Bayesian Estimation and Model Selection of Multivariate Linear Model with Polytomous Variables

This article provides a Bayesian analysis of the multivariate linear model with polytomous variables. The computational burden due to the intractable multiple integrals induced by the polytomous variables and the model is solved by augmenting the underlying latent continuous measurements of the observed polytomous data. A Gibbs sampler algorithm is implemented to produce the Bayesian estimate. A Bayes factor approach is proposed for model selection, and it is approximated by the Bayesian information criterion (BIC) via the bridge sampling. The proposed methodology is illustrated by examples using multivariate linear regression and multivariate two-way analysis of variance with real data.

[1]  Adrian E. Raftery,et al.  Bayesian Model Selection in Structural Equation Models , 1992 .

[2]  J. Berger.  Statistical Decision Theory and Bayesian Analysis , 1988 .

[3]  J. H. Schuenemeyer,et al.  Generalized Linear Models (2nd ed.) , 1992 .

[4]  U. Olsson.  On The Robustness Of Factor Analysis Against Crude Classification Of The Observations. , 1979, Multivariate behavioral research.

[5]  Jian Qing Shi,et al.  Maximum Likelihood Estimation of Two‐Level Latent Variable Models with Mixed Continuous and Polytomous Data , 2001 .

[6]  L. Wasserman,et al.  Computing Bayes Factors by Combining Simulation and Asymptotic Approximations , 1997 .

[7]  A. Gelfand,et al.  Bayesian Model Choice: Asymptotics and Exact Calculations , 1994 .

[8]  S. Chib.  Marginal Likelihood from the Gibbs Output , 1995 .

[9]  B. Muthén.  A general structural equation model with dichotomous, ordered categorical, and continuous latent variable indicators , 1984 .

[10]  Jian Qing Shi,et al.  Bayesian sampling‐based approach for factor analysis models with continuous and polytomous data , 1998 .

[11]  Peter M. Bentler,et al.  Full maximum likelihood analysis of structural equation models with polytomous variables , 1990 .

[12]  Wai-Yin Poon,et al.  Maximum likelihood estimation of multivariate polyserial and polychoric correlation coefficients , 1988 .

[13]  D. Rubin,et al.  Inference from Iterative Simulation Using Multiple Sequences , 1992 .

[14]  Xiao-Li Meng,et al.  SIMULATING RATIOS OF NORMALIZING CONSTANTS VIA A SIMPLE IDENTITY: A THEORETICAL EXPLORATION , 1996 .

[15]  G. C. Tiao,et al.  Bayesian inference in statistical analysis , 1973 .

[16]  Charles J. Geyer,et al.  Practical Markov Chain Monte Carlo , 1992 .

[17]  Scott L. Zeger,et al.  Generalized linear models with random e ects: a Gibbs sampling approach , 1991 .

[18]  Karl G. Jöreskog,et al.  Lisrel 8: Structural Equation Modeling With the Simplis Command Language , 1993 .

[19]  Sylvia Richardson,et al.  Inference and monitoring convergence , 1995 .

[20]  Peter M. Bentler,et al.  Structural equation models with continuous and polytomous variables , 1992 .

[21]  Xiao-Li Meng,et al.  Fitting Full-Information Item Factor Models and an Empirical Investigation of Bridge Sampling , 1996 .

[22]  John Geweke,et al.  Efficient Simulation from the Multivariate Normal and Student-t Distributions Subject to Linear Constraints and the Evaluation of Constraint Probabilities , 1991 .

[23]  L. Tierney,et al.  Fully Exponential Laplace Approximations to Expectations and Variances of Nonpositive Functions , 1989 .

[24]  Beth A. Reboussin,et al.  An estimating equations approach for the LISCOMP model , 1998 .

[25]  G. Schwarz.  Estimating the Dimension of a Model , 1978 .

[26]  P. McCullagh,et al.  Generalized Linear Models, 2nd Edn. , 1990 .

[27]  A. Raftery.  Choosing Models for Cross-Classifications , 1986 .

[28]  Donald Geman,et al.  Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[29]  S Y Lee,et al.  Hypothesis Testing and Model Comparison in Two-level Structural Equation Models , 2001, Multivariate behavioral research.

[30]  S Y Lee,et al.  Bayesian estimation and test for factor analysis model with continuous and polytomous data in several populations. , 2001, The British journal of mathematical and statistical psychology.

[31]  F. Drasgow,et al.  The polyserial correlation coefficient , 1982 .

[32]  R. Kass,et al.  Approximate Bayes Factors and Orthogonal Parameters, with Application to Testing Equality of Two Binomial Proportions , 1992 .

[33]  J. Berger,et al.  Testing Precise Hypotheses , 1987 .

[34]  Wai-Yin Poon,et al.  Maximum likelihood estimation of polyserial correlations , 1986 .

[35]  A. Zellner.  An Introduction to Bayesian Inference in Econometrics , 1971 .

[36]  Ulf Olsson,et al.  Maximum likelihood estimation of the polychoric correlation coefficient , 1979 .

[37]  Sylvia Richardson,et al.  Markov Chain Monte Carlo in Practice , 1997 .

[38]  Model Selection in Structural Equation Models With Continuous and Polytomous Data , 2001 .

[39]  L. Tierney,et al.  Accurate Approximations for Posterior Moments and Marginal Densities , 1986 .

[40]  P M Bentler,et al.  A two-stage estimation of structural equation models with continuous and polytomous variables. , 1995, The British journal of mathematical and statistical psychology.

[41]  S Y Lee,et al.  Statistical analysis of nonlinear structural equation models with continuous and polytomous data. , 2000, The British journal of mathematical and statistical psychology.