A Bayesian analysis of finite mixtures in the LISREL model

Psychometrika

Volumn 66, Issue 1, 2001, Pages 133-152

  Zhu, Hong Tu ^b   Lee, Sik Yum ^a  

^a CHINESE UNIVERSITY OF HONG KONG   (Hong Kong)

^b NONE

Author keywords

Bayesian analysis; Bayesian classification; Conditional distributions; Finite mixtures; Gibbs sampler; Goodness of ft assessment; LISREL models; Residual and outlier analyses

Indexed keywords

EID: 0035286078     PISSN: 00333123     EISSN: None     Source Type: Journal    
DOI: 10.1007/BF02295737     Document Type: Article

References (65)

1. Albert, J.H., & Chib, S. (1993). Bayesian analysis of binary and polychotomous response data. Journal of the American Statistical Association, 88, 669-679.
2. Albert, J.H., & Chib, S. (1995). Bayesian residual analysis for binary regression models. Biometrika, 82, 747-759.
3. Arminger, G., & Muthén, B.O. (1998). A Bayesian approach to nonlinear latent variable models using the Gibbs sampler and the Metropolis-Hastings algorithm. Psychometrika, 63, 271-300.
4. Barnett, V., & Lewis, T. (1984). Outliers in statistical data (2nd ed). Chichester: Wiley & Sons.
5. Bentler, P.M. (1983) Some contributions to efficient statistics for structural models: Specification and estimation of moment structures. Psychometrika, 48, 493-517.
6. Bentler, P.M. (1992). EQS: Structural equation program manual. Los Angeles, CA: BMDP Statistical Software.
7. Besag, J., Green, P., Higdon, D., & Mengersen, K. (1995). Bayesian computation and stochastic systems. Statistical Science, 10, 3-66.
8. Bollen, K.A., & Long, J.S. (Eds.). (1993). Testing structural equation models. Newbury Park, CA: Sage.
9. Broemeling, L.D. (1985). Bayesian analysis of linear models. New York, NY: Marcel Dekker.
10. Browne, M.W. (1984). Asymptotically distribution-free methods in the analysis of covariance structures, British Journal of Mathematical and Statistical Psychology, 37, 62-83.
11. Byrne, B.M. (1994). Structural equation modeling with EQS and EQS/Windows, Thousand Oaks, CA: Sage.
12. Chaloner, K. (1991). Bayesian residual analysis in the presence of censoring. Biometrika, 78, 637-644.
13. Chaloner, K., & Brant, R. (1988). A Bayesian approach to outlier detection and residual analysis. Biometrika, 75, 651-659.
14. Cook, R.D., & Weisberg, S. (1982). Residuals and influence in regression. London: Chapman and Hall.
15. Crawford, S.L., DeGroot, M.H., Kadane, J.B., & Small, M.J. (1992). Modeling lake chemistry distributions: Approximate Bayesian methods for estimating a finite mixture model. Techometrics, 34, 441-453.
16. Day, N.E. (1969). Estimating the components of a mixture of normal distributions, Biometrika, 56, 463-474.
17. 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.
18. Diebolt, J., & Robert, C.P. (1994). Estimation of finite mixture distributions through Bayesian sampling. Journal of the Royal Statistical Society, Series B, 56, 163-175.
19. Do, K., & McLachlan, G.J. (1984). Estimation of mixing proportions: A case study. Applied Statistics, 33, 134-140.
20. Dolan, C.V., & van der Maas, Han L.J. (1998). Fitting multivariate normal finite mixtures subject to structural equation modeling. Psychometrika, 63, 227-254.
21. Ganesalingam, S., & McLachlan, G.J. (1981). Some efficiency results for the estimation of the mixing proportion in a mixture of two normal distributions. Biometrics, 37, 23-33.
22. Gelfand, A.E., & Smith, A.F.M. (1990). Sampling-based approaches to calculating marginal densities. Journal of the American Statisiical Association, 85, 398-409.
23. Gelman, A., Carlin, J.B., Stern, H.S., & Rubin, D.B. (1995). Bayesian data analysis. London: Chapman and Hall.
24. Gelman, A., Meng, X.L., & Stern. H. (1996). Posterior predictive assessment of model fitness via realized discrepancies. Statistics Sinica. 6, 733-807.
25. Gelman, A., & Rubin, D.B. (1992). Inference from iterative simulation using multiple sequences. Statistical Science, 7, 457-472.
26. 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.
27. Geyer, C.J. (1992). Practical Markov chain Monte Carlo. Statistical Science, 7, 473-511.
28. Gilks, W.R., Richardson, S., & Spiegelhalter, D.J. (1996). Introducing Markov chain Monte Carlo. In W. R. Gilks, S. Richardson and D. J. Spiegelhalter (Eds.), Markov Chain Monte Carlo in practice (pp. 1-19). London: Chapman and Hall.
29. Hathaway, R.J. (1985). A constrained formulation of maximum-likelihood estimation for normal mixture distributions. The Annals of Statistics, 13, 795-800.
30. Hasselblad, V. (1966). Estimation of parameters for a mixture of normal distributions, Technometrics, 8, 431-444.
31. Hawkins, D.M. (1980). Identification of outliers. London: Chapman and Hall.
32. Hoyle, R. (Ed.). (1995). Structural equation modeling: Concepts, issues and applications. Thousand Oaks, CA: Sage.
