Hostname: page-component-745bb68f8f-5r2nc Total loading time: 0 Render date: 2025-01-07T19:01:46.407Z Has data issue: false hasContentIssue false
  • English
  • Français

Bayesian Factor Analysis for Multilevel Binary Observations

Published online by Cambridge University Press:  01 January 2025

Asim Ansari  and
Kamel Jedidi
Asim Ansari*
Affiliation:
Columbia University
Kamel Jedidi
Affiliation:
Columbia University
*
Requests for reprints should be sent to Asim Ansaxi, 517 Uris Hall, Columbia University, 3022 Broadway, New York, NY, 10027. E-mail: maa48@columbia.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Multilevel covariance structure models have become increasingly popular in the psychometric literature in the past few years to account for population heterogeneity and complex study designs. We develop practical simulation based procedures for Bayesian inference of multilevel binary factor analysis models. We illustrate how Markov Chain Monte Carlo procedures such as Gibbs sampling and Metropolis-Hastings methods can be used to perform Bayesian inference, model checking and model comparison without the need for multidimensional numerical integration. We illustrate the proposed estimation methods using three simulation studies and an application involving student's achievement results in different areas of mathematics.

Keywords

MCMC methodsbinary datamultilevel factor analysisGibbs samplingMetropolis-Hastings
Type
Original Paper
Information
Psychometrika , Volume 65 , Issue 4 , December 2000 , pp. 475 - 496
Copyright
Copyright © 2000 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.)

Footnotes

The authors thank Ian Westbury, University of Illinois at Urbana Champaign for kindly providing the SIMS data for the application.

