Testing Hypotheses about Correlation Matrices in General MANOVA Designs

TL;DR

This paper develops new statistical tests for hypotheses about correlation matrices in MANOVA designs, requiring minimal assumptions and including bootstrap methods to enhance small sample performance.

Contribution

It introduces a flexible approach for testing various correlation matrix hypotheses, including equal and specific structures, with theoretical justification and improved small sample performance.

Findings

New test statistics outperform existing methods in simulations

Bootstrap technique enhances small sample accuracy

Procedures for testing equal correlation and covariance matrices

Abstract

Correlation matrices are an essential tool for investigating the dependency structures of random vectors or comparing them. We introduce an approach for testing a variety of null hypotheses that can be formulated based upon the correlation matrix. Examples cover MANOVA-type hypothesis of equal correlation matrices as well as testing for special correlation structures such as, e.g., sphericity. Apart from existing fourth moments, our approach requires no other assumptions, allowing applications in various settings. To improve the small sample performance, a bootstrap technique is proposed and theoretically justified. Based on this, we also present a procedure to simultaneously test the hypotheses of equal correlation and equal covariance matrices. The performance of all new test statistics is compared with existing procedures through extensive simulations.