  Using a sample from a population to estimate the proportion of the population
with a certain category label is a broadly important problem. In the context of
microbiome studies, this problem arises when researchers wish to use a sample
from a population of microbes to estimate the population proportion of a
particular taxon, known as the taxon's relative abundance. In this paper, we
propose a beta-binomial model for this task. Like existing models, our model
allows for a taxon's relative abundance to be associated with covariates of
interest. However, unlike existing models, our proposal also allows for the
overdispersion in the taxon's counts to be associated with covariates of
interest. We exploit this model in order to propose tests not only for
differential relative abundance, but also for differential variability. The
latter is particularly valuable in light of speculation that dysbiosis, the
perturbation from a normal microbiome that can occur in certain disease
conditions, may manifest as a loss of stability, or increase in variability, of
the counts associated with each taxon. We demonstrate the performance of our
proposed model using a simulation study and an application to soil microbial
data.
