  In human microbiome studies, sequencing reads data are often summarized as
counts of bacterial taxa at various taxonomic levels specified by a taxonomic
tree. This paper considers the problem of analyzing two repeated measurements
of microbiome data from the same subjects. Such data are often collected to
assess the change of microbial composition after certain treatment, or the
difference in microbial compositions across body sites. Existing models for
such count data are limited in modeling the covariance structure of the counts
and in handling paired multinomial count data. A new probability distribution
is proposed for paired-multinomial count data, which allows flexible covariance
structure and can be used to model repeatedly measured multivariate count data.
Based on this distribution, a test statistic is developed for testing the
difference in compositions based on paired multinomial count data. The proposed
test can be applied to the count data observed on a taxonomic tree in order to
test difference in microbiome compositions and to identify the subtrees with
different subcompositions. Simulation results indicate that proposed test has
correct type 1 errors and increased power compared to some commonly used
methods. An analysis of an upper respiratory tract microbiome data set is used
to illustrate the proposed methods.
