CAUSAL INFERENCE IN STATISTICS

1. CAUSAL INFERENCE IN STATISTICS Judea Pearl University of California Los Angeles (www.cs.ucla.edu/~judea/jsm09)

2. OUTLINE • Inference: Statistical vs. Causal, • distinctions, and mental barriers • Unified conceptualization of counterfactuals, structural-equations, and graphs • Inference to three types of claims: • Effect of potential interventions • Attribution (Causes of Effects) • Direct and indirect effects • Frills

3. 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)

4. 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.

5. 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 e.g. Curing symptoms vs. curing diseases e.g. Analogy: mechanical deformation

6. Causal and statistical concepts do not mix. CAUSAL Spurious correlation Randomization / Intervention Confounding / Effect Instrumental variable Strong Exogeneity Explanatory variables 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)

7. Causal and statistical concepts do not mix. CAUSAL Spurious correlation Randomization / Intervention Confounding / Effect Instrumental variable Strong Exogeneity Explanatory variables 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.

8. Causal and statistical concepts do not mix. CAUSAL Spurious correlation Randomization / Intervention Confounding / Effect Instrumental variable Strong Exogeneity Explanatory variables 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.

9. Correct notation: Y := 2X X = 1 Y = 2 The solution WHY CAUSALITY NEEDS SPECIAL MATHEMATICS Scientific Equations (e.g., Hooke’s Law) are non-algebraic e.g., Length (Y) equals a constant (2) times the weight (X) Y = 2X X = 1 Process information Had X been 3, Y would be 6. If we raise X to 3, Y would be 6. Must “wipe out” X = 1.

10. (or) Y 2X X = 1 Y = 2 The solution WHY CAUSALITY NEEDS SPECIAL MATHEMATICS Scientific Equations (e.g., Hooke’s Law) are non-algebraic e.g., Length (Y) equals a constant (2) times the weight (X) Correct notation: X = 1 Process information Had X been 3, Y would be 6. If we raise X to 3, Y would be 6. Must “wipe out” X = 1.

11. 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!”

12. FAMILIAR CAUSAL MODEL ORACLE FOR MANIPILATION X Y Z INPUT OUTPUT

13. e.g., STRUCTURAL CAUSAL MODELS • Definition: A structural causal model is a 4-tuple • V,U, F, P(u), where • V = {V1,...,Vn} are endogeneas 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

14. I W Q P STRUCTURAL MODELS AND CAUSAL DIAGRAMS The functions vi = fi(v,u) define a graph vi = fi(pai,ui)PAi V \ ViUi U Example: Price – Quantity equations in economics U1 U2 PAQ

15. I W Q STRUCTURAL MODELS AND INTERVENTION Let X be a set of variables in V. The action do(x) sets X to constants x regardless of the factors which previously determined X. do(x) replaces all functions fi determining X with the constant functions X=x, to create a mutilated modelMx U1 U2 P

16. I W Q STRUCTURAL MODELS AND INTERVENTION Let X be a set of variables in V. The action do(x) sets X to constants x regardless of the factors which previously determined X. do(x) replaces all functions fi determining X with the constant functions X=x, to create a mutilated modelMx Mp U1 U2 P P = p0

17. The Fundamental Equation of Counterfactuals: 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.

18. 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.

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

20. THE FOUR NECESSARY STEPS OF CAUSAL ANALYSIS 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.

21. THE FOUR NECESSARY STEPS FOR EFFECT 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.

22. THE FOUR NECESSARY STEPS FOR EFFECT 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.

23. THE FOUR NECESSARY STEPS FOR POLICY ANALYSIS 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.

24. THE FOUR NECESSARY STEPS FOR POLICY ANALYSIS 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.

25. INFERRING THE EFFECT OF INTERVENTIONS • The problem:To predict the impact of a proposed intervention using data obtained prior to the intervention. • The solution (conditional): Causal Assumptions + Data   Policy Claims • Mathematical tools for communicating causal assumptions formally and transparently. • Deciding (mathematically) whether the assumptions communicated are sufficient for obtaining consistent estimates of the prediction required. • Deriving (if (2) is affirmative)a closed-form expression for the predicted impact • Suggesting (if (2)  is negative)a set of measurements and experiments that, if performed, would render a consistent estimate feasible.

26. THE FOUR NECESSARY STEPS FROM DEFINITION TO ASSUMPTIONS 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.

27. 2. Counterfactuals: 4. 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?

