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Topic Outline 1)Motivation 2)Representing/Modeling Causal Systems 3)Estimation and Updating 4)Model Search 5)Linear Latent Variable Models 6)Case Study: fMRI 1

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

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3 Outline 1. Measurement Models & Causal Inference 2. Strategies for Finding a Pure Measurement Model 3. Purify 4.MIMbuild 5. Build Pure Clusters 6. Examples a)Religious Coping b)Test Anxiety

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

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5 Needed: Ability to detect conditional independence among latent variables

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6 Lead and IQ Parental Resources IQ Lead Exposure PR ~ N( =10, = 3 ) Lead = *PR + 2 2 ~ N( =0, = 1.635) IQ = *PR + 3 3 ~ N( =0, = 15) 22 33 Lead _||_ IQ | PR

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7 Psuedorandom sample: N = 2,000 Parental Resources IQ Lead Exposure Independent Variable Coefficient Estimate p-valueScreened-off at.05? PR No Lead Yes Regression of IQ on Lead, PR

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8 Measuring the Confounder Lead Exposure Parental Resources IQ X1X1 X2X2 X3X3 11 22 33 X 1 = 1 * Parental Resources + 1 X 2 = 2 * Parental Resources + 2 X 3 = 3 * Parental Resources + 3 PR_Scale = (X 1 + X 2 + X 3 ) / 3

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9 Scales don't preserve conditional independence Lead Exposure Parental Resources IQ X1X1 X2X2 X3X3 PR_Scale = (X 1 + X 2 + X 3 ) / 3 Independent Variable Coefficient Estimate p-value Screened-off at.05? PR_scale No Lead No

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10 Indicators Don’t Preserve Conditional Independence Lead Exposure Parental Resources IQ X1X1 X2X2 X3X3 Independent Variable Coefficient Estimate p-valueScreened-off at.05? X1X No X2X No X3X No Lead No Regress IQ on: Lead, X 1, X 2, X 3

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11 Structural Equation Models Work Lead Exposure Parental Resources IQ X1X1 X2X2 X3X3 Structural Equation Model p-value =.499) Lead and IQ “screened off” by PR

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12 Local Independence / Pure Measurement Models For every measured item x i : x i _||_ x j | latent parent of x i

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13 Local Independence Desirable

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14 Correct Specification Crucial

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

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16 Correctly Specify Deviations from Local Independence

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17 Correctly Specifying Deviations from Local Independence is Often Very Hard

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18 Finding Pure Measurement Models - Much Easier

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Tetrad Constraints Fact: given a graph with this structure it follows that L XYZW W = 1 L + 1 X = 2 L + 2 Y = 3 L + 3 Z = 4 L + 4 Cov WX Cov YZ = ( 1 2 2 L ) ( 3 4 2 L ) = = ( 1 3 2 L ) ( 2 4 2 L ) = Cov WY Cov XZ WX YZ = WY XZ = WZ XY tetrad constraints

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Early Progenitors Charles Spearman (1904) Statistical Constraints Measurement Model Structure g m1 m2 r1r2 m1 * r1 = m2 * r2

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21 Impurities/Deviations from Local Independence defeat tetrad constraints selectively x1,x2 * x3,x4 = x1,x3 * x2,x4 x1,x2 * x3,x4 = x1,x4 * x2,x3 x1,x3 * x2,x4 = x1,x4 * x2,x3 x1,x2 * x3,x4 = x1,x3 * x2,x4 x1,x2 * x3,x4 = x1,x4 * x2,x3 x1,x3 * x2,x4 = x1,x4 * x2,x3

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22 Purify True Model Initially Specified Measurement Model

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Purify Iteratively remove item whose removal most improves measurement model fit (tetrads or 2 ) – stop when confirmatory fit is acceptable Remove x4 Remove z2

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24 Purify Detectibly Pure Subset of Items Detectibly Pure Measurement Model

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25 Purify

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How a pure measurement model is useful 1.Consistently estimate covariances/correlations among latents - test conditional independence with estimated latent correlations 2.Test for conditional independence among latents directly

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2. Test conditional independence relations among latents directly 21 21 = 0 L 1 _||_ L 2 | {Q 1, Q 2,..., Q n } Question: L 1 _||_ L 2 | {Q 1, Q 2,..., Q n }

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

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29 Purify & MIMbuild

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

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

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

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33 Build Pure Clusters

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34 Build Pure Clusters Qualitative Assumptions 1.Two types of nodes: measured (M) and latent (L) 2.M L (measured don’t cause latents) 3.Each m M measures (is a direct effect of) at least one l L 4.No cycles involving M Quantitative Assumptions: 1.Each m M is a linear function of its parents plus noise 2.P(L) has second moments, positive variances, and no deterministic relations

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

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

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37 Build Pure Clusters Algorithm Sketch: 1.Use particular rank (tetrad) constraints on the measured correlations to find pairs of items m j, m k that do NOT share a single latent parent 2.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. 3.Purify 4.Remove latents with no children

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38 Build Pure Clusters + MIMbuild

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

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40 Case Study: Stress, Depression, and Religion Masters Students (N = 127) 61 - item survey (Likert Scale) Stress: St 1 - St 21 Depression: D 1 - D 20 Religious Coping: C 1 - C 20 p = 0.00 Specified Model

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

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

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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): X 1 - X 20 : Exploratory Factor Analysis:

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44 Build Pure Clusters : Case Study : Test Anxiety

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

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

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47 Limitations In simulation studies, requires large sample sizes to be really reliable (~ ). 2 pure indicators must exist for a latent to be discovered and included Moderately computationally intensive (O(n 6 )). No error probabilities.

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

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49 References Tetrad: Spirtes, P., Glymour, C., Scheines, R. (2000). Causation, Prediction, and Search, 2 nd 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, 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 Kauffma n

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