A Novel Testing Approach for Differences Among Brain Connectomes

TL;DR

This paper introduces a new statistical testing method for brain connectome data modeled on the SPD manifold, leveraging manifold-aware models to improve power over traditional distance-based tests.

Contribution

It develops a manifold-valued Mahalanobis distribution and a novel ANOVA test that better captures data structure, outperforming existing distance-based methods.

Findings

The proposed test achieves higher statistical power.

Manifold-based models better capture brain connectome structures.

Theoretical properties of estimators are established.

Abstract

Statistical analysis on non-Euclidean spaces typically relies on distances as the primary tool for constructing likelihoods. However, manifold-valued data admits richer structures in addition to Riemannian distances. We demonstrate that simple, tractable models that do not rely exclusively on distances can be constructed on the manifold of symmetric positive definite (SPD) matrices, which naturally arises in brain connectivity analysis. Specifically, we highlight the manifold-valued Mahalanobis distribution, a parametric family that extends classical multivariate concepts to the SPD manifold. We develop estimators for this distribution and establish their asymptotic properties. Building on this framework, we propose a novel ANOVA test that leverages the manifold structure to obtain a test statistic that better captures the dimensionality of the data. We theoretically demonstrate that our test achieves superior statistical power compared to distance-based Fr\'echet ANOVA methods.