1Topic Outline Motivation Representing/Modeling Causal Systems Estimation and UpdatingModel SearchLinear Latent Variable ModelsCase Study: fMRI
2Richard Scheines Carnegie Mellon University DiscoveringPure Measurement ModelsRichard Scheines Carnegie Mellon UniversityRicardo Silva* University College LondonClark Glymour and Peter Spirtes Carnegie Mellon University
3Outline Measurement Models & Causal Inference Strategies for Finding a Pure Measurement ModelPurifyMIMbuildBuild Pure ClustersExamplesReligious CopingTest Anxiety
4Goals: What Latents are out there? Causal Relationships Among Latent ConstructsDepressionRelationshipSatisfactionDepressionRelationshipSatisfactionoror ?
5Needed: Ability to detect conditional independence among latent variables
6Lead and IQ e2 e3 Lead _||_ IQ | PR PR ~ N(m=10, s = 3) Parental ResourcesLeadExposureIQLead _||_ IQ | PRPR ~ N(m=10, s = 3)Lead = *PR + e2e2 ~ N(m=0, s = 1.635)IQ = *PR + e3e3 ~ N(m=0, s = 15)
7Psuedorandom sample: N = 2,000 Parental ResourcesLeadExposureIQRegression of IQ on Lead, PRIndependentVariableCoefficient Estimatep-valueScreened-offat .05?PR0.980.000NoLead-0.0880.378Yes
26How a pure measurement model is useful Consistently estimate covariances/correlations among latents - test conditional independence with estimated latent correlationsTest for conditional independence among latents directly
28MIMbuild Input: - Purified Measurement Model - Covariance matrix over set of pure itemsMIMbuildPC algorithm with independence testsperformed directly on latent variablesOutput: Equivalence class of structural modelsover the latent variables
30Goal 2: What Latents are out there? How should they be measured?
31Latents and the clustering of items they measure imply tetrad constraints diffentially
32Build Pure Clusters (BPC) Input:- Covariance matrix over set of original itemsBPC1) Cluster (complicated boolean combinations of tetrads)2) PurifyOutput: Equivalence class of measurement models over a pure subset of original Items
34Build 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 LNo cycles involving MQuantitative Assumptions:Each m M is a linear function of its parents plus noiseP(L) has second moments, positive variances, and no deterministic relations
35Build Pure Clusters Output - provably reliable (pointwise consistent): Equivalence class of measurement models over a pure subset of MFor example:True ModelOutput
36Build Pure Clusters Output Measurement models in the equivalence class are at most refinements, but never coarsenings or permuted clusterings.Output
37Build 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 parentAdd 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.PurifyRemove latents with no children
41Case Study: Stress, Depression, and Religion Build Pure Clusters
42Case Study: Stress, Depression, and Religion Assume Stress temporally prior:MIMbuild to find Latent Structure:p = 0.28
43Case Study : Test Anxiety Bartholomew and Knott (1999), Latent variable models and factor analysis12th Grade Males in British Columbia (N = 335)20 - item survey (Likert Scale items): X1 - X20:Exploratory Factor Analysis:
45Case Study : Test Anxiety Build Pure Clusters:Exploratory Factor Analysis:p-value = 0.00p-value = 0.47
46Case Study : Test Anxiety MIMbuildp = .43UninformativeScales: No Independencies or Conditional Independencies
47LimitationsIn simulation studies, requires large sample sizes to be really reliable (~ ).2 pure indicators must exist for a latent to be discovered and includedModerately computationally intensive (O(n6)).No error probabilities.
48Open Questions/Projects IRT models?Bi-factor model extensions?Appropriate incorporation of background knowledge
49References 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,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