ABSTRACT Joint analysis of multiple phenotypes can increase statistical power in genetic association studies. Principal component analysis, as a popular dimension reduction method, especially when the number of phenotypes is high dimensional, has been proposed to analyze multiple correlated phenotypes. It has been empirically observed that the first PC, which summarizes the largest amount of variance, can be less powerful than higher-order PCs and other commonly used methods in detecting genetic association signals. In this article, we investigate the properties of PCA-based multiple phenotype analysis from a geometric perspective by introducing a novel concept called principal angle. A particular PC is powerful if its principal angle is and is powerless if its principal angle is . Without prior knowledge about the true principal angle, each PC can be powerless. We propose linear, nonlinear, and data-adaptive omnibus tests by combining PCs. We demonstrate that the Wald test is a special quadratic PC-based test. We show that the omnibus PC test is robust and powerful in a wide range of scenarios. We study the properties of the proposed methods using power analysis and eigen-analysis. The subtle differences and close connections between these combined PC methods are illustrated graphically in terms of their rejection boundaries. Our proposed tests have convex acceptance regions and hence are admissible. The p-values for the proposed tests can be efficiently calculated analytically and the proposed tests have been implemented in a publicly available R package MPAT. We conduct simulation studies in both low- and high-dimensional settings with various signal vectors and correlation structures. We apply the proposed tests to the joint analysis of metabolic syndrome-related phenotypes with datasets collected from four international consortia to demonstrate the effectiveness of the proposed combined PC testing procedures. Supplementary materials for this article, including a standardized description of the materials available for reproducing the work, are available as an online supplement.

中文翻译：

---

多表型研究中主成分关联检验功效的几何透视

摘要 多种表型的联合分析可以增加遗传关联研究的统计功效。主成分分析作为一种流行的降维方法，特别是当表型数量为高维时，已被提出用于分析多个相关表型。根据经验观察到，总结最大方差的第一个 PC 可能不如高阶 PC 和其他常用的检测遗传关联信号的方法强大。在本文中，我们通过引入一个称为主角的新概念，从几何角度研究基于 PCA 的多表型分析的特性。一个特定的 PC 是强大的，如果它的主角是 ，如果它的主角是 ，则是无能为力的。在没有关于真实主角的先验知识的情况下，每台PC都可以无能为力。我们通过组合 PC 提出线性、非线性和数据自适应综合测试。我们证明 Wald 检验是一种特殊的基于二次 PC 的检验。我们表明，综合 PC 测试在各种场景中都是稳健且强大的。我们使用功效分析和特征分析来研究所提出方法的特性。这些组合 PC 方法之间的细微差异和密切联系以它们的拒绝边界的形式进行了图示。我们提出的测试具有凸接受区域，因此是可以接受的。可以有效地分析计算建议测试的 p 值，并且建议测试已在公开可用的 R 包 MPAT 中实现。我们使用各种信号向量和相关结构在低维和高维设置中进行模拟研究。我们将提议的测试应用于代谢综合征相关表型与从四个国际联盟收集的数据集的联合分析，以证明提议的组合 PC 测试程序的有效性。本文的补充材料，包括对可用于复制作品的材料的标准化描述，可作为在线补充材料获得。