  Microbiome data are complex in nature, involving high dimensionality,
compositionally, zero inflation, and taxonomic hierarchy. Compositional data
reside in a simplex that does not admit the standard Euclidean geometry. Most
existing compositional regression methods rely on transformations that are
inadequate or even inappropriate in modeling data with excessive zeros and
taxonomic structure. We develop a novel relative-shift regression framework
that directly uses compositions as predictors. The new framework provides a
paradigm shift for compositional regression and offers a superior biological
interpretation. New equi-sparsity and taxonomy-guided regularization methods
and an efficient smoothing proximal gradient algorithm are developed to
facilitate feature aggregation and dimension reduction in regression. As a
result, the framework can automatically identify clinically relevant microbes
even if they are important at different taxonomic levels. A unified
finite-sample prediction error bound is developed for the proposed regularized
estimators. We demonstrate the efficacy of the proposed methods in extensive
simulation studies. The application to a preterm infant study reveals novel
insights of association between the gut microbiome and neurodevelopment.
