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High-dimensional multivariate repeated measures analysis with unequal covariance matrices

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  • Harrar, Solomon W.
  • Kong, Xiaoli

Abstract

In this paper, test statistics for repeated measures design are introduced when the dimension is large. By large dimension is meant the number of repeated measures and the total sample size grow together but either one could be larger than the other. Asymptotic distribution of the statistics is derived for the equal as well as unequal covariance cases in the balanced as well as unbalanced cases. The asymptotic framework considered requires proportional growth of the sample sizes and the dimension of the repeated measures in the unequal covariance case. In the equal covariance case, one can grow at much faster rate than the other. The derivations of the asymptotic distributions mimic that of Central Limit Theorem with some important peculiarities addressed with sufficient rigor. Consistent and unbiased estimators of the asymptotic variances, which make efficient use of all the observations, are also derived. Simulation study provides favorable evidence for the accuracy of the asymptotic approximation under the null hypothesis. Power simulations have shown that the new methods have comparable power with a popular method known to work well in low-dimensional situation but the new methods have shown enormous advantage when the dimension is large. Data from Electroencephalograph (EEG) experiment is analyzed to illustrate the application of the results.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:jmvana:v:145:y:2016:i:c:p:1-21
    DOI: 10.1016/j.jmva.2015.11.012
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    References listed on IDEAS

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    1. Wang, Haiyan & Akritas, Michael G., 2010. "Rank test for heteroscedastic functional data," Journal of Multivariate Analysis, Elsevier, vol. 101(8), pages 1791-1805, September.
    2. Bathke, Arne C. & Harrar, Solomon W. & Wang, Haiyan & Zhang, Ke & Piepho, Hans-Peter, 2010. "Series of randomized complete block experiments with non-normal data," Computational Statistics & Data Analysis, Elsevier, vol. 54(7), pages 1840-1857, July.
    3. 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.
    4. Wang, Haiyan & Higgins, James & Blasi, Dale, 2010. "Distribution-free tests for no effect of treatment in heteroscedastic functional data under both weak and long range dependence," Statistics & Probability Letters, Elsevier, vol. 80(5-6), pages 390-402, March.
    5. 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.
    6. Srivastava, Muni S. & Yanagihara, Hirokazu, 2010. "Testing the equality of several covariance matrices with fewer observations than the dimension," Journal of Multivariate Analysis, Elsevier, vol. 101(6), pages 1319-1329, July.
    7. 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.
    8. Solomon W. Harrar & Jin Xu, 2014. "Analyzing Mean Profiles of Nonnormal Populations," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 43(17), pages 3553-3573, September.
    9. Huynh Huynh, 1978. "Some approximate tests for repeated measurement designs," Psychometrika, Springer;The Psychometric Society, vol. 43(2), pages 161-175, June.
    10. Chen, Song Xi & Qin, Yingli, 2010. "A Two Sample Test for High Dimensional Data with Applications to Gene-set Testing," MPRA Paper 59642, University Library of Munich, Germany.
    11. Haiyan Wang & Michael Akritas, 2010. "Inference from heteroscedastic functional data," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 22(2), pages 149-168.
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    1. Hyodo, Masashi & Nishiyama, Takahiro & Pavlenko, Tatjana, 2020. "Testing for independence of high-dimensional variables: ρV-coefficient based approach," Journal of Multivariate Analysis, Elsevier, vol. 178(C).

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