Rectified Fisher-Bingham Model for Compositional Data with Zeros

TL;DR

This paper proposes a novel Fisher-Bingham model for compositional data with zeros, enabling coherent likelihood estimation and improved detection of structured differences, especially in microbiota studies.

Contribution

It introduces a rectified Fisher-Bingham model with a Monte Carlo EM algorithm and a score test for compositional data with zeros, avoiding zero imputation.

Findings

The method closely approximates the true distribution in simulations.

It achieves higher power in detecting compositional changes with many zeros.

Application reveals microbiota shifts missed by standard methods.

Abstract

This paper introduces a rectified and renormalized Fisher-Bingham model for compositional data with zeros, motivated in part by the presence of zeros in microbiota studies. The approach represents compositions through a square-root transformation that maps data to the positive orthant of the unit sphere, and models them via a latent Fisher-Bingham followed by a deterministic transformation that induces exact zeros. This construction yields a coherent likelihood without requiring zero imputation or separate modeling of zero and nonzero components. Parameter estimation is performed using a Monte Carlo expectation-maximization algorithm that accommodates the latent structure. We further develop a score test for detecting structured differences in composition across groups, providing a parametric alternative to commonly used distance-based methods. Simulation studies demonstrate that the proposed method closely approximates the induced distribution and achieves higher power for detecting structured compositional changes, particularly when observations include many zero-valued components. An application to a dietary intervention study illustrates that the method identifies meaningful microbiota shifts not detected by standard approaches.