Hostname: page-component-89b8bd64d-n8gtw Total loading time: 0 Render date: 2026-05-08T04:37:06.622Z Has data issue: false hasContentIssue false
  • English
  • Français

A Robust Bayesian Approach for Structural Equation Models with Missing Data

Published online by Cambridge University Press:  01 January 2025

Sik-Yum Lee  and
Ye-Mao Xia
Sik-Yum Lee*
Affiliation:
The Chinese University of Hong Kong
Ye-Mao Xia
Affiliation:
The Chinese University of Hong Kong
*
Requests for reprints should be sent to Sik-Yum Lee, Department of Statistics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong. E-mail: sylee@sta.cuhk.edu.hk
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, normal/independent distributions, including but not limited to the multivariate t distribution, the multivariate contaminated distribution, and the multivariate slash distribution, are used to develop a robust Bayesian approach for analyzing structural equation models with complete or missing data. In the context of a nonlinear structural equation model with fixed covariates, robust Bayesian methods are developed for estimation and model comparison. Results from simulation studies are reported to reveal the characteristics of estimation. The methods are illustrated by using a real data set obtained from diabetes patients.

Keywords

robust Bayesian methodsnormal/independent distributionsnonlinear structural equation models with covariatesmodel comparison

Information

Type
Theory and Methods
Information
Psychometrika , Volume 73 , Issue 3 , September 2008 , pp. 343 - 364
Copyright
Copyright © 2008 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

