Abstract: In this work, we propose a longitudinal quantile regression framework that enables a robust characterization of heterogeneous covariate-response associations in the presence of high-dimensional compositional covariates and repeated measurements of both the response and the covariates. We develop a globally adaptive penalization procedure that can consistently identify covariate sparsity patterns across a continuum set of quantile levels. The proposed estimation procedure properly aggregates longitudinal observations over time, and satisfies the sum-zero coefficient constraint needed for a proper interpretation of the effects of compositional covariates. We establish the oracle rate of the uniform convergence and weak convergence of the resulting estimators, and further justify the proposed uniform selector of the tuning parameter in terms of achieving global model selection consistency. We derive an efficient algorithm by incorporating existing R packages to facilitate stable and fast computation. Our extensive simulation studies confirm our theoretical findings. We apply the proposed method to a longitudinal study of cystic fibrosis children, where the associations between the gut microbiome and other diet-related biomarkers are of interest.

Key words and phrases: Compositional covariates, globally adaptive penalization, longitudinal data, quantile regression.