Hostname: page-component-745bb68f8f-f46jp Total loading time: 0 Render date: 2025-01-08T10:00:00.236Z Has data issue: false hasContentIssue false
  • English
  • Français

Bayesian Analysis of Nonlinear Structural Equation Models with Nonignorable Missing Data

Published online by Cambridge University Press:  01 January 2025

Sik-Yum Lee  and
Nian-Sheng Tang
Sik-Yum Lee*
Affiliation:
The Chinese University of Hong Kong
Nian-Sheng Tang
Affiliation:
Yunnan University, Kunming
*
Requests for reprints should be sent to Professor S.Y. Lee, Department of Statistics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong. E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

A Bayesian approach is developed for analyzing nonlinear structural equation models with nonignorable missing data. The nonignorable missingness mechanism is specified by a logistic regression model. A hybrid algorithm that combines the Gibbs sampler and the Metropolis–Hastings algorithm is used to produce the joint Bayesian estimates of structural parameters, latent variables, parameters in the nonignorable missing model, as well as their standard errors estimates. A goodness-of-fit statistic for assessing the plausibility of the posited nonlinear structural equation model is introduced, and a procedure for computing the Bayes factor for model comparison is developed via path sampling. Results obtained with respect to different missing data models, and different prior inputs are compared via simulation studies. In particular, it is shown that in the presence of nonignorable missing data, results obtained by the proposed method with a nonignorable missing data model are significantly better than those that are obtained under the missing at random assumption. A real example is presented to illustrate the newly developed Bayesian methodologies.

Keywords

Bayes factorGibbs samplerMetropolis–Hastings algorithmmodel comparisonnonignorable missing datapath sampling
Type
Original Paper
Information
Psychometrika , Volume 71 , Issue 3 , September 2006 , pp. 541 - 564
Copyright
Copyright © 2006 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

This research is fully supported by a grant (CUHK 4243/03H) from the Research Grant Council of the Hong Kong Special Administration Region. The authors are thankful to the editor and reviewers for valuable comments for improving the paper, and also to ICPSR and the relevant funding agency for allowing the use of the data.

