated from V Therefore, X1 is a valid instrument

40. FIND EQUIVALENT MODELS • Model-equivalence implies d-separation-equivalence • d-separation tests should replace Lee & Herschberger rules Z X W Z X W   Y Y (a) (b) Reason: conditioned on Y, Path Z  Y  X  W is open in (a) and blocked in (b)

41. L T Z ? Z T CONFOUNDING EQUIVALENCE WHEN TWO MEASUREMENTS ARE EQUALLY VALUABLE W1 W4 W2 W3 V1 V2 X Y

42. 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. 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.

43. L Theorem: Adjustment for L replaces Adjustment for Z PROPENSITY SCORE ESTIMATOR (Rosenbaum & Rubin, 1983) Z1 Z2 P(y | do(x)) = ? Z4 Z3 Z5 X Z6 Y

44. WHAT PROPENSITY SCORE (PS) PRACTITIONERS NEED TO KNOW • The asymptotic bias of PS is EQUAL to that of ordinary adjustment (for same Z). • Including an additional covariate in the analysis CANSPOIL the bias-reduction potential of others. • In particular, instrumental variables tend to amplify bias. • Choosing sufficient set for PS, requires knowledge of the model.

45. RETROSPECTIVE COUNTERFACTUALS ETT – EFFECT OF TREATMENT ON THE TREATED • Regret: • I took a pill to fall asleep. • Perhaps I should not have? • Program evaluation: • What would terminating a program do to • those enrolled?

46. COUNTERFACTUAL DEFINITION EFFECT OF TREATMENT ON THE TREATED Define: Assume: Identify: Estimate: Express the target quantity Q as a function Q(M) that can be computed from any model M. 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.

47. W ETT - IDENTIFICATION Theorem (Shpitser-Pearl, 2009) ETT is identifiable in G iff P(y | do(x),w) is identifiable in G X Y Moreover, Complete graphical criterion

48. EFFECT DECOMPOSITION (direct vs. indirect effects) • Why decompose effects? • What is the definition of direct and indirect effects? • What are the policy implications of direct and indirect effects? • When can direct and indirect effect be estimated consistently from experimental and nonexperimental data?

49. WHY DECOMPOSE EFFECTS? • To understand how Nature works • To comply with legal requirements • To predict the effects of new type of interventions: • Signal routing, rather than variable fixing

50. Adjust for Z? No! No! • LEGAL IMPLICATIONS • OF DIRECT EFFECT Can data prove an employer guilty of hiring discrimination? X Z (Gender) (Qualifications) Y (Hiring) What is the direct effect of X on Y ? (averaged over z)