Robust Inference for Multivariate Growth Curve Models and High-Dimensional Extensions

背景：多元增长曲线模型（GCM），也称为广义方差分析（GMANOVA）模型，在纵向数据、增长曲线以及其他响应曲线(如：剂量反应曲线)。例如，由于感兴趣的结果的时间或剂量依赖性，当平均值被结构化时，模型就产生了，这在涉及纵向或剂量-反应数据的应用中经常出现。传统的GCM方法是在正态假设下发展起来的。然而，在实践中，通常会遇到分布偏态的结果。最近的一项模拟研究表明，在正态性假设下推导的估计量对偏离正态性很敏感，其中结果表明，当用于偏斜数据分析时，估计量与增加的偏差和均方误差（MSE）相关。对GCM的推断也是基于样本量(n)大于时间点数量(p)的假设，其中协方差矩阵被假设为正定。然而，在高维数据中，p往往大于n，导致样本协方差矩阵的奇异性，因此传统的方法不起作用。处理高维数据的方法是近年来发展起来的。尽管这些方法解释了在不同时间点所采取的测量之间的相关性，但这些方法未能解释时间依赖性，这通常促使纵向研究，研究人员感兴趣的是确定随时间的变化。在之前的工作中，我们考虑了两种方法。第一种方法涉及到对方差分析模型的转换，然后是经验贝叶斯方法。第二种方法涉及使用Moore-Penrose广义逆。尽管使用转换的方法提供了一个将时间依赖性纳入模型的框架，但该方法缺乏统计能力，导致估计器的偏差和MSE增加。另一方面，基于Moore-Penrose逆的方法提高了功率和精度；估计量通常与一个随机分布在零附近的偏差有关。然而，仿真结果表明，在奇异点附近（当np时），估计器的最优性和测试性能下降。该建议试图解决以往方法的局限性，并提供改进的高维纵向数据推断。本文的总体目标是提供GCM参数的鲁棒估计和高维扩展，具体目标是：1)驱动多元偏态正态分布下GCM模型参数的估计量；2)导出多元偏态正态分布下扩展GCM的估计量；3)提供高维场景下GCM的最优推断

Background: Multivariate Growth Curve Model (GCM), also known as the Generalized Analysis of Variance (GMANOVA) model, is useful in the analysis of longitudinal data, growth curves as well as other response curves (eg. dose-response curves). The model arises when the mean is structured due to, for instance, time or dose dependency of the outcome of interest, which is often the case in applications involving longitudinal or dose-response data. Traditional methods for the GCM are developed under the assumption of normality. In practice, however, it is common to encounter outcomes with skewed distributions. A recent simulation study reveals that estimators derived under the normality assumption are sensitive to departure from normality, where the results show that the estimators are associated with increased bias and mean squared error (MSE), when used in the analysis of skewed data. Inference for the GCM is also based on the assumption of larger sample size (n) than the number of time points (p), where the covariance matrix is assumed to be positive definite. In high-dimensional data, however, p is often larger than n, leading to singularity of the sample covariance matrix, and hence traditional approaches do not work. Methods for handling high-dimensional data have been developed in recent years. Although these methods account for correlations among measurements taken across the different time points, the methods fail to account for time dependency, which often motivates longitudinal studies, where researchers are interested to determine the change over time. In previous work, we considered two approaches. The first approach involves a transformation to the MANOVA model followed by an empirical Bayes approach. The second method involves use of the Moore-Penrose generalized inverse. Although the approach using a transformation provided a framework for incorporating time dependency in the model, the method lacks statistical power and lead to estimators with increased bias and MSE. On the other hand, the method based on Moore-Penrose inverse provided increased power and precision; and the estimators are in general associated with a bias randomly distributed around zero. Nevertheless, the simulation results show that the optimallity of the estimators and the performance of the test declines near singularity (when np). This proposal attempts to address the limitations of the previous method and provides improved inference for high-dimensional longitudinal data.The overall objective of this proposal is to provide robust estimators for the parameters of the GCM and provide high-dimensional extensionsSpecific Objectives are to 1) drive estimators for the model parameters of the GCM under multivariate skewed normal distribution 2) derive estimators for the extended GCM under multivariate skewed normal distribution 3) provide an optimal inference for the GCM under high-dimensional scenarios