  One important problem in microbiome analysis is to identify the bacterial
taxa that are associated with a response, where the microbiome data are
summarized as the composition of the bacterial taxa at different taxonomic
levels. This paper considers regression analysis with such compositional data
as covariates. In order to satisfy the subcompositional coherence of the
results, linear models with a set of linear constraints on the regression
coefficients are introduced. Such models allow regression analysis for
subcompositions and include the log-contrast model for compositional covariates
as a special case. A penalized estimation procedure for estimating the
regression coefficients and for selecting variables under the linear
constraints is developed. A method is also proposed to obtain de-biased
estimates of the regression coefficients that are asymptotically unbiased and
have a joint asymptotic multivariate normal distribution. This provides valid
confidence intervals of the regression coefficients and can be used to obtain
the $p$-values. Simulation results show the validity of the confidence
intervals and smaller variances of the de-biased estimates when the linear
constraints are imposed. The proposed methods are applied to a gut microbiome
data set and identify four bacterial genera that are associated with the body
mass index after adjusting for the total fat and caloric intakes.
