A Geometric Perspective on the Power of Principal Component Association Tests in Multiple Phenotype Studies

TL;DR

This paper introduces a geometric framework for evaluating PCA-based tests in multi-phenotype genetic studies, proposing robust omnibus tests that outperform traditional methods in power and applicability.

Contribution

It develops a novel geometric concept called principal angle and proposes new combined PCA tests that are robust, powerful, and computationally efficient for multi-phenotype analysis.

Findings

Omnibus PCA tests are robust across various scenarios.

Proposed methods outperform traditional PCA in power.

Tests are computationally efficient and implemented in R package MPAT.

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 paper, 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 $0^o$ and is powerless if its principal angle is $90^o$. Without prior knowledge about the true principal angle, each PC can be powerless. We propose linear, non-linear and data-adaptive omnibus tests by combining PCs. 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 {\it 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 data sets collected from four international consortia to demonstrate the effectiveness of the proposed combined PC testing procedures.