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

背景资料：* 多元增长曲线模型（GCM），也称为广义方差分析（GMANOVA）模型，可用于分析纵向数据、增长曲线以及其他响应曲线（例如，剂量-反应曲线）。 当平均值由于例如感兴趣的结果的时间或剂量依赖性而被结构化时，该模型出现，这在涉及纵向或剂量响应数据的应用中通常是这种情况。传统的大气环流模型是在正态性假设下发展起来的。然而，在实践中，经常会遇到具有偏态分布的结果。最近的一项模拟研究表明，正态性假设下得到的估计是敏感的偏离正态性，结果表明，估计与增加的偏差和均方误差（MSE），当用于分析偏态数据。** GCM的推断也基于样本量（n）大于时间点数量（p）的假设，其中协方差矩阵假设为正定。然而，在高维数据中，p通常大于n，导致样本协方差矩阵的奇异性，因此传统方法不起作用。近年来，已经开发了用于处理高维数据的方法。虽然这些方法考虑了不同时间点测量值之间的相关性，但这些方法未能考虑时间依赖性，这通常会激发纵向研究，研究人员有兴趣确定随时间的变化。在以前的工作中，我们考虑了两种方法。第一种方法涉及MANOVA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。 虽然使用转换的方法提供了一个框架，将时间依赖性的模型，该方法缺乏统计能力，并导致估计增加的偏见和MSE。 另一方面，基于Moore-Penrose逆的方法提供了增加的功效和精度;并且估计量通常与随机分布在零附近的偏差相关联。然而，模拟结果表明，估计的最优性和测试的性能下降附近的奇异性（当np）。该建议试图解决以前方法的局限性，并为高维纵向数据提供改进的推断。本提案的总体目标是为GCM的参数提供稳健的估计，并提供高维扩展 ** 具体目标是 * 1)多元偏态正态分布下GCM模型参数的驱动估计量 * 2)在多元偏态正态分布下推导扩展GCM的估计量 * 3)为高维情景下的GCM提供最优推理

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 extensions******Specific 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