Hostname: page-component-74656f9884-g96k6 Total loading time: 0 Render date: 2026-11-10T06:26:46.165Z Has data issue: false hasContentIssue false
  • English
  • Français

Bayesian Semiparametric Structural Equation Models with Latent Variables

Published online by Cambridge University Press:  01 January 2025

Mingan Yang  and
David B. Dunson
Mingan Yang*
Affiliation:
Saint Louis University
David B. Dunson
Affiliation:
Duke University
*
Requests for reprints should be sent to Mingan Yang, School of Public Health, Saint Louis University, St. Louis, MO 63104, USA. E-mail: mingany@yahoo.com
Get access Rights & Permissions [Opens in a new window]

Abstract

Structural equation models (SEMs) with latent variables are widely useful for sparse covariance structure modeling and for inferring relationships among latent variables. Bayesian SEMs are appealing in allowing for the incorporation of prior information and in providing exact posterior distributions of unknowns, including the latent variables. In this article, we propose a broad class of semiparametric Bayesian SEMs, which allow mixed categorical and continuous manifest variables while also allowing the latent variables to have unknown distributions. In order to include typical identifiability restrictions on the latent variable distributions, we rely on centered Dirichlet process (CDP) and CDP mixture (CDPM) models. The CDP will induce a latent class model with an unknown number of classes, while the CDPM will induce a latent trait model with unknown densities for the latent traits. A simple and efficient Markov chain Monte Carlo algorithm is developed for posterior computation, and the methods are illustrated using simulated examples, and several applications.

Keywords

Dirichlet processfactor analysislatent classlatent traitmixture modelnonparametric Bayesparameter expansion