References

Ansari, A., Jedidi, K., Dube, L. (2002). Heterogenous factor analysis models: A Bayesian approach. Psychometrika, 67, 49–78.CrossRefGoogle Scholar
Bayarri, M.J., Berger, J.O. (2000). P values for composite null models. Journal of the American Statistical Association, 95, 1127–1142.Google 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
DiCiccio, T.J., Kass, R.E., Raftery, A., Wasserman, L. (1997). Computing Bayes factors by combining simulation and asymptotic approximations. Journal of the American Statistical Association, 92, 903–915.CrossRefGoogle Scholar
Diggle, P., Kenward, M.G. (1994). Informative drop-out in longitudinal data analysis (with discussion). Applied Statistics, 43, 49–93.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
Gelman, A. (1996). Inference and monitoring convergence. In Gilks, W.R., Richardson, S., Speigelhalter, D.J. (Eds.), Markov chain Monte Carlo in practice (pp. 131–144). London: Chapman & Hall.Google Scholar
Gelman, A., Carlin, J.B., Stern, H.S., Rubin, D.B. (1995). Bayesian data analysis, London: Chapman & Hall.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
Gelman, A., Meng, X.L., Stern, H. (1996). Posterior predictive assessment of model fitness via realized discrepancies. Statistica Sinica, 6, 733–807.Google Scholar
Gelman, A., Roberts, G.O., & Gilks, W.R. (1995). Efficient Metropolis jumping rules. In Bernardo, J.M., Berger, J.O., Dawid, A.P., & Smith, A.F.M. (Eds.), Bayesian statistics, 5 (pp. 599–607). Oxford: Oxford University Press.Google 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
Geyer, C.J. (1992). Practical Markov chain Monte Carlo. Statistical Science, 7, 473–511.Google Scholar
Heckman, J. (1976). The common structure of statistical models of truncation sample selection and limited dependent variables, and a simple estimator for such models. Annals of Economic and Social Measurement, 5, 475–492.Google Scholar
Ibrahim, J.G., Chen, M.H., Lipsitz, S.R. (2001). Missing responses in generalised linear mixed models when the missing data mechanism is nonignorbale. Biometrika, 88, 551–564.CrossRefGoogle Scholar
Ibrahim, J.G., Lipsitz, S.R. (1996). Parameter estimation from incomplete data in binomial regression when the missing data mechanics is nonignorable. Biometrics, 52, 1070–1078.CrossRefGoogle Scholar
Jamshidian, M., Bentler, P.M. (1999). ML estimation of mean and covariance structures with missing data using complete data routines. Journal of Educational and Behavioral Statistics, 24, 21–41.CrossRefGoogle Scholar
Kass, R.E., Raftery, A.E. (1995). Bayes factors. Journal of the American Statistical Association, 90, 773–795.CrossRefGoogle Scholar
Kenny, D.A., Judd, C.M. (1984). Estimating the nonlinear and interactive effects of latent variables. Psychological Bulletin, 96, 201–210.CrossRefGoogle Scholar
Laird, N.M., Ware, J.H. (1982). Random effects models for longitudinal data. Biometrics, 38, 963–974.CrossRefGoogle ScholarPubMed
Lee, S.Y. (1986). Estimation for structural equation models with missing data. Psychometrika, 51, 93–99.CrossRefGoogle Scholar
Lee, S.Y., Song, X.Y. (2003). Maximum likelihood estimation and model comparison for mixtures of structural equation models with ignorable missing data. Journal of Classification, 20, 221–255.CrossRefGoogle Scholar
Lee, S.Y., Song, X.Y. (2004). Bayesian model comparison of nonlinear latent variable models with missing continuous and ordinal categorical data. British Journal of Mathematical and Statistical Psychology, 57, 131–150.CrossRefGoogle Scholar
Lee, S.Y., Song, X.Y., Lee, J.C.K. (2003). Maximum likelihood estimation of nonlinear structural equation models with ignorable missing data. Journal of Educational and Behavioral Statistics, 28, 111–134.CrossRefGoogle Scholar
Lee, S.Y., Zhu, H.T. (2000). Statistical analysis of nonlinear structural equation models with continuous and polytomous data. British Journal of Mathematical and Statistical Psychology, 53, 209–232.CrossRefGoogle Scholar
Lee, S.Y., Zhu, H.T. (2002). Maximum likelihood estimation of nonlinear structural equation models. Psychometrika, 67, 189–210.CrossRefGoogle Scholar
Little, R.J.A., Rubin, D.B. (1987). Statistical analysis with missing data, Dordrecht: Wiley.Google Scholar
Meng, X.L., van Dyk, D. (1997). The EM algorithm-an old folk-song sung to a fast new tune. Journal of the Royal Statistical Society, Series B, 59, 511–540.CrossRefGoogle Scholar
R Developmnet Core Team (2004). R: A language and environment for statistical computing. Vienna: R Foundation for Statistical Computing.Google Scholar
Raftery, A.E. (1996). Approximate Bayes factors and accounting for model uncertainty in generalised linear models. Biometrika, 83, 251–266.CrossRefGoogle Scholar
Richarson, S., Green, D.J. (1997). On Bayesian analysis of mixture with unknown numbers of components (with discussion). Journal of the Royal Statistical Society, 59, 731–792.CrossRefGoogle Scholar
Roberts, G.O. (1996). Markov chain concepts related to sampling algorithm. In Gilks, W.R., Richardson, S., Spiegelhalter, D.J. (Eds.), Markov chain Monte Carlo in practice (pp. 45–58). London: Chapman and Hall.Google Scholar
Rosenbaum, P.R., Rubin, D.B. (1985). Constructing a control group using multivariate matched sampling incorporating the propensity score. American Statistician, 39, 33–38.CrossRefGoogle Scholar
Schines, R., Hoijtink, H., Bromsma, A. (1999). Bayesian estimation and testing of structural equation models. Psychometrika, 64, 37–52.CrossRefGoogle Scholar
Schumacker, R.E., Marcoulides, G.A. (1998). Interaction and nonlinear effects in structural equation modelling, Hillsdale, NJ: Erlbaum.Google Scholar
Sibylle, S., Ligges, U., Gelman, A. (2005). R2WinBUGS: A package for running WinBUGS from R. Journal of Statistical Software, 12, 1–17.Google Scholar
Song, X.Y., Lee, S.Y. (2002). Analysis of structural equation model with ignorable missing continuous and polytomous data. Psychometrika, 67, 261–288.CrossRefGoogle Scholar
Tanner, M.A., Wong, W.H. (1987). The calculation of posterior distributions by data augmentation (with discussion). Journal of the American Statistical Association, 82, 528–550.CrossRefGoogle Scholar
Wall, M.M., Amemiya, Y. (2000). Estimation for polynomial structural equation models. Journal of the American Statistical Association, 95, 929–940.CrossRefGoogle Scholar
Wei, G.C., Tanner, M.A. (1990). A Monte Carlo implementation of the EM algorithm and the poor man’s data augmentation algorithm. Journal of the American Statistical Association, 85, 699–704.CrossRefGoogle Scholar
World Values Study Group (1994). World Values Survey, 1981–1984 and 1990–1993. ICPSR version. Ann Arbor, MI: Institute for Social Research [producer], 1994. Ann Arbor, MI: Interuniversity Consortium for Political and Social Research [distributor], 1994.Google Scholar