Direction-Projection-Permutation for High-Dimensional Hypothesis Tests

High-dimensional low sample size (HDLSS) data are becoming increasingly common in statistical applications. When the data can be partitioned into two classes, a basic task is to construct a classifier that can assign objects to the correct class. Binary linear classifiers have been shown to be especially useful in HDLSS settings and preferable to more complicated classifiers because of their ease of interpretability. We propose a computational tool called direction-projection-permutation (DiProPerm), which rigorously assesses whether a binary linear classifier is detecting statistically significant differences between two high-dimensional distributions. The basic idea behind DiProPerm involves working directly with the one-dimensional projections of the data induced by binary linear classifier. Theoretical properties of DiProPerm are studied under the HDLSS asymptotic regime whereby dimension diverges to infinity while sample size remains fixed. We show that certain variations of DiProPerm are consistent and that consistency is a nontrivial property of tests in the HDLSS asymptotic regime. The practical utility of DiProPerm is demonstrated on HDLSS gene expression microarray datasets. Finally, an empirical power study is conducted comparing DiProPerm to several alternative two-sample HDLSS tests to understand the advantages and disadvantages of each method.