Factor Augmented Inverse Regression and its Application to Microbiome Data Analysis

We investigate the relationship between count data that inform the relative abundance of features of a composition, and factors that influence the composition. Our work is motivated from microbiome studies aiming to extract microbial signatures that are predictive of host phenotypes based on data collected from a group of individuals harboring radically different microbial communities. We introduce multinomial Factor Augmented Inverse Regression (FAIR) of the count vector onto response factors as a general framework for obtaining low-dimensional summaries of the count vector that preserve information relevant to the response. By augmenting known response factors with random latent factors, FAIR extends multinomial logistic regression to account for overdispersion and general correlations among counts. The projections of the count vector onto the loading vectors represent additional contribution, in addition to the projections that result from response factors. The method of maximum variational likelihood and a fast variational expectation-maximization algorithm are proposed for approximate inference based on variational approximation, and the asymptotic properties of the resulting estimator are derived. Moreover, a hybrid information criterion and a group-lasso penalized criterion are proposed for model selection. The effectiveness of FAIR is illustrated through simulations and application to a microbiome dataset. Supplementary materials for this article are available online.