  The so-called gut-brain axis has stimulated extensive research on
microbiomes. One focus is to assess the association between certain clinical
outcomes and the relative abundances of gut microbes, which can be presented as
sub-compositional data in conformity with the taxonomic hierarchy of bacteria.
Motivated by a study for identifying the microbes in the gut microbiome of
preterm infants that impact their later neurobehavioral outcomes, we formulate
a constrained integrative multi-view regression, where the neurobehavioral
scores form multivariate response, the sub-compositional microbiome data form
multi-view feature matrices, and a set of linear constraints on their
corresponding sub-coefficient matrices ensures the conformity to the simplex
geometry. To enable joint selection and inference of sub-compositions/views, we
assume all the sub-coefficient matrices are possibly of low-rank, i.e., the
outcomes are associated with the microbiome through different sets of latent
sub-compositional factors from different taxa. We propose a scaled composite
nuclear norm penalization approach for model estimation and develop a
hypothesis testing procedure through de-biasing to assess the significance of
different views. Simulation studies confirm the effectiveness of the proposed
procedure. In the preterm infant study, the identified microbes are mostly
consistent with existing studies and biological understandings. Our approach
supports that stressful early life experiences imprint gut microbiome through
the regulation of the gut-brain axis.
