Regression for ordinal data on the simplex

We introduce a regression framework for ordinal compositional data that operates directly on the simplex and avoids the use of log-ratio transformations. The proposed model describes the relationship between compositional predictors and ordinal compositional responses through a column-stochastic linear operator, ensuring that fitted values remain valid compositions. Estimation is based on the minimization of the 1-Wasserstein distance, which incorporates the ordinal structure of the support through its ground metric. The resulting optimization problem admits an exact linear programming formulation, guaranteeing global optimality and computational feasibility. The framework accommodates both ordinal and nominal predictors, while explicitly preserving the ordered nature of the response categories. A simulation study based on Gaussian data-generating process evaluates the predictive performance of the method across different simplex dimensions and sample sizes. The results show stable and coherent behavior, highlighting the role of the Wasserstein metric in capturing ordinal structure within compositional regression.