Download presentation
Presentation is loading. Please wait.
Published byKristina Beatrice Scott Modified over 9 years ago
1
Challenges posed by Structural Equation Models Thomas Richardson Department of Statistics University of Washington Joint work with Mathias Drton, UC Berkeley Peter Spirtes, CMU
2
Overview n Challenges for Likelihood Inference n Problems in Model Selection and Interpretation n Partial Solution u sub-class of path diagrams: ancestral graphs
3
Problems for Likelihood Inference n Likelihood may be multimodal u e.g. the bi-variate Gaussian Seemingly Unrelated Regression (SUR) model: X1X1 X2X2 Y1Y1 Y2Y2 may have up to 3 local maxima. Consistent starting value does not guarantee iterative procedures will find the MLE.
4
Problems for Likelihood Inference n Discrete latent variable models are not curved exponential families C X1X1 X2X2 X3X3 X4X4 binary observed variables ternary latent class variable 15 parameters in saturated model 14 model parameters BUT model has 2d.f. (Goodman) Usual asymptotics may not apply
5
Problems for Likelihood Inference n Likelihood may be highly multimodal in the asymptotic limit u After accounting for label switching/aliasing C X1X1 X2X2 X3X3 X4X4 Why report one mode ? d.f. may vary as a function of model parameters
6
Problems for Model Selection n SEM models with latent variables are not curved exponential families Standard 2 asymptotics do not necessarily apply e.g. for LRTs u Model selection criteria such as BIC are not asymptotically consistent u The effective degrees of freedom may vary depending on the values of the model parameters
7
Problems for Model Selection n Many models may be equivalent: X1X1 X2X2 Y1Y1 Y2Y2 X1X1 X2X2 Y1Y1 Y2Y2 X1X1 X2X2 Y1Y1 Y2Y2 X1X1 X2X2 Y1Y1 Y2Y2
8
Problems for Model Selection X1X1 XpXp Y1Y1 YqYq X1X1 XpXp Y1Y1 YqYq n Models with different numbers of latents may be equivalent: u e.g. unrestricted error covariance within blocks
9
Problems for Model Selection n Models with different numbers of latents may be equivalent: u e.g. unrestricted error covariance within blocks X1X1 XpXp Y1Y1 YqYq X1X1 XpXp Y1Y1 YqYq Wegelin & Richardson (2001)
10
Two scenarios n A single SEM model is proposed and fitted. The results are reported.
11
Two scenarios n A single SEM model is proposed and fitted. The results are reported. n The researcher fits a sequence of models, making modifications to an original specification. u Model equivalence implies: F Final model depends on initial model chosen F Sequence of changes is often ad hoc F Equivalent models may lead to very different substantive conclusions u Often many equivalence classes of models give reasonable fit. Why report just one?
12
Partial Solution n Embed each latent variable model in a ‘larger’ model without latent variables characterized by conditional independence restrictions. n We ignore non-independence constraints and inequality constraints. Latent variable model Model imposing only independence constraints on observed variables Sets of distributions
13
ab t cd Toy Example: acbd ad ad c ad b ac d bd a G at dt bc t +others The Generating graph n Begin with a graph, and associated set of independences
14
ab t cd acbd ad ad c ad b ac d bd a G at dt bc t +others hidden: ‘Unobserved’ independencies in red Marginalization n Suppose now that some variables are unobserved n Find the independence relations involving only the observed variables Toy Example:
15
ab t cd acbd ad ad c ad b ac d bd a G at dt bc t +others hidden: ‘Unobserved’ independencies in red Marginalization n Suppose now that some variables are unobserved n Find the independence relations involving only the observed variables Toy Example:
16
ab t cd abcd acbd ad ad c ad b ac d bd a G G* ‘Graphical Marginalization’ n Now construct a graph that represents the conditional independence relations among the observed variables. n Bi-directed edges are required. represents Toy Example: all and only the distributions in which these independencies hold
17
Equivalence re-visited n Restrict model class to path diagrams including only observed variables characterized by conditional independence u Ancestral Graph Markov models n For such models we can: u Determine the entire class of equivalent models u Identify which features they have in common n Models are curved exponential: usual asymptotics do apply
18
A T AB C D AC BD AD AD C AD B AC D BD A A BCD Ancestral Graph
19
A V ABCD T AB C D U AC BD AD AD C AD B AC D BD A A BCD A BCD Equivalent ancestral graphs
20
A V ABCD T AB C D U Q A BC D P R AC BD AD AD C AD B AC D BD A A BCD A BCD A BCD Markov Equiv. Class of Graphs with Latent Variables Equivalent ancestral graphs
21
A V ABCD T AB C D U + infinitely many others Q A BC D P R AC BD AD AD C AD B AC D BD A A BCD A BCD A BCD A BCD N A BC D M R L Markov Equiv. Class of Graphs with Latent Variables Equivalence Classes Equivalent ancestral graphs
22
ABCD A V ABCD T AB C D U + infinitely many others Q A BC D P R AC BD AD AD C AD B AC D BD A A BCD A BCD A BCD A BCD N A BC D M R L Markov Equiv. Class of Graphs with Latent Variables Equivalence class of Ancestral Graphs Partial Ancestral Graph
23
ABCD A V ABCD T AB C D U + infinitely many others Q A BC D P R AC BD AD AD C AD B AC D BD A A BCD A BCD A BCD A BCD Equivalence class of Ancestral Graphs N A BC D M R L Markov Equiv. Class of Graphs with Latent Variables
24
Measurement models n If we have pure measurement models with several indicators per latent: u May apply similar search methods among the latent variables (Spirtes et al. 2001; Silva et al.2003)
25
Other Related Work n Iterative ML estimation methods exist u Guaranteed convergence F Multimodality is still possible Implemented in R package ggm (Drton & Marchetti, 2003) n Current work: u Extension to discrete data F Parameterization and ML fitting for binary bi-directed graphs already exist u Implementing search procedures in R
26
References n Richardson, T., Spirtes, P. (2002) Ancestral graph Markov models, Ann. Stat., 30: 962-1030 n Richardson, T. (2003) Markov properties for acyclic directed mixed graphs. Scand. J. Statist. 30(1), pp. 145-157 n Drton, M., Richardson T. (2003) A new algorithm for maximum likelihood estimation in Gaussian graphical models for marginal independence. UAI 03, 184-191 n Drton, M., Richardson T. (2003) Iterative conditional fitting in Gaussian ancestral graph models. UAI 04 130-137. n Drton, M., Richardson T. (2004) Multimodality of the likelihood in the bivariate seemingly unrelated regressions model. Biometrika, 91(2), 383-92. Marchetti, G., Drton, M. (2003) ggm package. Available from http://cran.r-project.org Marchetti, G., Drton, M. (2003) ggm package. Available from http://cran.r-project.org
Similar presentations
© 2024 SlidePlayer.com. Inc.
All rights reserved.