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Brain segmentation at different levels is generally represented ashierarchical trees. Brain regional atrophy at specific levels was found to bemarginally associated with Alzheimer's disease outcomes. In this study, wepropose an L1-type regularization for predictors that follow a hierarchicaltree structure. Considering a tree as a directed acyclic graph, we interpretthe model parameters from a path analysis perspective. Under this concept, theproposed 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 modelselection is also consistent. By applying to a brain structural magneticresonance imaging dataset acquired from the Alzheimer's Disease NeuroimagingInitiative, the proposed approach identifies brain regions where atrophy inthese regions demonstrates the declination in memory. With regularization onthe total effects, the findings suggest that the impact of atrophy on memorydeficits is localized from small brain regions but at various levels of brainsegmentation.

Identifying brain hierarchical structures associated with Alzheimer's disease using a regularized regression method with tree predictors