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

Xiao Zhang

doi:doi:10.11648/j.ajtas.20231201.11

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

Xiao Zhang

Published in American Journal of Theoretical and Applied Statistics (Volume 12, Issue 1)

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), 2024. 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
Y1  - 2023/03/28
PY  - 2023
N1  - https://doi.org/10.11648/j.ajtas.20231201.11
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
EP  - 12
PB  - Science Publishing Group
SN  - 2326-9006
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
IS  - 1
ER  -

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Author Information

Xiao Zhang

Department of Mathematical Sciences, Michigan Technological University, Houghton, USA

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Plain Text BibTeX RIS

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}
}

Copy | Download

TY  - JOUR
T1  - Bayesian Analysis of Multivariate Longitudinal Ordinal Data Using Multiple Multivariate Probit Models
AU  - Xiao Zhang
Y1  - 2023/03/28
PY  - 2023
N1  - https://doi.org/10.11648/j.ajtas.20231201.11
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
EP  - 12
PB  - Science Publishing Group
SN  - 2326-9006
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
IS  - 1
ER  -

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