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Bayesian Analysis of Multiple Group Nonlinear Structural Equation Models with Ordered Categorical and Dichotomous Variables: A Survey

Author Affiliations

  • 1Department of Mathematical Science, Faculty of Science, University Teknologi Malaysia, 81310, Skudai, Johor, MALAYSIA
  • 2Department of Operation Management Technique, Technical College of Management, Foundation of Technical Education, Mosul, IRAQ

Res. J. Mathematical & Statistical Sci., Volume 3, Issue (12), Pages 1-11, December,12 (2015)


This paper is designed to give a complete overview of the literature that is available, as it relates to application of the Bayesian analysis model to investigate multiple group nonlinear structural equation models, also known as SEMs, including those having ordered categorical, dichotomous and categorical-dichotomous mixed variables. It will also work to summarize Bayesian multiple group nonlinear SEMs with nonlinear covariate variables, and latent variables in the structural model and both linear covariant and latent variable sin the measurement models. More specifically, it will be suggested that using hidden continuous normal distribution, including both right and left censoring and truncation, and interval censoring and truncation, can improve the Bayesian approach to multiple group nonlinear structural equation models when solving problems using ordered categorical and dichotomous data.


  1. Khine M.S., Application of Structural Equation Modeling in Educational Research and Practice, Springer ISBN 946209330X, (2013)
  2. Byrne B.M., Structural Equation Modeling with Amos: Basic Concepts, Applications, and Programming, New York, NY [u.a.]: Routledge, ISBN 9780805863727 0805863729 9780805863734 0805863737, (2010)
  3. Wang J. and Wang X., Structural Equation Modeling: Applications Using Mplus, John Wiley and Sons, ISBN 1118356314, (2012)
  4. Muthén L.K. and Muthén B.O., Mplus: statistical analysis with latent variables, user’s guide; [Version 7], Muthén and Muthén (2012)
  5. Lee S.Y., Song X.Y., Skevington S. and Hao Y.T., Application of structural equation models to quality of life, Structural equation modeling, 12(3), 435–453, ISSN 1070-5511, (2005)
  6. Song X. and Lee S.Y., A multivariate probit latent variable model for analyzing dichotomous responses, Statistica Sinica. 15(3), 645. ISSN 1017-0405, (2005)
  7. Fox J., Teacher’s Corner: Structural Equation Modeling with the sem Package in R. Structural Equation Modeling: A Multidisciplinary Journal, 13(3), 465–486. ISSN 1070-5511 1532-8007, (2006)
  8. Asparouhov T. and Muth˙en B., Exploratory structural equation modeling, Structural Equation Modeling: A Multidisciplinary Journal, 16(3), 397–438, (2009)
  9. Yang Y. and Green S.B., A Note on Structural Equation Modeling Estimates of Reliability, Structural Equation Modeling: A Multidisciplinary Journal, 17(1), 66–81.ISSN 1070-5511 1532-8007, (2010)
  10. Iacobucci D., Structural Equations Modeling: Fit Indices, Sample Size, and Advanced Topics, Journal of Consumer Psychology, 20(1), 90–98. ISSN 10577408, (2010)
  11. Markus K.A., Structural Equations and Causal Explanations: Some Challenges for Causal SEM, Structural Equation Modeling: A Multidisciplinary Journal, 17(4), 654–676. ISSN 1070-5511 1532-8007, (2010)
  12. Hildreth L., Residual Analysis for Structural Equation Modeling, Ph.D. Thesis, (2013)
  13. Wu J.Y. and Kwok O.M., Using SEM to Analyze Complex Survey Data: A Comparison between DesignBased Single-Level and Model-Based Multilevel Approaches, Structural Equation Modeling: A Multidisciplinary Journal, 19(1), 16–35. ISSN 1070-5511 1532-8007, (2012)
  14. Paul W.L. and Anderson M.J., Causal modeling with multivariate species data, Journal of Experimental Marine Biology and Ecology, 448, 72–84. ISSN 0022-0981, (2013)
  15. Lee S.Y. and Song X.Y., Basic and Advanced Structural Equation Models for Medical and Behavioural Sciences, Hoboken: Wiley, ISBN 9780470669525 0470669527, (2012)
  16. Lee S.Y. and Song X.Y., Model Comparison of Nonlinear Structural Equation Models with Fixed Covariates, Psychometrik, 68(1), 27–47, (2003b)
  17. Lee S.Y. and Tang, N.S., Analysis of Nonlinear Structural Equation Models with Nonignorable Missing Covariates and Ordered Categorical Data, Statistica Sinica, 16(4), 1117. ISSN 1017-0405, (2006)
  18. Lee S.Y., Song X.Y., Cai J.H., So W.Y., Ma C.W. and Chan C.N.J., Non‐linear structural equation models with correlated continuous and discrete data, British Journal of Mathematical and Statistical Psychology, 62(2), 327-347, (2009)
