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Bayesian Analysis of Multivariate Longitudinal Ordinal Data Using Multiple Multivariate Probit Models

Received: 1 March 2023     Accepted: 17 March 2023     Published: 28 March 2023
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Abstract

Multivariate longitudinal ordinal data are often involved in longitudinal studies with each individual having more than one longitudinal ordinal measure. However, due to complicated correlation structures within each individual and no explicit likelihood functions, analyzing multivariate longitudinal ordinal data is quite challenging. In this paper, Markov chain Monte Carlo (MCMC) sampling methods are developed to analyze multivariate longitudinal ordinal data by extending multivariate probit (MVP) models for univariate longitudinal ordinal data to multiple multivariate probit models (MMVP) for multivariate longitudinal ordinal data. The identifiable MVP models require the covariance matrix of the latent multivariate normal variables underlying the longitudinal ordinal variables to be a correlation matrix, thus a Metropolis-Hastings (MH) algorithm is usually necessitated, which brings a rigorous task to develop efficient MCMC sampling methods. In contrast to the identifiable MVP models, the non-identifiable MVP models can be constructed to circumvent a MH algorithm to sample a correlation matrix by a Gibbs sampling to sample a covariance matrix, and hence improve the mixing and convergence of the MCMC components. Therefore, both the identifiable MMVP models and the non-identifiable MMVP models for multivariate longitudinal ordinal data are presented, and their corresponding MCMC sampling methods are developed. The performances of these methods are illustrated through simulation studies and an application using data from the Russia Longitudinal Monitoring Survey-Higher School of Economics (RLMS-HSE).

Published in American Journal of Theoretical and Applied Statistics (Volume 12, Issue 1)
DOI 10.11648/j.ajtas.20231201.11
Page(s) 1-12
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2023. Published by Science Publishing Group

Keywords

Multivariate Longitudinal Ordinal Data, MCMC, Multivariate Probit Model, Multiple Multivariate Probit Model, Identification

References
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  • APA Style

    Xiao Zhang. (2023). Bayesian Analysis of Multivariate Longitudinal Ordinal Data Using Multiple Multivariate Probit Models. American Journal of Theoretical and Applied Statistics, 12(1), 1-12. https://doi.org/10.11648/j.ajtas.20231201.11

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    ACS Style

    Xiao Zhang. Bayesian Analysis of Multivariate Longitudinal Ordinal Data Using Multiple Multivariate Probit Models. Am. J. Theor. Appl. Stat. 2023, 12(1), 1-12. doi: 10.11648/j.ajtas.20231201.11

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    AMA Style

    Xiao Zhang. Bayesian Analysis of Multivariate Longitudinal Ordinal Data Using Multiple Multivariate Probit Models. Am J Theor Appl Stat. 2023;12(1):1-12. doi: 10.11648/j.ajtas.20231201.11

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  • @article{10.11648/j.ajtas.20231201.11,
      author = {Xiao Zhang},
      title = {Bayesian Analysis of Multivariate Longitudinal Ordinal Data Using Multiple Multivariate Probit Models},
      journal = {American Journal of Theoretical and Applied Statistics},
      volume = {12},
      number = {1},
      pages = {1-12},
      doi = {10.11648/j.ajtas.20231201.11},
      url = {https://doi.org/10.11648/j.ajtas.20231201.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20231201.11},
      abstract = {Multivariate longitudinal ordinal data are often involved in longitudinal studies with each individual having more than one longitudinal ordinal measure. However, due to complicated correlation structures within each individual and no explicit likelihood functions, analyzing multivariate longitudinal ordinal data is quite challenging. In this paper, Markov chain Monte Carlo (MCMC) sampling methods are developed to analyze multivariate longitudinal ordinal data by extending multivariate probit (MVP) models for univariate longitudinal ordinal data to multiple multivariate probit models (MMVP) for multivariate longitudinal ordinal data. The identifiable MVP models require the covariance matrix of the latent multivariate normal variables underlying the longitudinal ordinal variables to be a correlation matrix, thus a Metropolis-Hastings (MH) algorithm is usually necessitated, which brings a rigorous task to develop efficient MCMC sampling methods. In contrast to the identifiable MVP models, the non-identifiable MVP models can be constructed to circumvent a MH algorithm to sample a correlation matrix by a Gibbs sampling to sample a covariance matrix, and hence improve the mixing and convergence of the MCMC components. Therefore, both the identifiable MMVP models and the non-identifiable MMVP models for multivariate longitudinal ordinal data are presented, and their corresponding MCMC sampling methods are developed. The performances of these methods are illustrated through simulation studies and an application using data from the Russia Longitudinal Monitoring Survey-Higher School of Economics (RLMS-HSE).},
     year = {2023}
    }
    

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  • TY  - JOUR
    T1  - Bayesian Analysis of Multivariate Longitudinal Ordinal Data Using Multiple Multivariate Probit Models
    AU  - Xiao Zhang
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    PY  - 2023
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    DO  - 10.11648/j.ajtas.20231201.11
    T2  - American Journal of Theoretical and Applied Statistics
    JF  - American Journal of Theoretical and Applied Statistics
    JO  - American Journal of Theoretical and Applied Statistics
    SP  - 1
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    PB  - Science Publishing Group
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    UR  - https://doi.org/10.11648/j.ajtas.20231201.11
    AB  - Multivariate longitudinal ordinal data are often involved in longitudinal studies with each individual having more than one longitudinal ordinal measure. However, due to complicated correlation structures within each individual and no explicit likelihood functions, analyzing multivariate longitudinal ordinal data is quite challenging. In this paper, Markov chain Monte Carlo (MCMC) sampling methods are developed to analyze multivariate longitudinal ordinal data by extending multivariate probit (MVP) models for univariate longitudinal ordinal data to multiple multivariate probit models (MMVP) for multivariate longitudinal ordinal data. The identifiable MVP models require the covariance matrix of the latent multivariate normal variables underlying the longitudinal ordinal variables to be a correlation matrix, thus a Metropolis-Hastings (MH) algorithm is usually necessitated, which brings a rigorous task to develop efficient MCMC sampling methods. In contrast to the identifiable MVP models, the non-identifiable MVP models can be constructed to circumvent a MH algorithm to sample a correlation matrix by a Gibbs sampling to sample a covariance matrix, and hence improve the mixing and convergence of the MCMC components. Therefore, both the identifiable MMVP models and the non-identifiable MMVP models for multivariate longitudinal ordinal data are presented, and their corresponding MCMC sampling methods are developed. The performances of these methods are illustrated through simulation studies and an application using data from the Russia Longitudinal Monitoring Survey-Higher School of Economics (RLMS-HSE).
    VL  - 12
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Author Information
  • Department of Mathematical Sciences, Michigan Technological University, Houghton, USA

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