IDEAS home Printed from https://ideas.repec.org/a/eee/csdana/v215y2026ics0167947325001641.html

Measuring multivariate regression association via spatial sign

Author

Listed:
  • Shih, Jia-Han
  • Chen, Yi-Hau

Abstract

A regression association measure is proposed for capturing predictability of a multivariate outcome Y=(Y1,…,Yd) from a multivariate covariate X=(X1,…,Xp). Motivated by existing measures, the conventional Kendall’s tau is first generalized to measure multivariate association between two random vectors. Then the predictability of Y from X is measured by the generalized multivariate Kendall’s tau between Y and Y′, where Y and Y′ share the same conditional distribution and are conditionally independent given X. The proposed regression association measure can be expressed as the proportion of the variance of a function of Y that can be explained by X, indicating that the measure has a direct interpretation in terms of predictability. Based on the proposed measure, a conditional regression association measure is further proposed, which can be utilized to perform variable selection. Since the proposed measures are based on Y and Y′, a simple nonparametric estimation method based on nearest neighbors is available. An R package, MRAM, has been developed for implementation. Simulation studies are carried out to assess the performance of the proposed methods and real data examples are analyzed for illustration.

Suggested Citation

  • Shih, Jia-Han & Chen, Yi-Hau, 2026. "Measuring multivariate regression association via spatial sign," Computational Statistics & Data Analysis, Elsevier, vol. 215(C).
    • Handle: RePEc:eee:csdana:v:215:y:2026:i:c:s0167947325001641
    DOI: 10.1016/j.csda.2025.108288
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167947325001641
    Download Restriction: Full text for ScienceDirect subscribers only.
    File URL: https://libkey.io/10.1016/j.csda.2025.108288?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to

    for a different version of it.

