  Detecting associations between microbial compositions and sample
characteristics is one of the most important tasks in microbiome studies. Most
of the existing methods apply univariate models to single microbial species
separately, with adjustments for multiple hypothesis testing. We propose a
Bayesian analysis for a generalized mixed effects linear model tailored to this
application. The marginal prior on each microbial composition is a Dirichlet
Process, and dependence across compositions is induced through a linear
combination of individual covariates, such as disease biomarkers or the
subject's age, and latent factors. The latent factors capture residual
variability and their dimensionality is learned from the data in a fully
Bayesian procedure. The proposed model is tested in data analyses and
simulation studies with zero-inflated compositions. In these settings, within
each sample, a large proportion of counts per microbial species are equal to
zero. In our Bayesian model a priori the probability of compositions with
absent microbial species is strictly positive. We propose an efficient
algorithm to sample from the posterior and visualizations of model parameters
which reveal associations between covariates and microbial compositions. We
evaluate the proposed method in simulation studies, and then analyze a
microbiome dataset for infants with type 1 diabetes which contains a large
proportion of zeros in the sample-specific microbial compositions.
