Bayesian Variable Shrinkage and Selection in Compositional Data Regression: Application to Oral Microbiome

Abstract

Microbiome studies generate multivariate compositional responses, such as taxa counts, which are strictly non-negative, bounded, residing within a simplex, and subject to unit-sum constraint. In presence of covariates (which can be moderate to high dimensional), they are popularly modeled via the Dirichlet-Multinomial (D-M) regression framework. In this paper, we consider a Bayesian approach for estimation and inference under a D-M compositional framework, and present a comparative evaluation of some state-of-the-art continuous shrinkage priors for efficient variable selection to identify the most significant associations between available covariates, and taxonomic abundance. Specifically, we compare the performances of the horseshoe and horseshoe+ priors (with the benchmark Bayesian lasso), utilizing Hamiltonian Monte Carlo techniques for posterior sampling, and generating posterior credible intervals. Our simulation studies using synthetic data demonstrate excellent recovery and estimation accuracy of sparse parameter regime by the continuous shrinkage priors. We further illustrate our method via application to a motivating oral microbiome data generated from the NYC-Hanes study. RStan implementation of our method is made available at the GitHub link: (https://github.com/dattahub/compshrink).

Keywords: Bayesian; Compositional data; Dirichlet; Generalized Dirichlet; Horseshoe; Large p; Shrinkage prior; Sparse probability vectors; Stick-breaking.

Fig. 1

Functional forms of common G-L priors, i.e., Cauchy, Horseshoe (HS), Horseshoe+ (HS+), and Laplace, near zero (left panel), and tails (right panel). While the x-axis represents values of $\lambda$, the y-axis are the values of $\pi (\lambda )$

Fig. 2

Taxononomic composition of top 5% OTUs for the NYC-Hanes Data

Fig. 3

Rank Correlation heatmap for the NYC-Hanes Data

Fig. 4

Selected ${\beta }_{i,j}$’s from fitting the three competing methods to the NYC-Hanes data

Fig. 5

Density plots of the dissimilarity measures corresponding to the horseshoe, horseshoe+, and Laplace prior (Bayesian Lasso) assumptions in our proposed D-M model

Fig. 6

Simulation scheme I(a) results, evaluating recovery of true non-zero associations (left panel), and estimation accuracy (right panel)

Fig. 7

Simulation scheme 1(b) results, comparing recovery of true non-zero associations, via the 95% credible interval method (upper panel), and the 2-means model (lower panel)

Fig. 8

Simulation scheme II results, presenting boxplots of estimation (left panel) and misclassification (right panel) errors, corresponding to the three shrinkage priors, i.e., horseshoe, horseshoe+ and Bayesian Lasso