[Paper Review] Fully distribution-free center-outward rank tests for multiple-output regression and MANOVA

Marc Hallin, Daniel Hlubinka+1 more|arXiv (Cornell University)|Jul 30, 2020

Statistical Methods and Inference38 references11 citations

TL;DR

This paper introduces the first fully distribution-free, parametrically efficient rank tests for multiple-output regression and MANOVA using center-outward ranks based on measure transportation. These tests achieve asymptotic normality and Hájek representation, outperforming pseudo-Gaussian and elliptical methods in non-elliptical, heavy-tailed data, as shown in simulations and an archeological dataset where they detected significant group differences where others failed.

ABSTRACT

Extending rank-based inference to a multivariate setting such as multiple-output regression or MANOVA with unspecified d-dimensional error density has remained an open problem for more than half a century. None of the many solutions proposed so far is enjoying the combination of distribution-freeness and efficiency that makes rank-based inference a successful tool in the univariate setting. A concept of center-outward multivariate ranks and signs based on measure transportation ideas has been introduced recently. Center-outward ranks and signs are not only distribution-free but achieve in dimension d > 1 the (essential) maximal ancillarity property of traditional univariate ranks, hence carry all the "distribution-free information" available in the sample. We derive here the Hájek representation and asymptotic normality results required in the construction of center-outward rank tests for multiple-output regression and MANOVA. When based on appropriate spherical scores, these fully distribution-free tests achieve parametric efficiency in the corresponding models.

Motivation & Objective

• To address the long-standing open problem of constructing fully distribution-free, efficient tests for multiple-output regression and MANOVA under unspecified multivariate error densities.
• To extend the classical Hájek approach of univariate rank-based inference to the multivariate setting, preserving distribution-freeness and efficiency.
• To overcome the limitations of existing multivariate rank concepts—such as componentwise ranks, spatial ranks, and depth-based ranks—that lack distribution-freeness, efficiency, or directional sensitivity.
• To establish the theoretical foundations (Hájek representation and asymptotic normality) for center-outward rank statistics in multivariate linear models.
• To demonstrate the practical superiority of the proposed tests over pseudo-Gaussian and elliptical rank-based methods in real and simulated data with non-Gaussian, heavy-tailed, and non-elliptical distributions.

Proposed method

• The method employs center-outward ranks and signs derived from measure transportation (Monge-Kantorovich optimal transport), which generalize univariate ranks to dimension d > 1 while preserving maximal ancillarity.
• Empirical center-outward distribution functions are constructed using a grid-based approximation of the transport map, ensuring Glivenko-Cantelli convergence.
• Linear rank statistics are formed as weighted sums of center-outward ranks, with weights derived from the Hájek projection of LAN central sequences to ensure asymptotic efficiency.
• Asymptotic normality of the test statistics is established via a Hájek representation theorem for center-outward ranks in multivariate linear models.
• Two variants of the test are implemented using different grid configurations (n_S=7, n_R=18 and vice versa), ensuring robustness to grid choice.
• The tests are applied to both bivariate and full 4-dimensional MANOVA models, with p-values computed under the null hypothesis of no treatment effect.

Experimental results

Research questions

• RQ1 Can fully distribution-free multivariate rank tests be constructed that achieve parametric efficiency in multiple-output regression and MANOVA models?
• RQ2 Do center-outward ranks based on optimal transport provide a valid and efficient alternative to existing multivariate rank concepts in non-elliptical, heavy-tailed distributions?
• RQ3 How do the proposed center-outward rank tests compare to pseudo-Gaussian and elliptical rank-based tests in terms of size and power under non-Gaussian error densities?
• RQ4 Can the Hájek representation and asymptotic normality results be established for center-outward rank statistics in multivariate linear models?
• RQ5 Does the method detect significant group differences in real-world data where standard methods fail, particularly when distributions are non-elliptical and heavy-tailed?

Key findings

• In the bivariate analysis of the natron glass dataset, the center-outward rank tests detected significant differences between groups when CoO was included, while Pillai’s trace test yielded non-significant p-values (e.g., 0.1217 for MgO-CoO).
• The Wilcoxon center-outward rank test with n_R=7 and n_S=18 produced p-values of 0.0000 for all bivariate models involving CoO, indicating strong evidence against the null hypothesis at α=0.05.
• In the full 4-dimensional MANOVA, the center-outward test yielded a p-value of 10^-15, indicating highly significant differences, whereas Pillai’s test reported 0.1553 and the elliptical rank test reported 0.5827.
• The elliptical Wilcoxon rank test failed to detect any difference at any level α ≤ 0.5, highlighting the limitations of elliptical assumptions in real-world data.
• The proposed method detected significant group differences in the archeological dataset where traditional pseudo-Gaussian and elliptical methods failed, suggesting potential revisions to prior conclusions on Byzantine-Islamic period trade.
• Theoretical results confirm the Hájek representation and asymptotic normality of center-outward rank statistics, enabling the construction of efficient, distribution-free tests in multivariate linear models.