  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.
