  A compositional tree refers to a tree structure on a set of random variables
where each random variable is a node and composition occurs at each non-leaf
node of the tree. As a generalization of compositional data, compositional
trees handle more complex relationships among random variables and appear in
many disciplines, such as brain imaging, genomics and finance. We consider the
problem of sparse regression on data that are associated with a compositional
tree and propose a transformation-free tree-based regularized regression method
for component selection. The regularization penalty is designed based on the
tree structure and encourages a sparse tree representation. We prove that our
proposed estimator for regression coefficients is both consistent and model
selection consistent. In the simulation study, our method shows higher accuracy
than competing methods under different scenarios. By analyzing a brain imaging
data set from studies of Alzheimer's disease, our method identifies meaningful
associations between memory declination and volume of brain regions that are
consistent with current understanding.
