Two-group Poisson-Dirichlet mixtures for multiple testing

Abstract

The simultaneous testing of multiple hypotheses is common to the analysis of high-dimensional data sets. The two-group model, first proposed by Efron, identifies significant comparisons by allocating observations to a mixture of an empirical null and an alternative distribution. In the Bayesian nonparametrics literature, many approaches have suggested using mixtures of Dirichlet Processes in the two-group model framework. Here, we investigate employing mixtures of two-parameter Poisson-Dirichlet Processes instead, and show how they provide a more flexible and effective tool for large-scale hypothesis testing. Our model further employs nonlocal prior densities to allow separation between the two mixture components. We obtain a closed-form expression for the exchangeable partition probability function of the two-group model, which leads to a straightforward Markov Chain Monte Carlo implementation. We compare the performance of our method for large-scale inference in a simulation study and illustrate its use on both a prostate cancer data set and a case-control microbiome study of the gastrointestinal tracts in children from underdeveloped countries who have been recently diagnosed with moderate-to-severe diarrhea.

Keywords: Bayesian nonparametrics; Poisson-Dirichlet process; microbiome analysis; multiple testing; two-group model.

FIGURE 1

Microbiome data case study: Histogram of 535 z-scores obtained from the case term (β₁) in the Negative-Binomial generalized linear mixed effects model. We superimpose the posterior probabilities of the events {γ_i = 1|z} and the threshold corresponding to a Bayesian FDR of 1%

FIGURE 2

Prostate data set: Histogram of 6033 z-scores obtained from a two-group comparison. We superimpose the posterior probabilities of the events {γ_i = 1|z} and the threshold corresponding to a Bayesian FDR of 20%