Applying Gibbs Sampler for Multivariate Hierarchical Linear Model

Among various HLM techniques, the Multivariate Hierarchical Linear Model (MHLM) is desirable to use, particularly when multivariate criterion variables are collected and the covariance structure has information valuable for data analysis. In order to reflect prior information or to obtain stable results when the sample size and the number of groups are not sufficiently large, the Bayes method has often been employed in hierarchical data analysis. In these cases, although the Markov Chain Monte Carlo (MCMC) method is a rather powerful tool for parameter estimation, Procedures regarding MCMC have not been formulated for MHLM. For this reason, this research presents concrete procedures for parameter estimation through the use of the Gibbs samplers. Lastly, several future topics for the use of MCMC approach for HLM is discussed.

• Satoshi Usami

• Gibbs sampler
• Hierarchical Linear Model
• Markov Chain Monte Carlo
• Multivariate Hierarchical Linear Model

[1] Fahrmeir, L., & Raach, A, (2007). A Bayesian Semiparametric Latent
Variable Model for Mixed response. Psychometrica ,72(3), 327-346
[2] Gelman, A. ´╝å Rubin, D. B. (1992). Inference from iterative simulation
using multiple sequences. Statistical Science, 7, 457-472.
[3] Goldstein, H. (1987). Multilevel models in education and social research.
London: Oxford University Press.
[4] Goldstein, H. (2003). Multilevel statistical models (3rd ed.). New York:
Oxford University Press.
[5] Harville, D. A. (2000). Matrix algebra from a statistician's perspective.
Springer.
[6] Hox, J. (2002). Multilevel analysis: Techniques and applications. Mahwah,
NJ: Erlbaum.
[7] Miyazaki, Y. (2007). Application of hierarchical linear models to
educational research and viewpoints of utilizing the results to educational
policies, Japanese Journal for Research on Testing 3 (1), 123-146
[8] Okumura, T. (2007). Sample size determination for hierarchical linear
models considering uncertainty in parameter estimates. Behaviormetrika,
34(2), 79-94.
[9] Omori, Y. (2001). Recent developments in Markov Chain Monte Carlo.
Japan Statistical Society, 31, 305-344.
[10] Magnus, J. R. ´╝å Neudecker, H. (1988). Matrix differential calculus with
applications in statistics and econometrics. Wiley.
[11] Plewis, I. (2005). Modeling behaviour with multivariate multilevel growth
curves. Methodology, 1(2), 71-80.
[12] Rabe-Hesketh, S., Skrondal, A., & Pickles, A. (2004). Generalized
multilevel structural equation modelling. Psychometrika 69, 167-190.
[13] Raudenbush, S. W., ´╝å Bryk, A. S. (2002). Hierarchical linear models.
Applications and data analysis methods. (2nd ed.). Sage.
[14] Seltzer, M. H. (1991). The use of data augmentation in fitting hierarchical
linear models to educational data. Unpublished doctoral dissertation,
University of Chicago.
[15] Seltzer, M. H. (1993). Sensitivity analysis for fixed effects in the
hierarchical model: A Gibbs sampling approach. Journal of Educational
Statistics, 18, 207-235.
[16] Seltzer, M. H., Wong, W. H., ´╝å Bryk, A. S. (1996). Bayesian analysis in
applications of hierarchical models: issues and methods. Journal of
Educational and Behavioral Statistics, 21, 131-167.
[17] Snijders, T. A. B., ´╝å Bosker, R. J. (1999). Multilevel analysis: An
introduction to basic and advanced multilevel modeling. London: Sage.
[18] Tate, R. L., ´╝å Pituch, K. A. (2007). Multivariate hierarchical linear
modeling in randomized field experiments. The Journal of Experimental
Education, 73(4), 317-337
[19] Thum, Y. M. (1997). Hierarchical linear models for multivariate outcomes.
Journal of Educational and Behavioral Statistics, 22, 77-108.