Multivariate log-contrast regression with sub-compositional predictors: Testing the association between preterm infants' gut microbiome and neurobehavioral outcomes.

To link a clinical outcome with compositional predictors in microbiome analysis, the linear log-contrast model is a popular choice, and the inference procedure for assessing the significance of each covariate is also available. However, with the existence of multiple potentially interrelated outcomes and the information of the taxonomic hierarchy of bacteria, a multivariate analysis method that considers the group structure of compositional covariates and an accompanying group inference method are still lacking. 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. The neurobehavioral scores form multivariate responses, the log-transformed sub-compositional microbiome data form multi-view feature matrices, and a set of linear constraints on their corresponding sub-coefficient matrices ensures the sub-compositional nature. We assume all the sub-coefficient matrices are possible of low-rank to enable joint selection and inference of sub-compositions/views. 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. We apply the method to the preterm infant study, and the identified microbes are mostly consistent with existing studies and biological understandings.