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

背景：多元增长曲线模型(GCM)，又称广义方差分析(GMANOVA)模型，可用于纵向数据、增长曲线以及其他响应曲线的分析。剂量-反应曲线)。例如，当由于感兴趣的结果的时间或剂量依赖性而构造平均值时，该模型出现，这在涉及纵向或剂量-反应数据的应用中通常是这种情况。传统的GCM方法是在正态假设下发展起来的。然而，在实践中，经常会遇到分布不对称的结果。最近的一项模拟研究表明，在正态假设下得到的估计量对偏离正态分布很敏感，结果表明，在分析倾斜数据时，估计量与增加的偏差和均方误差(MSE)有关。*GCM的推断也基于样本容量(N)大于时间点数目(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。近年来发展了处理高维数据的方法。尽管这些方法考虑了跨越不同时间点进行的测量之间的相关性，但这些方法未能考虑到时间依赖性，这往往会引发纵向研究，研究人员有兴趣确定随着时间的变化。在以前的工作中，我们考虑了两种方法。第一种方法涉及到马诺瓦模型的转换，然后是经验贝叶斯方法。第二种方法涉及使用Moore-Penrose广义逆。虽然使用变换的方法提供了将时间相关性纳入模型的框架，但该方法缺乏统计能力，并导致估计器的偏差和均方根误差增加。另一方面，基于摩尔-彭罗斯逆的方法提供了更高的功率和精度；估计器通常与在零附近随机分布的偏差相关联。然而，模拟结果表明，在奇点附近(当n≈p时)，估计器的最优性和检验的性能下降。这项建议试图解决以前方法的局限性，并为高维纵向数据提供更好的推断。*本建议的总体目标是为GCM的参数提供稳健的估计并提供高维扩展*具体目标是*1)多元偏态分布下GCM的模型参数的估计量*2)多元偏态正态分布下扩展的GCM的估计量*3)为高维情形下的GCM提供最优推断

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 n≈p). 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