Zhang Xiao
Department of Mathematical Sciences, Michigan Technological University, Houghton, MI, 49931, USA.
Aust N Z J Stat. 2024 Sep;66(3):325-346. doi: 10.1111/anzs.12421. Epub 2024 Aug 13.
Multivariate longitudinal ordinal and continuous data exist in many scientific fields. However, it is a rigorous task to jointly analyse them due to the complicated correlated structures of those mixed data and the lack of a multivariate distribution. The multivariate probit model, assuming there is a multivariate normal latent variable for each multivariate ordinal data, becomes a natural modeling choice for longitudinal ordinal data especially for jointly analysing with longitudinal continuous data. However, the identifiable multivariate probit model requires the variances of the latent normal variables to be fixed at 1, thus the joint covariance matrix of the latent variables and the continuous multivariate normal variables is restricted at some of the diagonal elements. This constrains to develop both the classical and Bayesian methods to analyse mixed ordinal and continuous data. In this investigation, we proposed three Markov chain Monte Carlo (MCMC) methods: Metropolis--Hastings within Gibbs algorithm based on the identifiable model, and a Gibbs sampling algorithm and parameter-expanded data augmentation based on the constructed non-identifiable model. Through simulation studies and a real data application, we illustrated the performance of these three methods and provided an observation of using non-identifiable model to develop MCMC sampling methods.
多变量纵向有序和连续数据存在于许多科学领域。然而，由于这些混合数据复杂的相关结构以及缺乏多变量分布，对它们进行联合分析是一项艰巨的任务。多变量概率单位模型假设每个多变量有序数据都有一个多变量正态潜变量，成为纵向有序数据尤其是与纵向连续数据联合分析的自然建模选择。然而，可识别的多变量概率单位模型要求潜正态变量的方差固定为1，因此潜变量和连续多变量正态变量的联合协方差矩阵在一些对角元素上受到限制。这限制了开发经典方法和贝叶斯方法来分析混合有序和连续数据。在本研究中，我们提出了三种马尔可夫链蒙特卡罗（MCMC）方法：基于可识别模型的吉布斯算法中的梅特罗波利斯-黑斯廷斯算法，以及基于构建的不可识别模型的吉布斯抽样算法和参数扩展数据增广。通过模拟研究和实际数据应用，我们展示了这三种方法的性能，并提供了使用不可识别模型开发MCMC抽样方法的观察结果。
