A Bayesian Dirichlet-Multinomial Test for Cross-Group Differences

Testing for differences within data sets is an important issue across various applications. Our work is primarily motivated by the analysis of microbiomial composition, which has been increasingly relevant and important with the rise of DNA sequencing. We first review classical frequentist tests that are commonly used in tackling such problems. We then propose a Bayesian Dirichlet-multinomial framework for modeling the metagenomic data and for testing underlying differences between the samples. A parametric Dirichlet-multinomial model uses an intuitive hierarchical structure that allows for flexibility in characterizing both the within-group variation and the cross-group difference and provides very interpretable parameters. A computational method for evaluating the marginal likelihoods under the null and alternative hypotheses is also given. Through simulations, we show that our Bayesian model performs competitively against frequentist counterparts. We illustrate the method through analyzing metagenomic applications using the Human Microbiome Project data.