The phylogenetic Kantorovich-Rubinstein metric for environmental sequence samples

TL;DR

This paper introduces a phylogenetic metric based on the Kantorovich-Rubinstein distance to compare microbial communities, providing a rigorous foundation, computational methods, and statistical testing procedures.

Contribution

It establishes that weighted UniFrac is equivalent to the KR distance, extends the metric with uncertainty incorporation and $L^p$ generalizations, and develops permutation tests with Gaussian process approximations.

Findings

Weighted UniFrac equals the KR distance between empirical distributions.

Extensions incorporate uncertainty and generalize to $L^p$ metrics.

Permutation p-values can be approximated via Gaussian process functionals.

Abstract

Using modern technology, it is now common to survey microbial communities by sequencing DNA or RNA extracted in bulk from a given environment. Comparative methods are needed that indicate the extent to which two communities differ given data sets of this type. UniFrac, a method built around a somewhat ad hoc phylogenetics-based distance between two communities, is one of the most commonly used tools for these analyses. We provide a foundation for such methods by establishing that if one equates a metagenomic sample with its empirical distribution on a reference phylogenetic tree, then the weighted UniFrac distance between two samples is just the classical Kantorovich-Rubinstein (KR) distance between the corresponding empirical distributions. We demonstrate that this KR distance and extensions of it that arise from incorporating uncertainty in the location of sample points can be written as a readily computable integral over the tree, we develop $L^p$ Zolotarev-type generalizations of the metric, and we show how the p-value of the resulting natural permutation test of the null hypothesis "no difference between the two communities" can be approximated using a functional of a Gaussian process indexed by the tree. We relate the $L^2$ case to an ANOVA-type decomposition and find that the distribution of its associated Gaussian functional is that of a computable linear combination of independent $\chi_1^2$ random variables.