Regression models for ordinal compositional data

2026-10-04

AI Summary

This study addresses the challenges of limited interpretability, missing ordinal structure, and difficulty in handling zero values when modeling ordinal compositional data. To this end, it proposes a transformation-free linear regression framework. The method employs column-stochastic matrices to preserve simplex geometry and utilizes a weighted 1-Wasserstein distance as the loss function to incorporate ordinal information. Variable interactions are effectively characterized through a deterministic constraint model and a novel ordinal tensor product, while a Wasserstein coefficient of determination and an order-preserving index are introduced as diagnostic tools. Global optimality is efficiently achieved via linear programming. Both simulation studies and empirical applications demonstrate that the proposed framework delivers robust predictive performance alongside high interpretability.

Abstract

We present a unified, transformation-free linear regression framework tailored for ordinal compositional data, such as distributions across educational levels or aggregate Likert-scale survey responses. Traditional log-ratio approaches often obscure interpretability, ignore the inherent ordering of categories, and struggle with boundary zero values. To overcome these limitations, we propose a deterministic model constrained to the space of column-stochastic transformation matrices, naturally preserving the fundamental geometry of the simplex. By adopting the weighted 1-Wasserstein distance as the loss function, our method directly embeds the ordinal nature of the data into the estimation process. We provide a computationally efficient and globally optimal solution explicitly formulated as a Linear Programming (LP) problem. The framework systematically addresses four distinct regression scenarios, introducing a novel ordinal tensor product to rigorously handle interactions across multiple compositional predictors. Furthermore, we equip the model with a suitable regularization strategy and novel diagnostic tools, including a Wasserstein-based coefficient of determination ($R^2_W$) and an Order Preservation Index (OPI). Extensive simulation studies and two empirical applications demonstrate the robust predictive performance and interpretability of the proposed methodology.

Problem

• ordinal compositional data
• linear regression
• log-ratio transformation
• boundary zeros
• Wasserstein distance

Innovation

• ordinal compositional data
• Wasserstein distance
• linear programming
• ordinal tensor product
• transformation-free regression