References

Albert, J., Chib, S. (1993). Bayesian analysis of binary and polychotomous response data. Journal of the American Statistical Association, 88, 669–679CrossRefGoogle Scholar
Albert, J., Chib, S. (1995). Bayesian residual analysis for binary response regression models. Biometrika, 82, 747–759CrossRefGoogle Scholar
Arminger, G., Muthén, B. (1998). A Bayesian approach to nonlinear latent variable models using the Gibbs sampler and the Metropolis-Hastings algorithm. Psychometrika, 63, 271–300CrossRefGoogle Scholar
Bartholomew, D.J. (1980). Factor analysis for categorical data. Journal of the Royal Statistical Society, Series B, 42, 293–321CrossRefGoogle Scholar
Bartholomew, D.J. (1981). Posterior analysis of the factor model. British Journal of Mathematical and Statistical Psychology, 34, 93–99CrossRefGoogle Scholar
Bartholomew, D.J. (1984). Scaling binary data using a factor model. Journal of the Royal Statistical Society, Series B, 46, 120–123CrossRefGoogle Scholar
Bartholomew, D.J. (1987). Latent variable models and factor analysis. New York, NY: Oxford University PressGoogle Scholar
Best, N.G., Cowles, M.K., Vines, S.K. (1995). CODA: Convergence diagnostics and output analysis software for Gibbs sampler output, Version 0.3.. Cambridge, UK: Biostatistics Unit-MRCGoogle Scholar
Bock, R.D., Aitkin, M. (1981). Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm. Psychometrika, 46, 443–445CrossRefGoogle Scholar
Bock, R.D., Gibbons, R.D. (1996). High-dimensional multivariate probit analysis. Biometrics, 52, 1183–1194CrossRefGoogle ScholarPubMed
Brooks, S.P., Gelman, A. (1998). General methods for monitoring convergence of iterative simulations. Journal of Computational and Graphical Statistics, 7, 434–455CrossRefGoogle Scholar
Brooks, S.P., & Roberts, G.O. (in press). Assessing convergence of Markov chain Monte Carlo algorithms. Journal of Computational and Graphical Statistics.Google Scholar
Chambers, R.G. (1982). Correlation coefficients from 2 × 2 tables and from biserial data. British Journal of Mathematical and Statistical Psychology, 35, 216–227CrossRefGoogle Scholar
Chen, Ming-Hui, Dey, D.K. (1998). Bayesian modeling of correlated binary responses via scale mixture of multivariate normal link functions. Sankhya, Series A, 60, 322–343Google Scholar
Chen, Ming-Hui, Schmeiser, B.W. (1993). Performance of the Gibbs, Hit-and-Run, and Metropolis Samplers. Journal of Computational and Graphical Statistics, 2, 251–272CrossRefGoogle Scholar
Chib, S., Greenberg, E. (1995). Understanding the Metropolis-Hastings Algorithm. American Statistician, 49, 327–335CrossRefGoogle Scholar
Chib, S., Greenberg, E. (1998). Analysis of Multivariate Probit Models. Biometrika, 85(2), 347–361CrossRefGoogle Scholar
Christofferson, A. (1975). Factor analysis of dichotomized variables. Psychometrika, 40, 5–32CrossRefGoogle Scholar
Cowles, M.K., Carlin, B.P. (1996). Markov chain Monte Carlo convergence diagnostics: a comparative review. Journal of the American Statistical Association, 91, 883–904CrossRefGoogle Scholar
Crosswhite, F.J., Dossey, J.A., Swafford, J.O., McKnight, C.C., Cooney, T.J. (1985). Second International Mathematics Study: Summary report for the United States. Champaign, IL: StipesGoogle Scholar
Chen, M.H., Dey, Dipak K. (1998). Bayesian analysis of correlated binary data models. Sankhya, Series A, 60, 322–343Google Scholar
Gelfand, A.E. (1996). Model determination using sampling-based methods. In Gilks, W.R., Richardson, S., Spiegelhalter, D. J. (Eds.), Markov chain Monte Carlo in practice (pp. 145–161). London: Chapman & HallGoogle Scholar
Gelfand, A.E., Smith, A.F.M. (1990). Sampling-based approaches to calculating marginal densities. Journal of the American Statistical Association, 85, 972–985CrossRefGoogle Scholar
Gelman, A., Carlin, J.B., Stern, H. S., Rubin, D. R. (1996). Posterior predictive assessment of model fitness (with discussion). Statistica Sinica, 6, 733–807Google Scholar
Gelman, A., Rubin, D.R. (1992). Inference from iterative simulation using multiple sequences (with discussion). Statistical Science, 7, 457–511CrossRefGoogle Scholar
Geman, S., Geman, D. (1984). Stochastic Relaxation, Gibbs Distributions and the Bayesian Restoration of Images. IEEE Transactions of Pattern Analysis and Machine Intelligence, 6, 721–741CrossRefGoogle ScholarPubMed
Geweke, J.et al. (1992). Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments (with discussion). In Bernardo, J. M.et al. (Eds.), Bayesian Statistics 4 (pp. 156–163). Oxford: Oxford University PressGoogle Scholar
Geyer, C.J. (1992). Practical Markov chain Monte Carlo (with discussion). Statistical Science, 7, 473–511Google Scholar
Goldstein, H., McDonald, R.P. (1988). A general model for the analysis of multilevel data. Psychometrika, 53, 455–467CrossRefGoogle Scholar
Kass, R.E., Raftery, A.E. (1995). Bayes factors. Journal of American Statistical Association, 90, 773–795CrossRefGoogle Scholar
Lee, S.-Y (1981). A Bayesian approach to confirmatory factor analysis. Psychometrika, 46, 153–160CrossRefGoogle Scholar
Longford, N.T., Muthén, B. (1992). Factor analysis for clustered observations. Psychometrika, 57, 581–597CrossRefGoogle Scholar
Mardia, K.V. (1970). Families of bivariate distributions. London: GriffinGoogle Scholar
Martin, J.K., McDonald, R.P. (1975). Bayesian estimation in unrestricted factor analysis; a treatment for Heywood cases. Psychometrika, 40, 505–517CrossRefGoogle Scholar
McDonald, R.P., Goldstein, H. (1989). Balanced versus unbalanced designs for linear structural relations in two-level data. British Journal of Mathematical and Statistical Psychology, 42, 214–232CrossRefGoogle Scholar
Muthén, B. (1978). Contributions to factor analysis of dichotomous variables. Psychometrika, 43, 551–560CrossRefGoogle Scholar
Muthén, B. (1979). A structural probit model with latent variables. Journal of the American Statistical Association, 74, 807–811Google Scholar
Muthén, B. (1984). A general structural equation model with dichotomous, ordered categorical and continuous latent variable indicators. Psychometrika, 49, 115–132CrossRefGoogle Scholar
Muthén, B. (1987). LISCOMP: Analysis of linear structural equations with a comprehensive measurement model. Mooresville, IN: Scientific SoftwareGoogle Scholar
Muthén, B. (1989). Latent variable modeling in heterogeneous populations. Psychometrika, 54, 557–585CrossRefGoogle Scholar
Muthén, B. (1994). Multilevel covariance structure analysis. Sociological Methods & Research, 22, 376–398CrossRefGoogle Scholar
Muthén, B., Christofferson, A. (1981). Simultaneous factor analysis of dichotomous variables in several groups. Psychometrika, 46, 407–419CrossRefGoogle Scholar
Muthén, B., Satorra, A. (1989). Multilevel aspects of varying parameters in structural models. In Bock, R. D. (Eds.), Multilevel analysis of educational data (pp. 87–99). New York, NY: Academic PressGoogle Scholar
Sahu, S.K. (1998). Bayesian estimation and model choice in item response models. Cardiff, Wales, UK: Cardiff University, School of MathematicsGoogle Scholar
Shi, J., Lee, S.-Y. (1997). A Bayesian estimation of factor score in confirmatory factor model with polytomous, censored or truncated data. Psychometrika, 62, 29–50CrossRefGoogle Scholar
Tanner, M.A., Wong, W.H. (1987). The calculation of posterior distributions by data augmentation (with discussion). Journal of American Statistical Association, 82, 528–550CrossRefGoogle Scholar
Tierney, L. (1994). Markov chains for exploring posterior distributions (with discussion). Annals of Statistics, 22, 1701–1762Google Scholar