Anderson, R.J., Critchley, J., Chan, J.C.N., Dockram, C.S., Lee, Z.S., Thomas, G.N., & Tomlinson, B.T. (2001). Factor analysis of the metabolic syndrome: Obesity vs insulin resistance as central abnormality. International Journal of Obesity, 25, 1782–1788.CrossRefGoogle ScholarPubMed
Andrews, D.F., & Mallows, C.L. (1974). Scale mixtures of normal distribution. Journal of the Royal Statistical Society, Series B, 36, 99–102.CrossRefGoogle Scholar
Ansari, A., & Jedidi, K. (2000). Bayesian factor analysis for multilevel binary observations. Psychometrika, 64, 475–496.CrossRefGoogle Scholar
Bentler, P.M., & Wu, E.J.C. (2002). EQS 6 for windows user’s guide, Encino: Multivariate Software.Google Scholar
Browne, M.W. (1987). Robustness of statistical influence in factor analysis and related models. Biometrika, 74, 375–384.CrossRefGoogle Scholar
Browne, M.W., & Shapiro, A. (1988). Robustness of normal theory methods in the analysis of linear latent variable models. British Journal of Mathematical and Statistical Psychology, 41, 193–208.CrossRefGoogle Scholar
Chen, M.H., Ibrahim, J.G., & Sinha, D. (2004). A new joint model for longitudinal and survival data with a cure fraction. Journal of Multivariate Analysis, 91, 18–34.CrossRefGoogle Scholar
Dempster, A.P., Laird, N.M., & Rubin, D.B. (1977). Maximum likelihood from incomplete data via the EM algorithm (with discussion). Journal of the Royal Statistical Society, Series B, 39, 1–38.CrossRefGoogle Scholar
Dunson, D.B. (2000). Bayesian latent variable models for clustered mixed outcomes. Journal of the Royal Statistical Society, Series B, 62, 355–366.CrossRefGoogle Scholar
Dunson, D.B., & Perreault, S.D. (2001). Factor analytic models of clustered multivariate data with informative censoring. Biometrics, 57, 302–308.CrossRefGoogle ScholarPubMed
Geisser, S., & Eddy, W. (1979). A predictive approach to model selection. Journal of the American Statistical Association, 74, 153–160.CrossRefGoogle Scholar
Gelfand, A.E., Smith, A.F.M., & Lee, T.M. (1992). Bayesian analysis of constrained parameters and truncated data problem using Gibbs sampling. Journal of the American Statistical Association, 87, 523–532.CrossRefGoogle Scholar
Gelman, A., & Meng, X.L. (1998). Simulating normalizing constants: From importance sampling to bridge sampling to path sampling. Statistical Science, 13, 163–185.CrossRefGoogle Scholar
Geman, S., & Geman, D. (1984). Stochastic relaxation, Gibbs distribution, and the Bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6, 721–741.CrossRefGoogle ScholarPubMed
Hastings, W.K. (1970). Monter Carlo sampling methods using Markov chains and their applications. Biometrika, 57, 97–109.CrossRefGoogle Scholar
Hobert, J.P., & Casella, G. (1996). The effect of improper priors on Gibbs sampling in hierarchical linear mixed models. Journal of the American Statistical Association, 91, 1461–1473.CrossRefGoogle Scholar
Ibrahim, J.G., Chen, M.H., & Sinha, D. (2001). Bayesian survival analysis, New York: Springer.CrossRefGoogle Scholar
Jöreskog, K.G., & Sörbom, D. (1996). LISREL 8: Structural equation modelling with the SIMPLIS command language, Hove: Scientific Software International.Google Scholar
Kano, Y. (1994). Consistency property of elliptical probability density function. Journal of Multivariate Analysis, 51, 139–147.CrossRefGoogle Scholar
Kano, Y., Berkane, M., & Bentler, P.M. (1993). Statistical inference based on pseudomaximum likelihood estimators in elliptical populations. Journal of the American Statistical Association, 88, 135–143.CrossRefGoogle Scholar
Kass, R.E., & Raftery, A.E. (1995). Bayes factor. Journal of the American Statistical Association, 90, 773–795.CrossRefGoogle Scholar
Lange, K.L., & Sinsheimer, J.S. (1993). Normal/independent distributions and their applications in robust regression. Journal of Computational and Graphical Statistics, 2, 175–198.CrossRefGoogle Scholar
Lange, K.L., Little, R.J.A., & Taylor, J.M.G. (1989). Robust statistical modelling using the t-distribution. Journal of the American Statistical Association, 84, 881–896.Google Scholar
Lee, S.Y., & Song, X.Y. (2003). Bayesian analysis of structural equation models with dichotomous variables. Statistics in Medicine, 22, 3073–3088.CrossRefGoogle ScholarPubMed
Lee, S.Y., & Song, X.Y. (2003). Model comparison of nonlinear structural equation models with fixed covariates. Psychometrika, 68, 27–47.CrossRefGoogle Scholar
Lee, S.Y., & Song, X.Y. (2004). Bayesian model comparison of nonlinear structural equation models with missing continuous and ordinal categorical data. British Journal of Mathematical and Statistical Psychology, 57, 131–150.CrossRefGoogle ScholarPubMed
Lee, S.Y., & Song, X.Y. (2004). Evaluation of the Bayesian and maximum likelihood approaches in analyzing structural equation models with small sample sizes. Multivariate Behavioral Research, 39, 653–686.CrossRefGoogle ScholarPubMed
Lee, S.Y., & Song, X.Y. (2004). Maximum likelihood analysis of a general latent variable with hierarchically mixed data. Biometrics, 60, 624–636.CrossRefGoogle ScholarPubMed
Lee, S.Y., & Song, X.Y. (2007). A unified maximum likelihood approach for analyzing structural equation models with missing non-standard data. Sociological Methods & Research, 35, 352–381.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
Little, R.J.A. (1988). Robust estimation of mean and covariance matrix from data with missing values. Applied Statistics, 37, 23–39.CrossRefGoogle Scholar
Little, R.J.A., & Rubin, D.B. (1987). Statistical analysis with missing data, New York: Wiley.Google Scholar
Liu, C.H. (1995). Missing data imputation using the multivariate t-distribution. Journal of Multivariate Analysis, 46, 198–206.CrossRefGoogle Scholar
Liu, C.H. (1996). Bayesian robust multivariate linear regression with incomplete data. Journal of the American Statistical Association, 91, 1219–1227.CrossRefGoogle Scholar
Liu, C.H., & Rubin, D.B. (1994). The ECME algorithm: A simple extension of EM and ECM with Faster Monotone convergence. Biometrika, 81, 633–648.CrossRefGoogle Scholar
Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., & Teller, E. (1953). Equations of state calculations by fast computing machines. Journal of Chemical Physics, 21, 1087–1092.CrossRefGoogle Scholar
Pettit, L.I., & Young, K.D.S. (1990). Measuring the effect of observations on the Bayes factor. Biometrika, 77, 455–466.CrossRefGoogle Scholar
Philippe, A. (1997). Simulation of right and left truncated gamma distributions by mixtures. Statistics and Computing, 7, 173–181.CrossRefGoogle Scholar
Raykov, T. (2007). Longitudinal analysis with regression among random effects: A latent variable modeling approach. Structural Equation Modeling: A Multidisciplinary Journal, 14, 146–169.CrossRefGoogle Scholar
Ritter, C., & Tanner, M. (1992). Facilitating the Gibbs sampler: The Gibbs stopper and the Griddy–Gibbs sampler. Journal of the American Statistical Association, 87, 861–868.CrossRefGoogle Scholar
Scheines, R., Hoijtink, H., & Boomsma, A. (1999). Bayesian estimation and testing of structural equation models. Psychometrika, 64, 37–52.CrossRefGoogle Scholar
Shapiro, A., & Browne, M.W. (1987). Analysis of covariance structural under elliptical distribution. Journal of the American Statistical Association, 82, 1092–1097.CrossRefGoogle Scholar
Song, X.Y., & Lee, S.Y. (2004). Bayesian analysis of two-level nonlinear structural equation models with continuous and polytomous data. British Journal of Mathematical and Statistical Psychology, 57, 29–52.CrossRefGoogle ScholarPubMed
Song, X.Y., Lee, S.Y. (2006). Model comparison of generalized linear mixed models. Statistics in Medicine, 25, 1685–1698.CrossRefGoogle ScholarPubMed
Song, X.Y., & Lee, S.Y. (2007). Bayesian analysis of latent variable models with nonignorable missing outcomes from exponential family. Statistics in Medicine, 26, 681–693.CrossRefGoogle ScholarPubMed
Song, X.Y., Lee, S.Y., Ng, M.C., So, W.Y., & Chan, J.C.N. (2007). Bayesian analysis of structural equation models with multinomial variables and an application to type 2 diabetic nephropathy. Statistics in Medicine, 26, 2348–2369.CrossRefGoogle Scholar
Wang, J.J., Qiao, Q., Mietting, M., Lappalainen, J., Hu, G., & Tuomilehto, J. (2004). The metabolic syndrome defined by factor analysis and incident type 2 diabetes in Chinese population with high postprandial glucose. Diabetes Care, 27, 2429–2437.CrossRefGoogle ScholarPubMed
Willoughby, M., Vandergift, N., Blair, C., & Granger, D.A. (2007). A structural equation modeling approach for the analysis of cortisol data collected using pre-post-post designs. Structural Equation Modeling: A Multidisciplinary Journal, 14, 125–145.Google Scholar