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