Discovering Pure Measurement Models in Causal Systems: A Case Study Using fMRI Data

1. Topic Outline Motivation Representing/Modeling Causal Systems Estimation and Updating Model Search Linear Latent Variable Models Case Study: fMRI

2. Discovering Pure Measurement Models Richard ScheinesCarnegie Mellon University Ricardo Silva*University College London Clark Glymour and Peter SpirtesCarnegie Mellon University

3. Outline • Measurement Models & Causal Inference • Strategies for Finding a Pure Measurement Model • Purify • MIMbuild • Build Pure Clusters • Examples • Religious Coping • Test Anxiety

4. Goals: • What Latents are out there? • Causal Relationships Among Latent Constructs Relationship Satisfaction Depression or Relationship Satisfaction Depression or ?

5. Needed: Ability to detect conditional independence among latent variables

6. Lead and IQ e2 e3 Parental Resources Lead Exposure IQ Lead _||_ IQ | PR e2 ~ N(m=0, s = 1.635) Lead = 15 -.5*PR + e2 PR ~ N(m=10, s = 3) e3 ~ N(m=0, s = 15) IQ = 90 + 1*PR + e3

7. Psuedorandom sample: N = 2,000 Parental Resources Lead Exposure IQ Regression of IQ on Lead, PR

8. Measuring the Confounder e1 e3 e2 X1 X2 X3 Parental Resources Lead Exposure IQ X1 = g1* Parental Resources + e1 X2 = g2* Parental Resources + e2 X3 = g3* Parental Resources + e3 PR_Scale = (X1 + X2 + X3) / 3

9. Scales don't preserve conditional independence X1 X2 X3 Parental Resources Lead Exposure IQ PR_Scale = (X1 + X2 + X3) / 3

10. Indicators Don’t Preserve Conditional Independence X1 X2 X3 Parental Resources Lead Exposure IQ Regress IQ on: Lead, X1, X2, X3

11. Structural Equation Models Work X1 X2 X3 Parental Resources Lead Exposure IQ b • Structural Equation Model • (p-value = .499) • Lead and IQ “screened off” by PR

12. Local Independence / Pure Measurement Models • For every measured item xi: • xi _||_ xj | latent parent of xi

13. Local Independence Desirable

14. Correct Specification Crucial

15. Strategies • Find a Locally Independent Measurement Model • Correctly specify the MM, including deviations from Local Independence

16. Correctly Specify Deviations from Local Independence

17. Correctly Specifying Deviations from Local Independence is Often Very Hard

18. Finding Pure Measurement Models - Much Easier

19. tetrad constraints CovWXCovYZ =(122L)(342L) ==(132L) (242L)= CovWYCovXZ WXYZ = WYXZ = WZXY Tetrad Constraints • Fact: given a graph with this structure • it follows that L W = 1L + 1 X = 2L + 2 Y = 3L + 3 Z = 4L + 4 1 4 2 3 W X Y Z

20. Early Progenitors Charles Spearman (1904) StatisticalConstraints Measurement Model Structure g m1 m2 r1 r2 rm1 * rr1 = rm2 * rr2

21. Impurities/Deviations from Local Independence defeat tetrad constraints selectively rx1,x2 * rx3,x4 = rx1,x3 * rx2,x4 rx1,x2 * rx3,x4 = rx1,x4 * rx2,x3 rx1,x3 * rx2,x4 = rx1,x4 * rx2,x3 rx1,x2 * rx3,x4 = rx1,x3 * rx2,x4 rx1,x2 * rx3,x4 = rx1,x4 * rx2,x3 rx1,x3 * rx2,x4 = rx1,x4 * rx2,x3

22. Purify True Model Initially Specified Measurement Model

23. Purify Iteratively remove item whose removal most improves measurement model fit (tetrads or c2) – stop when confirmatory fit is acceptable Remove x4 Remove z2

24. Purify Detectibly Pure Subset of Items Detectibly Pure Measurement Model

25. Purify

26. How a pure measurement model is useful Consistently estimate covariances/correlations among latents- test conditional independence with estimatedlatent correlations Test for conditional independence among latents directly

