  The analysis of human microbiome data is often based on dimension-reduced
graphical displays and clustering derived from vectors of microbial abundances
in each sample. Common to these ordination methods is the use of biologically
motivated definitions of similarity. Principal coordinate analysis, in
particular, is often performed using ecologically defined distances, allowing
analyses to incorporate context-dependent, non-Euclidean structure. Here we
describe how to take a step beyond ordination plots and incorporate this
structure into high-dimensional penalized regression models. Within this
framework, the estimate of a regression coefficient vector is obtained via the
joint eigen properties of multiple similarity matrices, or kernels. This allows
for multivariate regression models to incorporate both a matrix of microbial
abundances and, for instance, a matrix of phylogenetically-informed
similarities between the abundance profiles. Further, we show how this
regression framework can be used to address the compositional nature of
multivariate predictors comprised of relative abundances; that is, vectors
whose entries sum to a constant. We illustrate this regression framework with
several simulations using data from two recent studies on the gut and vaginal
microbiome. We conclude with an application to our own data, where we also
incorporate a significance test for the estimated coefficients that represent
associations between microbial abundance and a response.
