Richard Scheines Carnegie Mellon University

1. Searching for StatisticalCausal Models: Theory and Practice Richard ScheinesCarnegie Mellon University

2. Goals • Policy, Law, and Science: How can we use data to answer • subjunctive questions (effects of future policy interventions), or • counterfactual questions (what would have happened had things been done differently (law)? • scientific questions (what mechanisms run the world) • Rumsfeld Problem: Do we know what we don’t know: Can we tell when there is or is not enough information in the data to answer causal questions?

3. Early Progenitors Charles Spearman (1904) StatisticalConstraints CausalStructure g m1 m2 r1 r2 rm1 * rr1 = rm2 * rr2

4. Early Progenitors Sewall Wright (1920s,1930s) Graphical Model  Causal & Statistical Interpretation The Method of Path Coefficients (1934). Annals of Mathematical Statistics, 5, 161-215.

5. Social Sciences: 1940s  1970s • Factor Analysis • Instrumental VariableEstimators • Structural Equation Models • Simultaneous Equation Models • Economics: • Cowles Commission • Franklin Fisher • Art Goldberger • Clive Granger • Herb Simon • Haavelmo • R. Strotz • H. Wold • Sociology, Psychometrics, etc. • Hubert Blalock • Herb Costner • Otis Dudley Duncan • David Heise • David Kenny • Ken Bollen

6. Population level Counterfactuals

7. 1970s & 1980s:Graphical Models & Independence Structures • S. Lauritzen • J. Darroch • T. Speed • H. Kiiveri • N. Wermuth • D. Hausman • D. Papineau • P. Dawid • D. Cox • J. Robins • J. Whittaker • Judea Pearl 1988

8. 1988  1993:Axioms, Intervention, and Latent Variable Model Search • P. Spirtes, C. Glymour, and R. Scheines • Causal Markov Axiom • Full model of interventions, both surgical and non-surgical • Equivalence classes for latent variable models, with search

9. Modern Non-Parametric Theory of Statistical Causal Models Intervention & Manipulation Graphical Models Counterfactuals Constraints (Independence) CausalBayes Nets

10. Semantics of SCMs • Choice 1: Take direct causation as primitive, axiomatizeCausal systems over V  Probabilistic Independence Relations in P(V) • Choice 2: Define direct causation in terms of intervention, i.e., (hypothetical) treatment)

11. Choice 1: Causal Markov Axiom If G is a causal graph, and P a probability distribution over the variables in G, then in <G,P> satisfy the Markov Axiom iff: every variable V is independent of its non-effects, conditional on its immediate causes.

12. Causal Graphs Causal Graph G = {V,E} Each edge X  Y represents a direct causal claim: X is a direct cause of Y relative to V Chicken Pox

13. Causal Graphs Not Cause Complete Common Cause Complete

14. Pre-intervention graph “Soft” Intervention “Hard” Intervention Interventions & Causal Graphs Model an intervention by adding an “intervention” variable outside the original system as a direct cause of its target. Intervene on Income

15. Structural Equation Models 1. Structural Equations 2. Statistical Constraints Causal Graph Statistical CausalModel

16. Structural Equation Models • Structural Equations: One Assignment Equation for each variable V V := f(parents(V), errorV) for SEM (linear regression) f is a linear function • Statistical Constraints: Joint Distribution over the Error terms Causal Graph

17. Causal Graph SEM Graph (path diagram) Structural Equation Models Equations: Education := ed Income :=Educationincome Longevity :=EducationLongevity Statistical Constraints: (ed, Income,Income ) ~N(0,2) 2diagonal - no variance is zero

18. Two Routes to the Causal Markov Condition • Assumption 1: Weak Causal Markov Assumption V1,V2causally disconnectedV1 _||_ V2 • Assumption 2b: Determinism, e.g., Structural Equations • For each ViV, Vi := f(parents(Vi)) • Assumption 2a: • Causal Markov Axiom

19. Choice 2: Define Direct Causation from Intervention X is a direct cause of Y relative to S, iff z,x1  x2 P(Y | X set= x1 , Zset=z)  P(Y | X set= x2 , Zset=z) where Z = S - {X,Y} X is a cause of Y iff x1  x2 P(Y | X set= x1)  P(Y | X set= x2)

20. Modularity of Intervention/Manipulation Causal Graph Structural Equations: Education := ed Longevity :=f1(Education)Longevity Income := f2(Education)income Manipulated Structural Equations: Education := ed Longevity :=f1(Education)Longevity Income := f3(M1) Manipulated Causal Graph M1

21. Manipulation --> Causal Markov • Manipulation conception of causation and Modularity --> weak version of CMA • Zhang, Jiji and Spirtes, Peter (2007) Detection of Unfaithfulness and Robust Causal Inference. In [2007] LSE-Pitt Conference: Confirmation, Induction and Science (London, 8 - 10 March, 2007)http://philsci-archive.pitt.edu/archive/00003188/01/Detection_of_Unfaithfulness_and_Robust_Causal_Inference.pdf

22. Statistical Inference Background Knowledge - X2 before X3 SCM SearchStatistical DataCausal Structure

23. Faithfulness • Constraints on a probability distribution P generated by a causal structure G hold for all parameterizations of G. Revenues = aRate + cEconomy + eRev. Economy = bRate + eEcon. Faithfulness: a ≠ -bc