27. 2. Test conditional independence relations among latents directly Question: L1 _||_ L2 | {Q1, Q2, ..., Qn} b21 b21= 0  L1 _||_ L2 | {Q1, Q2, ..., Qn}

28. MIMbuild Input: - Purified Measurement Model - Covariance matrix over set of pure items MIMbuild PC algorithm with independence tests performed directly on latent variables Output: Equivalence class of structural models over the latent variables

29. Purify &MIMbuild

30. Goal 2: What Latents are out there? • How should they be measured?

31. Latents and the clustering of items they measure imply tetrad constraints diffentially

32. Build Pure Clusters (BPC) Input: - Covariance matrix over set of original items BPC 1) Cluster (complicated boolean combinations of tetrads) 2) Purify Output: Equivalence class of measurement models over a pure subset of original Items

33. Build Pure Clusters

34. Build Pure Clusters • Qualitative Assumptions • Two types of nodes: measured (M) and latent (L) • M L (measured don’t cause latents) • Each m  M measures (is a direct effect of) at least one l  L • No cycles involving M • Quantitative Assumptions: • Each m  M is a linear function of its parents plus noise • P(L) has second moments, positive variances, and no deterministic relations

35. Build Pure Clusters Output - provably reliable (pointwise consistent): Equivalence class of measurement models over a pure subset of M For example: TrueModel Output

36. Build Pure Clusters Measurement models in the equivalence class are at most refinements, but never coarsenings or permuted clusterings. Output

37. Build Pure Clusters • Algorithm Sketch: • Use particular rank (tetrad) constraints on the measured correlations to find pairs of items mj, mk that do NOT share a single latent parent • Add a latent for each subset S of M such that no pair in S was found NOT to share a latent parent in step 1. • Purify • Remove latents with no children

38. Build Pure Clusters + MIMbuild

39. Case Studies Stress, Depression, and Religion (Lee, 2004) Test Anxiety (Bartholomew, 2002)

40. Specified Model Case Study: Stress, Depression, and Religion • Masters Students (N = 127) 61 - item survey (Likert Scale) • Stress: St1 - St21 • Depression: D1 - D20 • Religious Coping: C1 - C20 p = 0.00

41. Case Study: Stress, Depression, and Religion Build Pure Clusters

42. Case Study: Stress, Depression, and Religion • Assume Stress temporally prior: • MIMbuild to find Latent Structure: p = 0.28

43. Case Study : Test Anxiety Bartholomew and Knott (1999), Latent variable models and factor analysis 12th Grade Males in British Columbia (N = 335) 20 - item survey (Likert Scale items): X1 - X20: Exploratory Factor Analysis:

44. Case Study : Test Anxiety Build Pure Clusters:

45. Case Study : Test Anxiety Build Pure Clusters: Exploratory Factor Analysis: p-value = 0.00 p-value = 0.47

46. MIMbuild Scales: No Independencies or Conditional Independencies p = .43 Uninformative Case Study : Test Anxiety

47. Limitations • In simulation studies, requires large sample sizes to be really reliable (~ 400-500). • 2 pure indicators must exist for a latent to be discovered and included • Moderately computationally intensive (O(n6)). • No error probabilities.

48. Open Questions/Projects • IRT models? • Bi-factor model extensions? • Appropriate incorporation of background knowledge

49. References • Tetrad: www.phil.cmu.edu/projects/tetrad_download • Spirtes, P., Glymour, C., Scheines, R. (2000). Causation, Prediction, and Search, 2nd Edition, MIT Press. • Pearl, J. (2000). Causation: Models of Reasoning and Inference, Cambridge University Press. • Silva, R., Glymour, C., Scheines, R. and Spirtes, P. (2006) “Learning the Structure of Latent Linear Structure Models,” Journal of Machine Learning Research, 7, 191-246. • Learning Measurement Models for Unobserved Variables, (2003). Silva, R., Scheines, R., Glymour, C., and Spirtes. P., in Proceedings of the Nineteenth Conference on Uncertainty in Artificial Intelligence , U. Kjaerulff and C. Meek, eds., Morgan Kauffman