    References listed on IDEAS

    as
    1. Puccetti, Giovanni & Scarsini, Marco, 2010. "Multivariate comonotonicity," Journal of Multivariate Analysis, Elsevier, vol. 101(1), pages 291-304, January.
    2. Nabarun Deb & Bodhisattva Sen, 2023. "Multivariate Rank-Based Distribution-Free Nonparametric Testing Using Measure Transportation," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 118(541), pages 192-207, January.
    3. Jia-Han Shih & Yi-Hau Chen, 2025. "A Class of Regression Association Measures based on Concordance," The American Statistician, Taylor & Francis Journals, vol. 79(3), pages 320-329, July.
    4. H Shi & M Drton & F Han, 2022. "On the power of Chatterjee’s rank correlation [Adaptive test of independence based on HSIC measures]," Biometrika, Biometrika Trust, vol. 109(2), pages 317-333.
    5. Zhexiao Lin & Fang Han, 2024. "On the failure of the bootstrap for Chatterjee’s rank correlation," Biometrika, Biometrika Trust, vol. 111(3), pages 1063-1070.
    6. Holger Dette & Karl F. Siburg & Pavel A. Stoimenov, 2013. "A Copula-Based Non-parametric Measure of Regression Dependence," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 40(1), pages 21-41, March.
    7. Shih, Jia-Han & Emura, Takeshi, 2021. "On the copula correlation ratio and its generalization," Journal of Multivariate Analysis, Elsevier, vol. 182(C).
    8. Hofert, Marius & Oldford, Wayne & Prasad, Avinash & Zhu, Mu, 2019. "A framework for measuring association of random vectors via collapsed random variables," Journal of Multivariate Analysis, Elsevier, vol. 172(C), pages 5-27.
    9. De Keyser, Steven & Gijbels, Irène, 2025. "High-dimensional copula-based Wasserstein dependence," Computational Statistics & Data Analysis, Elsevier, vol. 204(C).
    10. Sourav Chatterjee, 2021. "A New Coefficient of Correlation," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 116(536), pages 2009-2022, October.
    11. Z Lin & F Han, 2023. "On boosting the power of Chatterjee’s rank correlation," Biometrika, Biometrika Trust, vol. 110(2), pages 283-299.
    12. H Dette & M Kroll, 2025. "A simple bootstrap for Chatterjee’s rank correlation," Biometrika, Biometrika Trust, vol. 112(1), pages 3070-3102.
    13. repec:plo:pone00:0170238 is not listed on IDEAS
    14. Azadkia, Mona & Chatterjee, Sourav, 2021. "A simple measure of conditional dependence," LSE Research Online Documents on Economics 125584, London School of Economics and Political Science, LSE Library.
    15. Grothe, Oliver & Schnieders, Julius & Segers, Johan, 2014. "Measuring association and dependence between random vectors," Journal of Multivariate Analysis, Elsevier, vol. 123(C), pages 96-110.
    16. Denuit, Michel & Lambert, Philippe, 2005. "Constraints on concordance measures in bivariate discrete data," Journal of Multivariate Analysis, Elsevier, vol. 93(1), pages 40-57, March.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Mona Azadkia & Leihao Chen & Fang Han, 2025. "Bias correction for Chatterjee's graph-based correlation coefficient," Papers 2508.09040, arXiv.org, revised Jan 2026.
    2. De Keyser, Steven & Gijbels, Irène, 2025. "High-dimensional copula-based Wasserstein dependence," Computational Statistics & Data Analysis, Elsevier, vol. 204(C).
    3. Fang Han, 2024. "An Introduction to Permutation Processes (version 0.5)," Papers 2407.09664, arXiv.org.
    4. Zhexiao Lin & Fang Han, 2023. "On the failure of the bootstrap for Chatterjee's rank correlation," Papers 2303.14088, arXiv.org, revised Apr 2023.
    5. Wang, Li & Zhou, Hongyi & Ma, Weidong & Yang, Ying, 2025. "A conditional distribution function-based measure for independence and K-sample tests in multivariate data," Journal of Multivariate Analysis, Elsevier, vol. 205(C).
    6. Zhang, Qingyang, 2023. "On the asymptotic null distribution of the symmetrized Chatterjee’s correlation coefficient," Statistics & Probability Letters, Elsevier, vol. 194(C).
    7. Ansari Jonathan & Rockel Marcus, 2024. "Dependence properties of bivariate copula families," Dependence Modeling, De Gruyter, vol. 12(1), pages 1-36.
    8. F. Durante & A. Gatto & F. Ravazzolo, 2024. "Understanding relationships with the Aggregate Zonal Imbalance using copulas," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 33(2), pages 513-554, April.
    9. Saminger-Platz Susanne & Kolesárová Anna & Šeliga Adam & Mesiar Radko & Klement Erich Peter, 2024. "On comprehensive families of copulas involving the three basic copulas and transformations thereof," Dependence Modeling, De Gruyter, vol. 12(1), pages 1-36.
    10. Feng, Junlong & Lee, Sokbae, 2025. "Individual welfare analysis: Random quasilinear utility, independence, and confidence bounds," Journal of Econometrics, Elsevier, vol. 247(C).
    11. Yihui He & Fang Han, 2023. "On propensity score matching with a diverging number of matches," Papers 2310.14142, arXiv.org, revised Nov 2023.
    12. Fuchs Sebastian & Limbach Carsten, 2026. "A dimension reduction for extreme types of directed dependence," Dependence Modeling, De Gruyter, vol. 14(1), pages 1-14.
    13. Jianhua Mei & Fu Ouyang & Thomas T. Yang, 2025. "Dimension Reduction for Conditional Density Estimation with Applications to High-Dimensional Causal Inference," Papers 2507.22312, arXiv.org, revised Oct 2025.
    14. Wang, Ya & Fang, Linjiajie & Jing, Bingyi, 2024. "Adapted Chatterjee correlation coefficient," Statistics & Probability Letters, Elsevier, vol. 215(C).
    15. Ziming Lin & Fang Han, 2024. "On the consistency of bootstrap for matching estimators," Papers 2410.23525, arXiv.org, revised Nov 2024.
    16. Mordant, Gilles & Segers, Johan, 2022. "Measuring dependence between random vectors via optimal transport," Journal of Multivariate Analysis, Elsevier, vol. 189(C).
    17. Liqi Xia & Ruiyuan Cao & Jiang Du & Jun Dai, 2025. "Consistent complete independence test in high dimensions based on Chatterjee correlation coefficient," Statistical Papers, Springer, vol. 66(1), pages 1-32, January.
    18. Yanqin Fan & Marc Henry, 2020. "Vector copulas," Papers 2009.06558, arXiv.org, revised Apr 2021.
    19. Alfred Galichon & Marc Henry, 2026. "An econometrician's guide to optimal transport," Papers 2604.04227, arXiv.org.
    20. Fuchs, Sebastian & Di Lascio, F. Marta L. & Durante, Fabrizio, 2021. "Dissimilarity functions for rank-invariant hierarchical clustering of continuous variables," Computational Statistics & Data Analysis, Elsevier, vol. 159(C).

    More about this item

    Keywords

    ;
    ;
    ;
    ;

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:csdana:v:215:y:2026:i:c:s0167947325001641. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/csda .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.