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Analysis of high-dimensional repeated measures designs: The one sample case

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  • Rauf Ahmad, M.
  • Werner, C.
  • Brunner, E.

Abstract

A one sample statistic is derived for the analysis of repeated measures design when the data are multivariate normal and the dimension, d, can be large compared to the sample size, n, i.e. d>n. Quadratic and bilinear forms are used to define the statistic based on Box's approximation [Box, G.E.P., 1954. Some theorems on quadratic forms applied in the study of analysis of variance problems I: Effect of inequality of variance in the one-way classification. Annals of Mathematical Statistics 25 (2), 290-302]. The statistic has an approximate distribution, even for moderately large n. One of the main advantages of the statistic is that it can be used both for unstructured and factorially structured repeated measures designs. In the asymptotic derivations, it is assumed that n-->[infinity] while d remains finite and fixed. However, it is demonstrated through simulations that for n as small as 10, the new statistic very closely approximates the target distribution, unaffected by even large values of . The application is illustrated using a sleep lab example with .

Suggested Citation

  • Rauf Ahmad, M. & Werner, C. & Brunner, E., 2008. "Analysis of high-dimensional repeated measures designs: The one sample case," Computational Statistics & Data Analysis, Elsevier, vol. 53(2), pages 416-427, December.
    • Handle: RePEc:eee:csdana:v:53:y:2008:i:2:p:416-427
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    References listed on IDEAS

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    1. Bathke, Arne C. & Harrar, Solomon W. & Madden, Laurence V., 2008. "How to compare small multivariate samples using nonparametric tests," Computational Statistics & Data Analysis, Elsevier, vol. 52(11), pages 4951-4965, July.
    2. Michael G. Kenward, 1987. "A Method for Comparing Profiles of Repeated Measurements," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 36(3), pages 296-308, November.
    3. Harrar, Solomon W. & Bathke, Arne C., 2008. "Nonparametric methods for unbalanced multivariate data and many factor levels," Journal of Multivariate Analysis, Elsevier, vol. 99(8), pages 1635-1664, September.
    4. 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.
    5. Brunner, Edgar & Munzel, Ulrich & Puri, Madan L., 1999. "Rank-Score Tests in Factorial Designs with Repeated Measures," Journal of Multivariate Analysis, Elsevier, vol. 70(2), pages 286-317, August.
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    2. Zhou, Bu & Guo, Jia, 2017. "A note on the unbiased estimator of Σ2," Statistics & Probability Letters, Elsevier, vol. 129(C), pages 141-146.
    3. 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.
    4. Muni S. Srivastava & Hirokazu Yanagihara & Tatsuya Kubokawa, 2014. "Tests for Covariance Matrices in High Dimension with Less Sample Size," CIRJE F-Series CIRJE-F-933, CIRJE, Faculty of Economics, University of Tokyo.
    5. M. Ahmad, 2014. "A $$U$$ -statistic approach for a high-dimensional two-sample mean testing problem under non-normality and Behrens–Fisher setting," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 66(1), pages 33-61, February.

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