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Bayesian analysis of multivariate t linear mixed models using a combination of IBF and Gibbs samplers


  • Wang, Wan-Lun
  • Fan, Tsai-Hung


The multivariate linear mixed model (MLMM) has become the most widely used tool for analyzing multi-outcome longitudinal data. Although it offers great flexibility for modeling the between- and within-subject correlation among multi-outcome repeated measures, the underlying normality assumption is vulnerable to potential atypical observations. We present a fully Bayesian approach to the multivariate t linear mixed model (MtLMM), which is a robust extension of MLMM with the random effects and errors jointly distributed as a multivariate t distribution. Owing to the introduction of too many hidden variables in the model, the conventional Markov chain Monte Carlo (MCMC) method may converge painfully slowly and thus fails to provide valid inference. To alleviate this problem, a computationally efficient inverse Bayes formulas (IBF) sampler coupled with the Gibbs scheme, called the IBF-Gibbs sampler, is developed and shown to be effective in drawing samples from the target distributions. The issues related to model determination and Bayesian predictive inference for future values are also investigated. The proposed methodologies are illustrated with a real example from an AIDS clinical trial and a careful simulation study.

Suggested Citation

  • Wang, Wan-Lun & Fan, Tsai-Hung, 2012. "Bayesian analysis of multivariate t linear mixed models using a combination of IBF and Gibbs samplers," Journal of Multivariate Analysis, Elsevier, vol. 105(1), pages 300-310.
  • Handle: RePEc:eee:jmvana:v:105:y:2012:i:1:p:300-310
    DOI: 10.1016/j.jmva.2011.10.006

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    References listed on IDEAS

    1. Guo-Liang Tian & Ming Tan & Kai Wang Ng, 2007. "An exact non-iterative sampling procedure for discrete missing data problems," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 61(2), pages 232-242.
    2. David J. Spiegelhalter & Nicola G. Best & Bradley P. Carlin & Angelika van der Linde, 2002. "Bayesian measures of model complexity and fit," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 64(4), pages 583-639.
    3. Wang, Wan-Lun & Fan, Tsai-Hung, 2010. "ECM-based maximum likelihood inference for multivariate linear mixed models with autoregressive errors," Computational Statistics & Data Analysis, Elsevier, vol. 54(5), pages 1328-1341, May.
    4. Tan, Ming & Tian, Guo-Liang & Wang Ng, Kai, 2006. "Hierarchical models for repeated binary data using the IBF sampler," Computational Statistics & Data Analysis, Elsevier, vol. 50(5), pages 1272-1286, March.
    5. Tian, Guo-Liang & Tan, Ming, 2003. "Exact statistical solutions using the inverse Bayes formulae," Statistics & Probability Letters, Elsevier, vol. 62(3), pages 305-315, April.
    6. Barndorff-Nielsen, O. & Schou, G., 1973. "On the parametrization of autoregressive models by partial autocorrelations," Journal of Multivariate Analysis, Elsevier, vol. 3(4), pages 408-419, December.
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    Cited by:

    1. D. Concordet & R. Servien, 2014. "Individual prediction regions for multivariate longitudinal data with small samples," Biometrics, The International Biometric Society, vol. 70(3), pages 629-638, September.
    2. Getachew A. Dagne, 2016. "Bayesian segmental growth mixture Tobit models with skew distributions," Computational Statistics, Springer, vol. 31(1), pages 121-137, March.


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