Multivariate longitudinal data is often encountered in the jobs of statisticians and practitioners. It is challenging to model the covariance matrix due to the complex structure of correlations among multiple responses. For this modeling task, several effective Cholesky decomposition based methods have been studied. However, direct interpretation of the covariation structure among multiple responses is still less well investigated to the best of our knowledge. In this paper, we propose a joint mean-variance correlation modeling method based on the triangular angles parameterization (TAP) for the correlation matrix of bivariate longitudinal data. The proposed unconstrained parameterization is able to automatically eliminate the positive definiteness constraint of the correlation matrix and leads to the aforementioned direct interpretation. Furthermore, the variance matrix is diagonal rather than block-diagonal, so the positive-definiteness constraint of this matrix can be easily satisfied. The entries of the proposed decomposition are modeled by regression models, and the maximum likelihood estimators of regression parameters are obtained. The resulting estimators are shown to be consistent and asymptotically normal. Simulations and a study of poplar growth illustrate that the proposed method performs well.

中文翻译：

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二元纵向数据相关矩阵的三角角参数化

在统计学家和从业者的工作中经常会遇到多元纵向数据。由于多重响应之间相关性的复杂结构，对协方差矩阵进行建模具有挑战性。对于此建模任务，已经研究了几种基于Cholesky分解的有效方法。但是，就我们所知，对多个响应之间的协方差结构的直接解释仍未得到很好的研究。本文针对双变量纵向数据的相关矩阵，提出了一种基于三角角参数化（TAP）的联合均值-方差相关建模方法。所提出的无约束参数化能够自动消除相关矩阵的正定性约束，并导致上述直接解释。此外，方差矩阵是对角线而不是块对角线，因此可以轻松满足该矩阵的正定约束。通过回归模型对建议分解的条目进行建模，并获得回归参数的最大似然估计。结果表明，估计量是一致的并且渐近正态。仿真和杨树生长研究表明，该方法效果良好。