  Motivated by regression analysis for microbiome compositional data, this
paper considers generalized linear regression analysis with compositional
covariates, where a group of linear constraints on regression coefficients are
imposed to account for the compositional nature of the data and to achieve
subcompositional coherence. A penalized likelihood estimation procedure using a
generalized accelerated proximal gradient method is developed to efficiently
estimate the regression coefficients. A de-biased procedure is developed to
obtain asymptotically unbiased and normally distributed estimates, which leads
to valid confidence intervals of the regression coefficients. Simulations
results show the correctness of the coverage probability of the confidence
intervals and smaller variances of the estimates when the appropriate linear
constraints are imposed. The methods are illustrated by a microbiome study in
order to identify bacterial species that are associated with inflammatory bowel
disease (IBD) and to predict IBD using fecal microbiome.
