Restoring the Duality between Principal Components of a Distance Matrix and Linear Combinations of Predictors, with Application to Studies of the Microbiome

Appreciation of the importance of the microbiome is increasing, as sequencing technology has made it possible to ascertain the microbial content of a variety of samples. Studies that sequence the 16S rRNA gene, ubiquitous in and nearly exclusive to bacteria, have proliferated in the medical literature. After sequences are binned into operational taxonomic units (OTUs) or species, data from these studies are summarized in a data matrix with the observed counts from each OTU for each sample. Analysis often reduces these data further to a matrix of pairwise distances or dissimilarities; plotting the first two or three principal components (PCs) of this distance matrix often reveals meaningful groupings in the data. However, once the distance matrix is calculated, it is no longer clear which OTUs or species are important to the observed clustering; further, the PCs are hard to interpret and cannot be calculated for subsequent observations. We show how to construct approximate decompositions of the data matrix that pair PCs with linear combinations of OTU or species frequencies, and show how these decompositions can be used to construct biplots, select important OTUs and partition the variability in the data matrix into contributions corresponding to PCs of an arbitrary distance or dissimilarity matrix. To illustrate our approach, we conduct an analysis of the bacteria found in 45 smokeless tobacco samples.