  19. Lee S.Y. and Song X.Y., Maximum Likelihood Analysis of a Two-Level Nonlinear Structural Equation Model With Fixed Covariates, Journal of Educational and Behavioral Statistics, 30(1), 1–26. ISSN 1076-9986, (2005)
  20. Henseler J. and Chin W.W., A Comparison of Approaches for the Analysis of Interaction Effects Between Latent Variables Using Partial Least Squares Path Modeling, Structural Equation Modeling: A Multidisciplinary Journal, 17(1), 82– 109. ISSN 1070-5511 1532-8007, (2010)
  21. Wen Z., Marsh H.W. and Hau K.T., Structural Equation Models of Latent Interactions: An Appropriate Standardized Solution and Its Scale-Free Properties, Structural Equation Modeling: A Multidisciplinary Journal, 17(1), 1–22. ISSN 1070-5511 1532-8007, (2010)
  22. Codd C.L., Nonlinear Structural Equation Models: Estimation and Applications, Ph.D. Thesis, The Ohio State University, (2011)
  23. Pek J., Losardo D. and Bauer D.J., Confidence Intervals for a Semiparametric Approach to Modeling Nonlinear Relations among Latent Variables, Structural Equation Modeling: A Multidisciplinary Journal, 18(4), 537–553. ISSN 1070-55111532-8007, (2011)
  24. Lee S.Y., Bayesian Analysis of Nonlinear Structural Equation Models with Nonignorable Missing Data, Psychometrika, 71(3), 541–564. ISSN 0033-3123 1860-0980, (2006)
  25. Rabe-Hesketh S., Skrondal A. and Pickles A., Generalized multilevel structural equation modeling, Psychometrika, 69(2), 167–190. ISSN 0033-3123, (2004)
  26. Song X.Y. and Lee S.Y., A Maximum Likelihood Approach for Multisample Nonlinear Structural Equation Models with Missing Continuous and Dichotomous Data, Structural Equation Modeling, 13(3), 325–351. ISSN 1070- 5511, (2006b)
  27. Koh K.H. and Zumbo B.D., Multi-Group Confirmatory Factor Analysis for Testing Measurement Invariance in Mixed Item Format Data, Journal of Modern Applied Statistical Methods, 7(2), 12. ISSN 1538-9472, (2008)
  28. Song X.Y., Lee S.Y. and Hser Y., A two-level structural equation model approach for analyzing multivariate longitudinal responses, Statistics in medicine, 27(16), 3017–3041. ISSN 1097-0258, (2008)
  29. Kline R.B., Principles and Practice of Structural Equation Modeling. New York: Guilford Publications, ISBN 9781606238769 1606238760, (2011)
  30. Muth˙en B. and Asparouhov T., Latent Variable Analysis with Categorical Outcomes: Multiple-Group and Growth Modeling in Mplus, Mplus Web Notes, 4(5), 1–22 (2002)
  31. Brown T.A., Confirmatory Factor Analysis for Applied Research, (2006)
  32. Skrondal A. and Rabe-Hesketh S., Structural equation modeling: categorical variables, Wiley Online Library, (2005)
  33. Montfort K.V., Mooijaart A. and Meijerink F., Estimating Structural Equation Models with Non-Normal Variables by Using Transformations, Statistica Neerlandica, 63(2), 213–226. ISSN 00390402 14679574, (2009)
  34. Deniz E., Bozdogan H. and Katragadda S., Structural equation modeling (SEM) of categorical and mixed-data using the Novel Gifi transformations and information complexity (ICOMP) criterion, Journal of the School of Business Administration, Istanbul University, 40(1), 86– 123. ISSN 1303-1732, (2011)
  35. Kim E.S. and Yoon M., Testing Measurement Invariance: A Comparison of Multiple-Group Categorical CFA and IRT, Structural Equation Modeling: A Multidisciplinary Journal, 18(2), 212–228. ISSN 1070-5511 1532-8007, (2011)
  36. Poon W.Y. and Wang H.B., Latent variable models with ordinal categorical covariates, Statistics and Computing, 22(5), 1135–1154. ISSN 0960-3174, (2012)
  37. Rhemtulla M., Brosseau Liard P.E. and Savalei V., When Can Categorical Variables be Treated as Continuous? A Comparison of Robust Continuous and Categorical Sem Estimation Methods under Suboptimal Conditions, Psychological Methods, 17(3), 354. ISSN 1939-1463, (2012)
  38. Song X.Y. and Lee S.Y., Bayesian Analysis of Two-level Nonlinear Structural Equation Models with Continuous and polytomous data, British Journal of Mathematical and Statistical Psychology, 57(1), 29–52, (2004)
  39. Lee S.Y., Song X.Y. and Tang N.S., Bayesian Methods for Analyzing Structural Equation Models with Covariates, Interaction, and Quadratic Latent Variables, Structural Equation Modeling: A Multidisciplinary Journal, 14(3), 404– 434. ISSN 1070-5511 1532-8007, (2007)
  40. Song X.Y. and Lee S.Y., Bayesian analysis of latent variable models with non-ignorable missing outcomes from exponential family, Statistics in Medicine, 26(3), 681–693, (2007)
