### Article

## Analysis of repeated measures designs for high dimensional data

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Published: | September 6, 2007 |
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We consider a group of n independent subjects which are repeatedly observed at d fixed time points. Typically, the aim of such a trial is to investigate whether there is a time effect, i.e. whether the time profile is flat. In addition, if the time profiles are observed under

different conditions, then we have a × d (structured) repeated measures for each subject. This set-up is known as profile analysis. If a × d < n, then well-established procedures are available for the analysis of this design assuming a multivariate normal distribution of the n observed vectors.

In recent years, the case of a × d > n, or even a × d >> n has gained considerable attention and several new procedures have been suggested for the analysis of such data, e.g. the generalized Hotelling’s T2 statistics [Ref. 3]. Here we consider a high dimensional version of the ANOVA-Type statistic [Ref. 2] which is based on the Box-approximation [Ref. 1]. Bilinear forms are suggested as estimators of the scaling factor and the degrees of freedom of the approximating Chi-square distribution. These estimators are unbiased and ratio-consistent for large n, irrespective of the dimension a × d of the underlying multivariate normal distribution. Simulations demonstrate that even for n = 10 the pre-assigned level of the test is well maintained. Moreover, this procedure does not depend on the structure of the underlying covariance matrix. Comparing this procedure with the generalized Hotelling’s T2 statistic, it turns out that it has much higher power which of course depends on the structure of the covariance matrix. The new procedure is applied to an example from a psychiatric study in a sleep lab.

### References

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- Box GEP. Some theorems on quadratic forms applied in the study of analysis of variance problems II. Annals of Mathematical statistics. 1954;25(3):484-98.
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- Brunner E, Puri ML. Nonparametric methods in factorial designs. Statistical Papers. 2001;42:1-52.
- 3.
- Srivastava MS. Some tests concerning the covariance matrix in high dimensional data. J. of Japanese Statistical Society. 2005;35:251-72.