Identifying Brain Hierarchical Structures Associated with Alzheimer's Disease Using a Regularized Regression Method with Tree Predictors

Yi Zhao, Bingkai Wang, Chin-Fu Liu, Andreia V. Faria, Michael I. Miller, Brian Caffo, Xi Luo

Abstract:

Brain segmentation at different levels is generally represented as hierarchical
trees. Brain regional atrophy at specific levels was found to be marginally associated with Alzheimer’s disease outcomes. In this study, we propose an 𝓁1-type
regularization for predictors that follow a hierarchical tree structure. Considering a tree as a directed acyclic graph, we interpret the model parameters from
a path analysis perspective. Under this concept, the proposed penalty regulates
the total effect of each predictor on the outcome. With regularity conditions,
it is shown that under the proposed regularization, the estimator of the model
coefficient is consistent in 𝓁2-norm and the model selection is also consistent.
When applied to a brain sMRI dataset acquired from the Alzheimer’s Disease
Neuroimaging Initiative (ADNI), the proposed approach identifies brain regions
where atrophy in these regions demonstrates the declination in memory. With
regularization on the total effects, the findings suggest that the impact of atrophy
on memory deficits is localized from small brain regions, but at various levels of
brain segmentation. Data used in preparation of this paper were obtained from
the ADNI database.