Tree-aggregated regression for compositional data with measurement errors

TL;DR

This paper introduces TARCO, a novel tree-aggregated regression method that corrects for measurement errors in high-dimensional compositional data, improving estimation accuracy and interpretability in applications like microbiome studies.

Contribution

The paper develops a new convex optimization approach that accounts for hierarchical measurement error interactions in tree-aggregated compositional regression, with theoretical guarantees and scalable algorithms.

Findings

TARCO outperforms existing methods in simulation studies.

The method achieves better support recovery and interpretability.

Application to microbiome data demonstrates practical advantages.

Abstract

High-dimensional compositional covariates, often derived from count data, are subject to measurement error and are frequently analyzed after aggregation along a prespecified tree to improve interpretability in applications such as microbiome studies. Existing approaches typically handle either tree-guided compositional regression or errors-in-variables correction, but they do not account for the hierarchical contamination induced by their interaction. We show that tree aggregation turns leaf-level measurement error into level-dependent, correlated contamination across aggregated nodes, which inflates bias, weakens concentration rates for corrected estimating quantities, and leads to unstable variable selection for naive approaches. We propose Tree-Aggregated Regression with Correction for Observation Error (TARCO), which integrates bias-corrected estimating quantities with a tree-aware positive semidefinite stabilization and sparse regularization, with tuning selected by cross-validation based on the corrected objective. The resulting convex program can be solved with scalable algorithms. We establish finite-sample bounds for prediction and estimation errors and prove sign consistency under conditions that explicitly reflect tree heterogeneity. The guarantees persist when the measurement-error covariance is replaced by a consistent estimator. Simulations across multiple tree depths and a microbiome application demonstrate improved estimation accuracy, support recovery, and aggregation-level interpretability compared with methods that ignore the interaction between tree aggregation and measurement error.