28. IDENTIFIABILITY Definition: Let Q(M) be any quantity defined on a causal model M, andlet A be a set of assumption. Q is identifiable relative to A iff P(M1) = P(M2) ÞQ(M1) = Q(M2) for all M1, M2, that satisfy A.

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

30. G THE PROBLEM OF CONFOUNDING 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?

31. G Gx Moreover, P(y | do(x)) =åP(y | x,z) P(z) (“adjusting” for Z) z ELIMINATING CONFOUNDING BIAS 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

32. EFFECT OF INTERVENTION BEYOND ADJUSTMENT Theorem (Tian-Pearl 2002) We can identifyP(y|do(x)) if there is no child Z of X connected to X by a confounding path. G Z1 Z2 Z3 Z4 Z5 X Y Z6

33. No, no! EFFECT OF WARM-UP ON INJURY (After Shrier & Platt, 2008)

34. EFFECT OF INTERVENTION COMPLETE IDENTIFICATION • Complete calculus for reducing P(y|do(x), z) to expressions void of do-operators. • Complete graphical criterion for identifying causal effects (Shpitser and Pearl, 2006). • Complete graphical criterion for empirical testability of counterfactuals (Shpitser and Pearl, 2007).

35. COUNTERFACTUALS AT WORK 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?

36. THE FOUR NECESSARY STEPS 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.

37. 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

38. G Gx Moreover, ETT “Standardized morbidity” ETT - THE BACK-DOOR CRITERION is identifiable in G if there is a set Z of variables such that Zd-separates X from Y in Gx. Z1 Z1 Z2 Z2 Z Z3 Z3 Z4 Z5 Z5 Z4 X X Z6 Y Y Z6

39. 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.

40. 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

41. WHAT PROPENSITY SCORE (PS) PRACTITIONERS NEED TO KNOW • The assymptotic bias of PS is EQUAL to that of ordinary adjustment (for same Z). • Including an additional covariate in the analysis CAN SPOIL the bias-reduction potential of others. • Choosing sufficient set for PS, requires knowledge about the model.

42. WHICH COVARIATES MAY / SHOULD BE ADJUSTED FOR? B1 Assignment Hygiene Age Treatment Outcome M B2 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:   AgeB1, M, B2, Follow-up, Assignment without Age Cost, Hygiene, {Assignment + Age},   {Hygiene + Age + B1} , more . . .

43. WHICH COVARIATES MAY / SHOULD BE ADJUSTED FOR? B1 Assignment Hygiene Age Treatment Outcome M B2 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:   AgeB1, M, B2, Follow-up, Assignment without Age Cost, Hygiene, {Assignment + Age},   {Hygiene + Age + B1} , more . . .

44. WHAT PROPENSITY SCORE (PS) PRACTITIONERS NEED TO KNOW • The assymptotic bias of PS is EQUAL to that of ordinary adjustment (for same Z). • Including an additional covariate in the analysis CAN SPOIL the bias-reduction potential of others. • Choosing sufficient set for PS, requires knowledge about the model. • 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.

45. 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

46. inconsistency: x = 0 Yx=0 = Y Y = xY1 + (1-x) Y0 “SUPER” DISTRIBUTION IN N-R MODEL X 0 0 0 1 Y 0 1 0 0 Z 0 1 0 0 Yx=0 0 1 1 1 Yx=1 1 0 0 0 Xz=0 0 1 0 0 Xz=1 0 0 1 1 Xy=0 0 1 1 0 U u1 u2 u3 u4

47. THE FOUR NECESSARY STEPS IN POTENTIAL-OUTCOME FRAMEWORK Define: Assume: Identify: Estimate: Express the target quantity Q as a counterfactual formula Formulate causal assumptions using the distribution: Determine if Q is identifiable. Estimate Q if it is identifiable; approximate it, if it is not.

48. EFFECT OF WARM-UP ON INJURY IN POTENTIAL-OUTCOME FRAMEWORK Z: Team motivation Coach Pre-game proprioception Fitness level Genetics Neuromuscular fatigue Connective tissue disorder Tissue weakness X Y Warm-up Injury

49. Yx X | Zjudgmental &  Try it:X  Y  Z ? Yx X | Zjudgmental &  opaque X, Y and Yx? 2) TYPICAL INFERENCE IN N-R MODEL Find P*(Yx=y) given covariate Z, Assume ignorability: Yx X | Z  Assume consistency: X=x  Yx=Y Problems: 1) opaque Is consistency the only connection between Is consistency the only connection between

50. 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. Every theorem in SCM is a theorem in Potential-Outcome Model, and conversely.