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Some high-dimensional tests for a one-way MANOVA

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  • Schott, James R.

Abstract

A statistic is proposed for testing the equality of the mean vectors in a one-way multivariate analysis of variance. The asymptotic null distribution of this statistic, as both the sample size and the number of variables go to infinity, is shown to be normal. Thus, this test can be used when the number of variables is not small relative to the sample size. In particular, it can be used when the number of variables exceeds the degrees of freedom for error, a situation in which standard MANOVA tests are invalid. A related statistic, also having an asymptotic normal distribution, is developed for tests concerning the dimensionality of the hyperplane formed by the population mean vectors. The finite sample size performances of the normal approximations are evaluated in a simulation study.

Suggested Citation

  • Schott, James R., 2007. "Some high-dimensional tests for a one-way MANOVA," Journal of Multivariate Analysis, Elsevier, vol. 98(9), pages 1825-1839, October.
    • Handle: RePEc:eee:jmvana:v:98:y:2007:i:9:p:1825-1839
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    1. Srivastava, Muni S. & Fujikoshi, Yasunori, 2006. "Multivariate analysis of variance with fewer observations than the dimension," Journal of Multivariate Analysis, Elsevier, vol. 97(9), pages 1927-1940, October.
    2. Schott, James R., 2006. "A high-dimensional test for the equality of the smallest eigenvalues of a covariance matrix," Journal of Multivariate Analysis, Elsevier, vol. 97(4), pages 827-843, April.
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    1. Zhang, Jin-Ting & Zhou, Bu & Guo, Jia, 2022. "Linear hypothesis testing in high-dimensional heteroscedastic one-way MANOVA: A normal reference L2-norm based test," Journal of Multivariate Analysis, Elsevier, vol. 187(C).
    2. Zhang, Jin-Ting & Guo, Jia & Zhou, Bu, 2017. "Linear hypothesis testing in high-dimensional one-way MANOVA," Journal of Multivariate Analysis, Elsevier, vol. 155(C), pages 200-216.
    3. Pini, Alessia & Sørensen, Helle & Tolver, Anders & Vantini, Simone, 2023. "Local inference for functional linear mixed models," Computational Statistics & Data Analysis, Elsevier, vol. 181(C).
    4. Jiang Hu & Zhidong Bai & Chen Wang & Wei Wang, 2017. "On testing the equality of high dimensional mean vectors with unequal covariance matrices," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 69(2), pages 365-387, April.
    5. Tianming Zhu & Jin-Ting Zhang, 2022. "Linear hypothesis testing in high-dimensional one-way MANOVA: a new normal reference approach," Computational Statistics, Springer, vol. 37(1), pages 1-27, March.
    6. T. Górecki & Ł. Smaga, 2017. "Multivariate analysis of variance for functional data," Journal of Applied Statistics, Taylor & Francis Journals, vol. 44(12), pages 2172-2189, September.
    7. Watanabe, Hiroki & Hyodo, Masashi & Nakagawa, Shigekazu, 2020. "Two-way MANOVA with unequal cell sizes and unequal cell covariance matrices in high-dimensional settings," Journal of Multivariate Analysis, Elsevier, vol. 179(C).
    8. Harrar, Solomon W. & Kong, Xiaoli, 2016. "High-dimensional multivariate repeated measures analysis with unequal covariance matrices," Journal of Multivariate Analysis, Elsevier, vol. 145(C), pages 1-21.
    9. Bodnar, Olha & Touli, Elena Farahbakhsh, 2023. "Exact test theory in Gaussian graphical models," Journal of Multivariate Analysis, Elsevier, vol. 196(C).
    10. M. Rauf Ahmad, 2019. "A unified approach to testing mean vectors with large dimensions," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 103(4), pages 593-618, December.
    11. Zhang, Jin-Ting & Zhu, Tianming, 2022. "A new normal reference test for linear hypothesis testing in high-dimensional heteroscedastic one-way MANOVA," Computational Statistics & Data Analysis, Elsevier, vol. 168(C).
    12. Huiqin Li & Jiang Hu & Zhidong Bai & Yanqing Yin & Kexin Zou, 2017. "Test on the linear combinations of mean vectors in high-dimensional data," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 26(1), pages 188-208, March.
    13. Chakraborty, Nilanjan & Sakhanenko, Lyudmila, 2023. "Novel multiplier bootstrap tests for high-dimensional data with applications to MANOVA," Computational Statistics & Data Analysis, Elsevier, vol. 178(C).
    14. Schott, James R., 2007. "A test for the equality of covariance matrices when the dimension is large relative to the sample sizes," Computational Statistics & Data Analysis, Elsevier, vol. 51(12), pages 6535-6542, August.
    15. Ley, Christophe & Paindaveine, Davy & Verdebout, Thomas, 2015. "High-dimensional tests for spherical location and spiked covariance," Journal of Multivariate Analysis, Elsevier, vol. 139(C), pages 79-91.
    16. Bodnar, Taras & Dette, Holger & Parolya, Nestor, 2019. "Testing for independence of large dimensional vectors," MPRA Paper 97997, University Library of Munich, Germany, revised May 2019.
    17. Harrar, Solomon W. & Kong, Xiaoli, 2022. "Recent developments in high-dimensional inference for multivariate data: Parametric, semiparametric and nonparametric approaches," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    18. Muni S. Srivastava & Tatsuya Kubokawa, 2011. "Tests for Multivariate Analysis of Variance in High Dimension Under Non-Normality," CIRJE F-Series CIRJE-F-831, CIRJE, Faculty of Economics, University of Tokyo.
    19. Zhao, Junguang & Xu, Xingzhong, 2016. "A generalized likelihood ratio test for normal mean when p is greater than n," Computational Statistics & Data Analysis, Elsevier, vol. 99(C), pages 91-104.
    20. Jin-Ting Zhang & Bu Zhou & Jia Guo, 2022. "Testing high-dimensional mean vector with applications," Statistical Papers, Springer, vol. 63(4), pages 1105-1137, August.
    21. Pini, Alessia & Stamm, Aymeric & Vantini, Simone, 2018. "Hotelling’s T2 in separable Hilbert spaces," Journal of Multivariate Analysis, Elsevier, vol. 167(C), pages 284-305.
    22. Cai, T. Tony & Xia, Yin, 2014. "High-dimensional sparse MANOVA," Journal of Multivariate Analysis, Elsevier, vol. 131(C), pages 174-196.
    23. Davy Paindaveine & Thomas Verdebout, 2013. "Universal Asymptotics for High-Dimensional Sign Tests," Working Papers ECARES ECARES 2013-40, ULB -- Universite Libre de Bruxelles.
    24. Srivastava, Muni S. & Kubokawa, Tatsuya, 2013. "Tests for multivariate analysis of variance in high dimension under non-normality," Journal of Multivariate Analysis, Elsevier, vol. 115(C), pages 204-216.
    25. J. Cuesta-Albertos & M. Febrero-Bande, 2010. "A simple multiway ANOVA for functional data," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 19(3), pages 537-557, November.

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