  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. By applying to a brain structural magnetic
resonance imaging dataset acquired from the Alzheimer's Disease Neuroimaging
Initiative, 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.
