A Bayesian method for detecting pairwise associations in compositional data

Compositional data consist of vectors of proportions normalized to a constant sum from a basis of unobserved counts. The sum constraint makes inference on correlations between unconstrained features challenging due to the information loss from normalization. However, such correlations are of long-standing interest in fields including ecology. We propose a novel Bayesian framework (BAnOCC: Bayesian Analysis of Compositional Covariance) to estimate a sparse precision matrix through a LASSO prior. The resulting posterior, generated by MCMC sampling, allows uncertainty quantification of any function of the precision matrix, including the correlation matrix. We also use a first-order Taylor expansion to approximate the transformation from the unobserved counts to the composition in order to investigate what characteristics of the unobserved counts can make the correlations more or less difficult to infer. On simulated datasets, we show that BAnOCC infers the true network as well as previous methods while offering the advantage of posterior inference. Larger and more realistic simulated datasets further showed that BAnOCC performs well as measured by type I and type II error rates. Finally, we apply BAnOCC to a microbial ecology dataset from the Human Microbiome Project, which in addition to reproducing established ecological results revealed unique, competition-based roles for Proteobacteria in multiple distinct habitats.Author summary: Data from many fields are available primarily in the form of proportions, also referred to as compositions, which impose mathematical constraints on identifying interactions among components in the underlying systems. In particular, correlations cannot be calculated directly from proportions or from count data that give rise to them. Methods that work around this difficulty generally do so by imposing strong assumptions about the distribution of underlying data or associated correlations, and these in turn often prevent quantifying uncertainty in the resulting estimates of correlation. We developed a statistical model (BAnOCC: Bayesian Analysis of Compositional Covariance) that both estimates correlations between counts or proportions and provides a posterior distribution for each correlation that quantifies how uncertain the estimate is. BAnOCC does well at controlling the number of false positives in simulated data and can be practically applied to a wide range of proportional data types.