Microbiome compositional analysis with logistic-tree normal models

Modern microbiome compositional data are often high-dimensional and exhibit complex dependency among the microbial taxa. However, existing statistical models for such data either do not adequately account for the dependency among the microbial taxa or lack computational scalability with respect to the number of taxa. This presents challenges in important applications such as association analysis between microbiome compositions and disease risk in which valid statistical analysis requires appropriately incorporating the "variance components" or "random effects" in the microbiome composition. We introduce a generative model, called the "logistic-tree normal" (LTN) model, that addresses this need. LTN marries two popular classes of models-namely, log-ratio normal (LN) and Dirichlet-tree (DT)-and inherits the key benefits of each. LTN incorporates the tree-based binomial decomposition as the DT does, but it jointly models the corresponding binomial probabilities using a (multivariate) logistic-normal distribution as in LN models. It therefore allows rich covariance structures as LN, along with computational efficiency realized through a Polya-Gamma augmentation on the binomial models associated with the tree splits. Accordingly, Bayesian inference on LTN can readily proceed by Gibbs sampling. LTN also allows common techniques for effective inference on high-dimensional data to be readily incorporated. We construct a general mixed-effects model using LTN to characterize compositional random effects, which allows flexible taxa covariance. We demonstrate its use in testing association between microbiome composition and disease risk as well as in estimating the covariance among taxa. We carry out an extensive case study using this LTN-enriched compositional mixed-effects model to analyze a longitudinal dataset from the T1D cohort of the DIABIMMUNE project.