33. Jedidi, K., Jagpal, H.S., & DeSarbo, W.S. (1997a). Finite-mixture structural equation models for response-based segmentation and unobserved heterogeneity. Marketing Science, 16, 39-59.
34. Jedidi, K., Jagpal, H.S., & DeSarbo, W.S. (1997b). STEMM: A general finite mixture structural equation model. Journal of Classification, 14, 23-50.
35. Jöreskog, K.G. (1978). Structural analysis of covariance and correlation matrices. Psychometrika, 43, 443-477.
36. Jöreskog, K.G., & Sorbom, D. (1996). LISREL 8: Structural equation modeling with the SIMPLIS command language. Hove and London: Scientific Software International.
37. Lee, S.Y. (1981). A Bayesian approach to confirmatory factor analysis. Psychometrika, 46, 153-160.
38. Lee, S.Y., & Poon, W.Y. (1992). Two-level analysis of covariance structures for unbalanced designs with small level-one samples. British Journal of Mathematical and Statistical Psychology, 45, 109-124.
39. Lee, S.Y., & Tsui, K.L. (1982). Covariance structure analysis in several populations. Psychometrika, 57, 409-426.
40. Lindsay, B.G. (1989). Moment matrices: Applications in mixtures. The Annals of Statistics, 17, 722-740.
41. Lindsay, B.G., & Basak, P. (1993). Multivariate normal mixtures: A fast consistent method of moments. Journal of the American Statistical Association, 88, 468-476.
42. Lindley, D.V., & Smith, A.F.M. (1972). Bayes estimates for the linear model (with discussion). Journal of the Royal Statistical Society, Series B, 34, 1-42.
43. Louis, T.A. (1982). Finding the observed information matrix when using EM algorithm. Journal of the Royal Statistical Society, Series B, 44, 226-233.
44. Meng, X.L. (1994). Posterior predictive p-values. The Annals of Statistics, 22, 1142-1160.
45. Mood, A.M., Graybill, F.A., & Boes, D.C. (1974). Introduction to the theory of statistics (3rd ed.). London: McGraw-Hill.
46. Muthén, B. (1989). Latent variable modeling in heterogeneous populations. Psychometrika, 54, 557-585.
47. Raiffa, H., & Schlaifer, L.R. (1961). Applied statistical secision theory (2nd ed.). Boston, MA: Harvard, Graduate School of Business Administration.
48. Redner, R.A., & Walker, H.F. (1984). Mixture densities, maximum likelihood and the EM algorithm. SIAM Review, 26, 195-239.
49. Richardson, S., & Green, P.J. (1997). On Bayesian analysis of mixtures with an unknown number of components (with discussion). Journal of the Royal Statistical Society, Series B, 59, 731-792.
50. Robert, C.P. (1996). Mixtures of distributions: Inference and estimation. In W.R. Gilks, S. Richardson, & D.J. Spiegelhalter (Eds.), Markov Chain Monte Carlo in practice (pp. 163-187). London: Chapman and Hall.
51. Roeder, K., & Wasserman, L. (1997). Practical Bayesian density estimation using mixtures of normals. Journal of the American Statistical Association, 92, 894-902.
52. Rubin, D.B. (1984). Bayesianly justifiable and relevant frequency calculations for the applied statistician. The Annals of Statistics, 12, 1151-1172.
53. Scheines, R., Hoijtink, H., & Boomsma, A. (1999). Bayesian estimation and testing of structural equation models. Psychometrika, 64, 37-52.
54. Shi, J.Q., & Lee, S.Y. (1998). Bayesian sampling-based approach for factor analysis models with continuous and polytomous data. British Journal of Mathematical and Statistical Psychology, 51, 233-252.
55. Smith, A.F.M., & Makov, U.E. (1978). A quasi-Bayes sequential procedure for mixtures. Journal of the Royal Statistical Society, Series B, 40, 106-112.
56. Titterington, D.M., Smith, A.F.M., & Makov, U.E. (1985). Statistical analysis of finite mixture distributions. Chichester: Wiley & Sons.
57. van Buuren, S. (1997). Fitting ARMA time series by structural equation models. Psychometrika, 62, 215-236.
58. Weiss, R.E. (1994). Residuals and outliers in Bayesian random effects models (Tech. Rep.). Los Angeles, CA: UCLA, School of Public Health, Department of Biostatistics.
59. Weiss, R.E., & Lazaro, C.G. (1992). Residual plots for repeated measures. Statistics in Medicine, 11, 115-124.
60. World Values Survey, 1981-1984 and 1990-1993 (ICPSR version). (1994). Ann Arbor, MI: Institute for Social Research [producer], Inter-university Consortium for Political and Social Research [distributor].
61. Yuan, K.E., & Bentler, P.M. (1997). Mean and covariance structures analysis: theoretical and practical improvements. Journal of the American Statistical Association, 92, 767-774.
62. Yung, Y.F. (1997). Finite mixtures in confirmatory factor-analysis models. Psychometrika, 62, 297-330.
63. Zeger, S.L., & Karim, M.R. (1991). Generalized linear models with random effects: A gibbs sampling approach. Journal of the American Statistical Association, 86, 79-86.
64. Zellner, A. (1971). An Introduction to Bayesian Inference in Econometrics, New York, NY: Wiley & Sons.
65. Zhu, H.T., & Lee, S.Y. (1999). Statistical analysis of nonlinear factor analysis models. British Journal of Mathematical and Statistical Psychology, 52, 225-242.