24. Equivalence Classes • Equivalence: • Independence (M1╞ X _||_ Y | Z  M2╞ X _||_ Y | Z) • Distribution (q1q2M1(q1) = M2(q2)) • Independence (d-separation equivalence) • DAGs : Patterns • PAGs : Latent variable models • Intervention Equivalence Classes • Measurement Model Equivalence Classes • Linear Non-Gaussian Model Equivalence Classes

25. Patterns

26. Patterns: What the Edges Mean

27. PAGs: Partial Ancestral Graphs

28. PAGs: Partial Ancestral Graphs What PAG edges mean.

29. Overview of Search Methods • Constraint Based Searches • TETRAD (SGS, PC, FCI) • Very fast – max N ~ 1,000 • Pointwise Consistent • Scoring Searches • Scores: BIC, AIC, etc. • Search: Hill Climb, Genetic Alg., Simulated Annealing • Difficult to extend to latent variable models • Meek and Chickering Greedy Equivalence Class (GES) • Very slow – max N ~ 30-40

30. Tetrad Demo http://www.phil.cmu.edu/projects/tetrad_download/

31. Does Foreign Investment in 3rd World Countries cause Repression? Case Study 1: Foreign Investment Timberlake, M. and Williams, K. (1984). Dependence, political exclusion, and government repression: Some cross-national evidence. American Sociological Review 49, 141-146. N = 72 PO degree of political exclusivity CV lack of civil liberties EN energy consumption per capita (economic development) FI level of foreign investment

32. Case Study 1: Foreign Investment Correlations po fi en fi -.175 en -.480 0.330 cv 0.868 -.391 -.430

33. Case Study 1: Foreign Investment Regression Results po = .227*fi - .176*en + .880*cv SE (.058) (.059) (.060) t 3.941 -2.99 14.6 Interpretation: increases in foreign investment increases political exclusion

34. Case Study 1: Foreign Investment Alternatives No model with testable constraint (df > 0) in which FI has a positive effect on PO

35. Case Study 2: Welfare Reform Single Mothers’ Self-Efficacy, Parenting in the Home Environment, and   Children’s Development in a Two-Wave Study (Social Work Research, 29, 1, 7-20) Aurora Jackson, Richard Scheines

36. Case Study 2: Welfare Reform Sampling Scheme • Longitudinal Data • Time 1: 1996-97 (N = 188) • Time 2: 1998-99 (N = 178) • Single black mothers in NYC • Current and former welfare recipients • With a child who was 3 – 5 at time 1, and 6 to 8 at time 2

37. Case Study 2: Welfare Reform Constructs/Scales/Measures • Employment Status • Perceived Self-efficacy • Depressive Symptoms • Quality of Mother/Father Relationship • Father/Child Contact • Quality of Home Environment • Behavior Problems • Cognitive Development

38. Case Study 2: Welfare Reform Background Knowledge • Tier 1: • Employment Status • Tier 2: • Depression • Self-efficacy • Mother/Father Relationship • Father/Child Contact • Mother’s Parenting/HOME • Tier 3: • Negative Behaviors • Cognitive Development Over 22 million path models consistent with these constraints

39. Case Study 2: Welfare Reform Conceptual Model c2 = 22.3, df = 20, p = .32 Tetrad Equivalence Class c2 = 18.87, df = 19, p = .46

40. Case Study 2: Welfare Reform Points of Agreement: • Mother’s Self-Efficacy mediates the effect of Employment on all other variables. • HOME environment mediates the effect of all other factors on outcomes: Cog. Develop and Prob. Behaviors Conceptual Model Points of Disagreement: • Depression key cause vs. only an effect Tetrad

41. Case Study 3: Online Courseware Online Course in Causal & Statistical Reasoning

42. Case Study 3: Online Courseware Variables • Pre-test (%) • Print-outs (% modules printed) • Quiz Scores (avg. %) • Voluntary Exercises (% completed) • Final Exam (%) • 9 other variables

43. 2002 2003 Case Study 3: Online Courseware Printing and Voluntary Comprehension Checks: 2002 --> 2003

44. Case Study 4: Charitable Giving Variables • Tangibility/Concreteness (Exp manipulation) • Imaginability (likert 1-7) • Impact (avg. of 2 likerts) • Sympathy (likert) • Donation ($)

45. Case Study 4: Charitable Giving Theoretical Model study 1 (N= 94) df = 5, c2 = 52.0, p= 0.0000 study 2 (N= 115) df = 5, c2 = 62.6, p= 0.0000

46. Case Study 4: Charitable Giving GES Outputs study 1:df = 5, c2 = 5.88, p= 0.32 study 2:df = 5, c2 = 8.23, p= 0.14 study 1:df = 5, c2 = 3.99, p= 0.55 study 2:df = 5, c2 = 7.48, p= 0.18

47. The Causal Theory Formation Problem for Latent Variable Models Given observations on a number of variables, identify the latent variables that underlie these variables and the causalrelations among these latent concepts. Example: Spectral measurements of solar radiation intensities. Variables are intensities at each measured frequency. Example: Quality of a Child’s Home Environment, Cumulative Exposure to Lead, Cognitive Functioning

48. The Most Common Automatic Solution: Exploratory Factor Analysis • Chooses “factors” to account linearly for as much of the variance/covariance of the measured variables as possible. • Great for dimensionality reduction • Factor rotations are arbitrary • Gives no information about the statistical and thus the causal dependencies among any real underlying factors. • No general theory of the reliability of the procedure

49. Other Solutions • Independent Components, etc • Background Theory • Scales

50. Key Causal Question Other Solutions: Background Theory Specified Model Thus, key statistical question: Lead _||_ Cog | Home ?