PCoA

Principal Coordinates Analysis (PCoA), also known as metric multidimensional scaling, is a multivariate ordination method used to visualize relationships among samples based on a matrix of pairwise dissimilarities. Unlike principal component analysis (PCA), which operates on raw data with Euclidean distances, PCoA can incorporate any distance or dissimilarity metric, such as Bray-Curtis, Jaccard, or UniFrac.

Procedure and interpretation: Start with a symmetric matrix D of pairwise distances among n samples. Choose an appropriate distance metric reflecting differences among samples. PCoA then performs double-centering of the squared distance matrix to produce a Gram matrix B = -1/2 J D^2 J, where J centers the data. Eigen-decomposition of B yields eigenvalues and eigenvectors; coordinates for each sample in a k-dimensional space are given by the first k eigenvectors scaled by the square roots of their corresponding eigenvalues. The resulting axes represent the major sources of variation, and plots of the first few axes provide a low-dimensional visualization of the multivariate relationships.

Notes and considerations: Negative eigenvalues can occur when the distance matrix is non-Euclidean; in such cases one may apply corrections (Cailliez or Lingoes) or use a non-metric multidimensional scaling approach. The choice of distance metric strongly influences the results, and data preprocessing (normalization, rarefaction, or transformations) can affect the outcome. For compositional data, transformations that account for relative abundances are often recommended before distance calculation.

Applications and software: PCoA is widely used in ecology and microbiome studies to explore beta diversity and visualize similarities among communities. Implementations exist in R (vegan, ape), Python (scikit-bio), and other platforms, facilitating interpretation of complex multivariate patterns.