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

Richard Scheines Carnegie Mellon University Discovering Pure Measurement Models Richard Scheines Carnegie Mellon University Ricardo Silva* University College London Clark Glymour and Peter Spirtes Carnegie Mellon University

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

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

Needed: Ability to detect conditional independence among latent variables

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

Psuedorandom sample: N = 2,000 Parental Resources Lead Exposure IQ Regression of IQ on Lead, PR Independent Variable Coefficient Estimate p-value Screened-off at .05? PR 0.98 0.000 No Lead -0.088 0.378 Yes

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

Scales don't preserve conditional independence Lead Exposure Parental Resources IQ X1 X2 X3 PR_Scale = (X1 + X2 + X3) / 3 Independent Variable Coefficient Estimate p-value Screened-off at .05? PR_scale 0.290 0.000 No Lead -0.423

Indicators Don’t Preserve Conditional Independence Lead Exposure Parental Resources IQ X1 X2 X3 Regress IQ on: Lead, X1, X2, X3 Independent Variable Coefficient Estimate p-value Screened-off at .05? X1 0.22 0.002 No X2 0.45 0.000 X3 0.18 0.013 Lead -0.414

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

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

Local Independence Desirable

Correct Specification Crucial

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

Correctly Specify Deviations from Local Independence

Correctly Specifying Deviations from Local Independence is Often Very Hard

Finding Pure Measurement Models - Much Easier

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 WXYZ = WYXZ = WZXY tetrad constraints CovWXCovYZ = (122L) (342L) = = (132L) (242L) = CovWYCovXZ

Early Progenitors g rm1 * rr1 = rm2 * rr2 Charles Spearman (1904) Statistical Constraints  Measurement Model Structure g m1 m2 r1 r2 rm1 * rr1 = rm2 * rr2 1

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

Purify True Model Initially Specified Measurement Model

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

Purify Detectibly Pure Subset of Items Detectibly Pure Measurement Model

Purify

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

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

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

Purify & MIMbuild

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

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

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

Build Pure Clusters

Build Pure Clusters Qualitative Assumptions Quantitative 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

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

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

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

Build Pure Clusters + MIMbuild

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

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 Specified Model p = 0.00

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

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

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:

Case Study : Test Anxiety Build Pure Clusters:

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

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

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.

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

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