Logistic-tree normal model for microbiome compositions

We introduce a probabilistic model, called the "logistic-tree normal" (LTN), for microbiome compositional data. The LTN marries two popular classes of models -- the logistic-normal (LN) and the Dirichlet-tree (DT) -- and inherits the key benefits of both. LN models are flexible in characterizing rich covariance structure among taxa but can be computationally prohibitive in face of high dimensionality (i.e., when the number of taxa is large) due to its lack of conjugacy to the multinomial sampling model. On the other hand, DT avoids this issue by decomposing the multinomial sampling model into a collection of binomials, one at each split of the phylogenetic tree of the taxa, and adopting a conjugate beta model for each binomial probability, but at the same time the DT incurs restrictive covariance among the taxa. In contrast, the LTN model decomposes the multinomial model into binomials as the DT does, but it jointly models the corresponding binomial probabilities using a (multivariate) LN distribution instead of betas. It therefore allows rich covariance structures as the LN models, while the decomposition of the multinomial likelihood allows conjugacy to be restored through the P\'olya-Gamma augmentation. Accordingly, Bayesian inference on the LTN model can readily proceed by Gibbs sampling. Moreover, the multivariate Gaussian aspect of the model allows common techniques for effective inference on high-dimensional data -- such as those based on sparsity and low-rank assumptions in the covariance structure -- to be readily incorporated. Depending on the goal of the analysis, the LTN model can be used either as a standalone model or embedded into more sophisticated models. We demonstrate its use in estimating taxa covariance and in mixed-effects modeling. Finally, we carry out a case study using an LTN-based mixed-effects model to analyze a longitudinal dataset from the DIABIMMUNE project.