Novel p-Value Based Multiple Testing Methods for Variable Selection with False Discovery Rate Control

多重检验是现代科学研究中遇到的最常见的统计挑战之一。该项目旨在解决多种测试方法应用中的一些长期存在的问题。其中一个问题是在大量变量中发现对感兴趣的结果有重要影响的变量。由于变量的未知相互依赖性，标准的多种测试方法不适用就是这样一个问题。正在开发的方法旨在提供新的方法来发现重要的变量，无论变量如何相互依赖，并保证平均而言，只有一小部分受控的不重要变量最终成为错误的发现。一个应用实例是在鉴定遗传变异中，在成千上万的遗传变异中，遗传变异可以影响某种疾病。新方法可以帮助识别与治疗干预相对更相关的基因。这些方法的发展背后的基本理论和方法论思想将扩展到解决类似的问题，在其他实验环境中的多种测试方法。该项目的研究成果将纳入课程，有利于本科生和研究生的培训，该研究项目的重点是解决与多重测试有关的重要理论和方法问题。例如，具有高斯噪声的多元线性回归设置下的特征/变量选择，在数据科学中起着重要作用，并且是科学调查中普遍存在的统计框架，通常被框定为多重测试问题。一个基于p值的多重检验方法，无论错误率被认为是控制错误发现的重要解释变量，捕捉解释变量的相关矩阵，而不会失去对错误率的控制，将是最理想的。不幸的是，这样的方法还有待开发的非渐近设置。类似地，对于具有非对角相关矩阵的多变量高斯均值的同时检验的相关问题，在误差率的控制下，基于p值的多重检验方法充分捕获相关信息而不失去对该速率的控制，在文献中基本上是不存在的。这些挑战将通过研究来应对，这些研究交叉了关于多重推理的两个开创性想法：1）使用基于p值的多重检验方法来控制错误发现; 2）使用设计矩阵的仿制品在线性回归设置中进行变量选择。具体地说，该项目旨在开发新的基于p值的错误发现率和其他强大的错误率控制多种测试方法，用于1）在低维和高维设置中具有高斯噪声的多元线性回归中的变量选择;和2）多元高斯均值与一般非线性的同时检验，该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Multiple testing is one of the most common statistical challenges encountered in modern scientific investigations. This project aims at resolving some longstanding issues with application of multiple testing methods. One of these issues arises in the context of discovering, among a large collection of variables, those that are important influences on an outcome of interest. Inapplicability of standard multiple testing methods due to the unknown interdependency of the variables is such an issue. The methods under development aim to provide new approaches to discovering important variables no matter how the variables depend on each other, with the guarantee that, on average, only a small, controlled fraction of unimportant variables end up as false discoveries. An example application is in the identification of genetic variants which, among many thousands of them, can influence a certain disease. The new methods can aid in identifying genes as being relatively more relevant for therapeutic intervention. The fundamental theoretical and methodological ideas behind the development of these methods will be extended towards resolving similar issues with multiple testing methods in other experimental settings as well. The research to be carried out in the project will be incorporated into courses, benefiting the training of undergraduates and graduate students.This research project is focused on addressing important theoretical and methodological issues related to multiple testing. For instance, feature/variable selection under the setting of multiple linear regression with Gaussian noise, which plays an important role in data science and is a ubiquitous statistical framework in scientific investigations, is often framed as a multiple testing problem. A p-value based multiple testing method, irrespective of what error rate is being considered to control the falsely discovered important explanatory variables, capturing the correlation matrix of the explanatory variables in full without losing control over the error rate, would be most ideal. Unfortunately, such methods are yet to be developed in a non-asymptotic setting. Similarly, for the related problem of simultaneous testing of multivariate Gaussian means with non-diagonal correlation matrix, subject to a control of an error rate, a p-value based multiple testing method fully capturing the correlation information without losing control over that rate is largely absent from the literature. The challenges will be met by research that cross-fertilizes two seminal ideas on multiple inference: 1) the use of p-value based multiple testing methods to control false discoveries; and 2) the use of the knockoff of the design matrix for variable selection in linear regression settings. Concretely, the project aims at developing novel p-value based false discovery rate and other powerful error rates controlling multiple testing methods for 1) variable selection in multiple linear regression with Gaussian noise, both in low- and high-dimensional settings; and 2) simultaneous testing of multivariate Gaussian means with a general non-diagonal covariance matrix.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.