Analysis of High Dimensional Repeated Measures Designs: The One- and Two-Sample Test Statistics

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• Publisher: Cuvillier Verlag
• Year: 2008
• eBook ISBN:
  9783736927346
• Print ISBN:
  9783867277341
• Edition: 1

All models are wrong; only some are useful. (G. E. P. Box)In this project, we have analyzed some useful models, based on an approximation introducedby G. E. P. Box; hence, the next few chapters map an odyssey wherein Box and hisadage go hand in hand. In a nutshell, one- and two-sample test statistics are developedfor the analysis of repeated measures designs when the dimension, d, can be large comparedto the sample size, n (d > n).The statistics do not depend on any specific structure of the covariance matrix andcan be used in a variety of situations: they are valid for testing any general linear hypothesis, are equally applicable to the design set up of profile analysis and to the usualmultivariate structure, are invariant to an orthogonal linear transformation, and are alsovalid when the data are not high dimensional.The test statistics, a modification of the ANOVA-type statistic (Brunner, 2001), arebased on Box's approximation (Box, 1954a), and follow a Â2f -distribution. The estimators, the building blocks of the test statistics, are composed of quadratic and symmetricbilinear forms, and are proved to be unbiased, L2-consistent and uniformly bounded indimension, d. This last property of estimators helps us in the asymptotic derivations inthat we need not let both n and d approach infinity. We let n! 1, while keep d fixed, such that the approximation of the distribution of the test statistic to the Â2 distributionremains accurate when d > n, or even d >> n.The performance of the statistics is evaluated through simulations and it is shownthat, for n as small as 10 or 20, the approximation is quite accurate, whatever be d. Thestatistic is also applied to a number of real data sets for numerical illustrations.