Compositional Data Analysis

The problem

Compositional data carry only relative information. Proportions, relative abundances, and probability vectors all sum to one, so a change in one part forces changes in the others, and standard methods applied to raw proportions can produce spurious correlations.

The review focuses on the high-dimensional regime, with hundreds to thousands of parts. There most parts are zero or nearly zero and a few carry most of the signal. Log-ratio methods are undefined at zero, and the Dirichlet distribution can neither concentrate on a few parts nor represent parts that rise and fall together.

The review

To our knowledge it is the first review of high-dimensional compositional data analysis from a Bayesian perspective. It surveys open problems in transformations, model evaluation, compositional regression, sparsity-inducing priors, positive correlation structure, and prior sensitivity. I wrote three of those sections.

Transformations

The additive, centered, and isometric log-ratio transformations each map compositions into ordinary coordinates, and each gives something up. The first depends on a reference part, the second has a singular covariance, and the third depends on a basis that is rarely meaningful. Zeros break all three. The choice decides the geometry a model is fit, regularized, and evaluated in, so it is a modeling decision rather than preprocessing.

Compositional regression

Compositions can appear as covariates, as responses, or on both sides. The linear log-contrast model constrains its coefficients to sum to zero, which is exactly what lets a model about absolute abundances be fit to proportions without knowing the totals. It also changes what a coefficient means. Every effect is a reallocation of one part against the others. From there I follow the regularized extensions used when parts outnumber samples, where uncertainty quantification is still underdeveloped.

Model evaluation

No single loss fits every compositional problem. Log-ratio losses measure relative accuracy but break at zero, likelihood-based criteria score the whole predictive distribution, and ecological dissimilarities such as Bray-Curtis answer a different question again. A small absolute error in a rare part can be a large log-ratio error, so the criterion, the transformation, and the zero-handling rule should be reported together.

A separate study of river bacteria

At Winona State I conducted the analysis for a study of bacterial communities at eleven sites on the Upper Yamuna in India and the Upper Mississippi in the United States. It uses relative abundances, but it is not a compositional data analysis and does not use the log-ratio methods above.

k-means clustering found five community types that followed geography, and similarity percentage (SIMPER) analysis identified the 29 taxa that separate them. I then correlated those taxa with eight land-cover classes at three nested scales, a 5-kilometer buffer, a 50-kilometer buffer, and the whole upstream watershed, controlling the false discovery rate. No association survived within 5 kilometers, twenty did at 50 kilometers, and eighteen did over the whole watershed. That suggests river bacteria are more useful as indicators of conditions across the watershed than of conditions at the bank.