Finding Differentially Covarying Needles in a Temporally Evolving Haystack: A Scan Statistics Perspective

TL;DR

This paper introduces a method combining parametric modeling and scan statistics on graph structures to detect subtle, group-specific differences in the temporal evolution of covariance matrices, with applications to Alzheimer's disease research.

Contribution

It develops a novel framework for identifying differential covariance patterns across groups by generalizing scan statistics to graph structures and analyzing error rates.

Findings

Identified group differences in covariance structures related to Alzheimer's risk factors.

Demonstrated the method's ability to detect signals missed by full-graph analysis.

Provided theoretical bounds on error rates for the proposed statistical tests.

Abstract

Recent results in coupled or temporal graphical models offer schemes for estimating the relationship structure between features when the data come from related (but distinct) longitudinal sources. A novel application of these ideas is for analyzing group-level differences, i.e., in identifying if trends of estimated objects (e.g., covariance or precision matrices) are different across disparate conditions (e.g., gender or disease). Often, poor effect sizes make detecting the differential signal over the full set of features difficult: for example, dependencies between only a subset of features may manifest differently across groups. In this work, we first give a parametric model for estimating trends in the space of SPD matrices as a function of one or more covariates. We then generalize scan statistics to graph structures, to search over distinct subsets of features (graph partitions) whose temporal dependency structure may show statistically significant group-wise differences. We theoretically analyze the Family Wise Error Rate (FWER) and bounds on Type 1 and Type 2 error. On a cohort of individuals with risk factors for Alzheimer's disease (but otherwise cognitively healthy), we find scientifically interesting group differences where the default analysis, i.e., models estimated on the full graph, do not survive reasonable significance thresholds.