Matrix variate p-value in MANOVA

TL;DR

This paper develops a new matrix variate p-value approach for MANOVA, leveraging beta distributions under various algebraic systems, providing exact tests that improve upon traditional approximations.

Contribution

It introduces a novel matrix variate p-value framework applicable to MANOVA, utilizing exact distributions for more accurate multivariate hypothesis testing.

Findings

Exact matrix variate p-values outperform approximate criteria.

The method applies across real, complex, quaternion, and octonion cases.

Demonstrated improved accuracy in classical MANOVA scenarios.

Abstract

The distribution functions of the matricvariate beta type I and II distributions are studied under real normed division algebras. The unified approach for real, complex, quaternions and octonions, also considers general properties and highlights the potential application of the exact emerging upper probabilities $P(\mathbf{B} > \mathbf{\Omega})$ and $P(\mathbf{F} > \mathbf{\nabla})$. In this setting, the matrix probabilities arise naturally as univariate extensions into the so termed matrix variate $p$-values. Then, a new criterion for the general multivariate linear hypothesis test can be proposed under a simple heuristic interpretation. The new technique can be applied in a number of classical statistical tests. In particular, the multivariate analysis of variance (MANOVA) is illustrated in two well known scenarios, and the performance of our exact method is compared with the existing approximated criteria.