  41. Lee S.Y. and Xia Y.M., A Robust Bayesian Approach for Structural Equation Models with Missing Data, Psychometrika, 73(3), 343–364, ISSN 0033-3123 1860- 0980, (2008)
  42. Song, X.Y. and Lee S.Y., A Bayesian Approach for Analyzing Hierarchical Data with Missing Outcomes Through Structural Equation Models, Structural Equation Modeling, 15(2), 272–300. ISSN 1070-5511, (2008)
  43. Lee, S.Y. and Song X.Y., On Bayesian Estimation and Model Comparison of an Integrated Structural Equation Model, Computational Statistics and Data Analysis, 52(10), 4814–4827. ISSN 01679473, (2008)
  44. Song X.Y., Xia Y.M. and Lee S.Y., Bayesian semi Parametric analysis of structural equation models with mixed continuous and unordered categorical variables, Statistics in Medicine, 28(17), 2253–2276, (2009)
  45. Cai J.H. and Song X.Y., Bayesian analysis of mixtures in structural equation models with non‐ignorable missing data, British Journal of Mathematical and Statistical Psychology, 63(3), 491-508, (2010)
  46. Stokes Riner A., Residual Diagnostic Methods forBayesian Structural Equation Models, Ph.D. Thesis.University of Rochester, (2009)
  47. Asparouhov T. and Muth˙en B., Bayesian analysis oflatent variable models using Mplus, Retrieved June, 17,2014. (2010)
  48. Yang M. and Dunson D.B., Bayesian SemiparametricStructural Equation Models with Latent Variables, Psychometrika, 75(4), 675–693. ISSN 0033-3123 1860-0980, (2010)
  49. Song X.Y., Lu Z.H., Hser Y.I. and Lee S.Y., A BayesianApproach for Analyzing Longitudinal Structural EquationModels, Structural Equation Modeling: AMultidisciplinary Journal, 18(2), 183–194. ISSN 1070-5511 1532-8007.(2011a)
  50. Song X.Y., Xia Y.M., Pan J.H. and Lee S.Y., ModelComparison of Bayesian Semiparametric and ParametricStructural Equation Models, Structural EquationModeling: A Multidisciplinary Journal, 18(1), 55–72.ISSN 1070-5511 1532-8007. (2011b)
  51. Wang Y.F. and Fan T.H., A Bayesian analysis on timeseries structural equation models, Journal of StatisticalPlanning and Inference, 141(6), 2071–2078. ISSN 0378-3758. (2011)
  52. Song X.Y., Tang N.S. and Chow S.M., A Bayesianapproach for generalized random Coefficient structuralequation Models for Longitudinal data with AdjacentTime effects, Computational Statistics and Data Analysis, 56(12), 4190– 4203. ISSN 0167-9473, (2012)
  53. Song X.Y. and Lee S.Y., A Tutorial on the BayesianApproach for Analyzing Structural Equation Models, Journal of Mathematical Psychology, 56(3), 135–148.ISSN 00222496, (2012)
  54. Chen J., Liu P. and Song, X.Y., Bayesian Diagnostics ofTransformation Structural Equation Models, Computational Statistics and Data Analysis, 68, 111– 128.ISSN 0167-9473, (2013)
  55. Yanuar F., Ibrahim K. and Jemain A.A., BayesianStructural Equation Modeling for the Health Index, Journal of Applied Statistics, 40(6), 1254–1269, ISSN0266-4763 1360-0532, (2013)
  56. Lee S.Y., Structural Equation Modeling: A BayesianApproach, Chichester, England; Hoboken, NJ: Wiley.ISBN 9780470024232 0470024232, (2007)
  57. Lee S.Y. and Song X.Y., Bayesian Analysis of StructuralEquation Models with Dichotomous Variables, Statisticsin Medicine, 22(19), 3073–3088. ISSN 1097- 0258,(2003a)
  58. Song X.Y. and Lee S.Y., Bayesian analysis of structuralequation models with nonlinear covariates and latent variables, Multivariate Behavioral Research, 41(3), 337–365. ISSN 0027-3171, (2006a)
  59. Lee S.Y., Song, X.Y. and Cai J.H., A Bayesian Approachfor Nonlinear Structural Equation Models withDichotomous Variables Using Logit and Probit Links, Structural Equation Modeling, 17(2), 280–302. ISSN1070-5511, (2010)
  60. Cai, J.-H., Song, X.-Y. and Lee, S.-Y, Bayesian Analysisof Nonlinear Structural Equation Models with MixedContinuous, Ordered and Unordered Categorical andNonignorable Missing Data, Statistics and its Interface, 1,99–114, (2008)
  61. Li, Y. and Yang, A., A Bayesian Criterion-Based Statisticfor Model Selection of Structural Equation Models withOrdered Categorical Data, International Journal ofModeling and Optimization, 1(2), (2011)
  62. Lu, B., Song, X.-Y. and Li, X.-D., Bayesian Analysis ofMulti-Group Nonlinear Structural Equation Models withApplication to Behavioral Finance, Quantitative Finance,12(3), 477–488, ISSN 1469-7688 1469-7696, (2012)
  63. Song X.Y., Lu Z.H., Cai J.H. and Ip E. H.S., A BayesianModeling Approach for Generalized SemiparametricStructural Equation Models, Psychometrika, 78(4), 624–647. ISSN 0033-3123, (2013)