A Novel Testing Approach for Differences Among Brain Connectomes
Nicolas Escobar-Velasquez, Jaroslaw Harezlak
TL;DR
The paper addresses comparing groups of brain connectomes modeled as SPD matrices by moving beyond distance-only analyses. It develops the Mahalanobis Affine Invariant Riemannian Metric (MAIRM) and a Riemannian generalization of Wilks' Lambda, establishing estimators, asymptotic distributions, and Euclidean limits. It proves a power advantage over Fréchet ANOVA under a Riemannian normal model, with a non-centrality parameter that scales as $\delta_{Novel}=\frac{n\theta^2}{g\sigma^2}$ versus $\delta_{Fréchet}=O(\theta^4)$ for small location shifts. The framework enables efficient, geometry-aware inference on brain-connectome data, though it relies on stronger distributional assumptions and invites future work on model-checking, covariance heterogeneity, covariates, and computation. Overall, the approach provides a principled method to exploit SPD geometry for more powerful group-difference testing in neuroimaging.
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 Fréchet ANOVA methods.