Information

Type
Original Paper
Information
Psychometrika , Volume 75 , Issue 4 , December 2010 , pp. 675 - 693
Copyright
Copyright © 2010 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Ansari, A., Iyengar, R. (2006). Semiparametric Thurstonian models for recurrent choice: a Bayesian analysis. Psychometrika, 71, 631–657.CrossRefGoogle Scholar
Blackwell, D., MacQueen, J.B. (1973). Ferguson distribution via Polya Urn schmes. Annals of Statistics, 2, 353–355.Google Scholar
Bollen, K.A. (1989). Structural equations with latent variables, New York: Wiley.CrossRefGoogle Scholar
Brown, E.R., Ibrahim, J.G. (2003). A Bayesian semiparametric joint hierarchical model for longitudinal and survival data. Biometrics, 59, 221–228.CrossRefGoogle ScholarPubMed
Burr, D., Doss, H. (2005). A Bayesian semiparametric model for random-effects meta-analysis. Journal of the American Statistical Association, 100, 242–251.CrossRefGoogle Scholar
Bush, C.A., MacEachern, S.N. (1996). A semiparametric Bayesian model for randomised block designs. Biometrika, 83, 275–285.CrossRefGoogle Scholar
Dunson, D.B. (2006). Bayesian dynamic modeling of latent trait distributions. Biostatistics, 7, 551–568.CrossRefGoogle ScholarPubMed
Escobar, M.D., West, M. (1995). Bayesian density estimation and inference using mixtures. Journal of the American Statistical Association, 90, 577–588.CrossRefGoogle Scholar
Fahrmeir, L., Raach, A. (2007). A Bayesian semiparametric latent variable model for mixed responses. Psychometrika, 72, 327–346.CrossRefGoogle Scholar
Ferguson, T.S. (1973). A Bayesian analysis of some nonparametric problems. Annals of Statistics, 2, 209–230.Google Scholar
Ferguson, T.S. (1974). Prior distributions on spaces of probability measures. Annals of Statistics, 2, 615–629.CrossRefGoogle Scholar
Fokoue, E. (2005). Mixtures of factor analyzers: an extension with covariates. Journal of Multivariate Analysis, 95, 370–384.CrossRefGoogle Scholar
Fokoue, E., Titterington, D.M. (2003). Mixtures of factor analysers: Bayesian estimation and inference by stochastic simulation. Machine Learning, 50, 73–94.CrossRefGoogle Scholar
Gelman, A. (2004). Parameterization and Bayesian modeling. Journal of the American Statistical Association, 99, 537–545.CrossRefGoogle Scholar
Ghosh, J., Dunson, D.B. (2009). Default priors and efficient posterior computation in Bayesian factor analysis. Journal of Computational and Graphical Statistics, 18, 306–320.CrossRefGoogle ScholarPubMed
Ishwaran, H., James, L.F. (2001). Gibbs samplling methods for stickbreaking priors. Journal of the American Statistical Association, 96, 161–173.CrossRefGoogle Scholar
Ishwaran, H., Takahara, G. (2002). Independent and identically distributed Monte Carlo algorithms for semiparametric linear mixed models. Journal of the American Statistical Association, 97, 1154–1166.CrossRefGoogle Scholar
Jedidi, K., Jagpal, H.S., DeSarbo, W.S. (1997). Finite-mixture structural equation models for response-based segmentation and unobserved heterogeneity. Marketing Science, 16, 39–59.CrossRefGoogle Scholar
Jöreskog, K.G., Sörbom, D. (1986). Lisrel VI: Analysis of linear structural relationships by maximum likelihood, instrumental variables, and least squares methods, Mooresville: Scientific Software.Google Scholar
Kleinman, K.P., Ibrahim, J.G. (1998). A semiparametric Bayesian approach to the random effects model. Biometrics, 54, 921–938.CrossRefGoogle Scholar
Lee, S.Y., Xia, Y.M. (2006). Maximum likelihood methods in treating outliers and symmetrically heavy-tailed distributions for nonlinear structural equation models with missing data. Psychometrika, 71, 565–585.CrossRefGoogle Scholar
Lee, S.Y., Lu, B., Song, X.Y. (2008). Semiparametric Bayesian analysis of structural equation models with fixed covariates. Statistics in Medicine, 27, 2341–2360.CrossRefGoogle ScholarPubMed
Li, Y., Müller, P., & Lin, X. (in press). Center-adjusted inference for a nonparametric Bayesian random effect distribution. Statistica Sinica.Google Scholar
Liu, C.H., Rubin, D.B., Wu, Y.N. (1998). Parameter expansion to accelerate EM: the PX-EM algorithm. Biometrika, 85, 755–770.CrossRefGoogle Scholar
Liu, J.S., Wu, Y.N. (1999). Parameter expansion for data augmentation. Journal of the American Statistical Association, 94, 1264–1274.CrossRefGoogle Scholar
Lubke, G.H., Muthén, B. (2005). Investigating population heterogeneity with factor mixture models. Psychological Methods, 10, 21–39.CrossRefGoogle ScholarPubMed
McLachlan, G.J., Bean, R.W., Jones, L.B.T. (2007). Extension of the mixture of factor analyzers model to incorporate the multivariate t-distribution. Computational Statistics & Data Analysis, 51, 5327–5338.CrossRefGoogle Scholar
Müller, P., Rosner, G.L. (1997). A Bayesian population model with hierarchical mixture priors applied to blood count data. Journal of the American Statistical Association, 92, 1279–1292.Google Scholar
Ohlssen, D.I., Sharples, L.D., Spiegelhalter, D.J. (2007). Flexible random-effects models using Bayesian semi-parametric models: applications to institutional comparisons. Statistics in Medicine, 26, 2088–2112.CrossRefGoogle ScholarPubMed
Palomo, J., Dunson, D.B., Bollen, K.A. (2007). Bayesian structural equation modeling. In Lee, S.-Y. (Eds.), Handbook of latent variable and related methods (pp. 163–188). Amsterdam: Elsevier.Google Scholar
Papaspiliopoulos, O., Roberts, G.O. (2008). Retrospective Markov chain Monte Carlo methods for Dirichlet process hierarchical models. Biometrika, 95, 169–186.CrossRefGoogle Scholar
Sethuraman, J. (1994). A constructive definition of the Dirichlet process. Statistica Sinica, 4, 639–650.Google Scholar
Zhu, H.T., Lee, S.Y. (2001). A Bayesian analysis of finite mixtures in the LISREL model. Psychometrika, 66, 133–152.CrossRefGoogle Scholar
Walle, F. (1980). Education and the demographic transition in Switzerland. Population and Development Review, 3, 463–472.CrossRefGoogle Scholar
Yang, M., Dunson, D.B., Baird, D. (2010). Semiparametric Bayes hierarchical models with mean and variance constraints. Computational Statistics & Data Analysis, 54, 2172–2186.CrossRefGoogle ScholarPubMed