This work has been motivated by a longitudinal data set on HIV CD4 T+ cell counts from Livingstone district, Zambia. The corresponding histogram plots indicate lack of symmetry in the marginal distributions and the pairwise scatter plots show non-elliptical dependence patterns. The standard linear mixed model for longitudinal data fails to capture these features. Thus it seems appropriate to consider a more general framework for modeling such data. In this article, we consider generalized linear mixed models (GLMM) for the marginals (e.g. Gamma mixed model), and temporal dependency of the repeated measurements is modeled by the copula corresponding to some skew-elliptical distributions (like skew-normal/skew-t). Our proposed class of copula based mixed models simultaneously takes into account asymmetry, between-subject variability and non-standard temporal dependence, and hence can be considered extensions to the standard linear mixed model based on multivariate normality. We estimate the model parameters using the IFM (inference function of margins) method, and also describe how to obtain standard errors of the parameter estimates. We investigate the finite sample performance of our procedure with extensive simulation studies involving skewed and symmetric marginal distributions and several choices of the copula. We finally apply our models to the HIV data set and report the findings.

翻译：本研究受赞比亚利文斯顿地区HIV CD4 T+细胞计数纵向数据集启发。相应的直方图显示边际分布缺乏对称性，成对散点图呈现非椭圆依赖模式。标准线性混合模型无法捕捉这些特征，因此有必要构建更通用的建模框架。本文针对边际分布采用广义线性混合模型（如Gamma混合模型），并通过对应偏斜椭圆分布（如偏斜正态/偏斜t分布）的Copula对重复测量的时间依赖性进行建模。我们提出的Copula混合模型可同时处理非对称性、个体间变异及非标准时间依赖性，因此可视为基于多元正态性的标准线性混合模型的扩展。采用边际推断函数（IFM）方法估计模型参数，并推导参数估计的标准误计算过程。通过涵盖偏态与对称边际分布及多种Copula选择的广泛模拟研究，验证了方法的有限样本性能。最后将模型应用于HIV数据集并进行结果报告。