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

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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 introduced
by G. E. P. Box; hence, the next few chapters map an odyssey wherein Box and his
adage go hand in hand. In a nutshell, one- and two-sample test statistics are developed
for the analysis of repeated measures designs when the dimension, d, can be large compared
to the sample size, n (d > n).
The statistics do not depend on any specific structure of the covariance matrix and
can 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 usual
multivariate structure, are invariant to an orthogonal linear transformation, and are also
valid when the data are not high dimensional.
The test statistics, a modification of the ANOVA-type statistic (Brunner, 2001), are
based on Box’s approximation (Box, 1954a), and follow a Â2
f -distribution. The estimators,
the building blocks of the test statistics, are composed of quadratic and symmetric
bilinear forms, and are proved to be unbiased, L2-consistent and uniformly bounded in
dimension, d. This last property of estimators helps us in the asymptotic derivations in
that 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 distribution
remains accurate when d > n, or even d >> n.
The performance of the statistics is evaluated through simulations and it is shown
that, for n as small as 10 or 20, the approximation is quite accurate, whatever be d. The
statistic is also applied to a number of real data sets for numerical illustrations.