Analysis of high-dimensional one group repeated measures designs

Abstract

We propose a novel one sample test for repeated measures designs and derive its limit distribution for the situation where both the sample size n as well as the dimension d of the observations go to infinity. This covers the high-dimensional case with $d>n$. The tests are based on a quadratic form which involve new unbiased and dimension-stable estimators of different traces of the underlying unrestricted covariance structure. It is shown that the asymptotic distribution of the statistic may be standard normal, standardized ${\chi }^2$-distributed, or even of weighted ${\chi }^2$-form in some situations. To this end, we suggest an approximation technique which is asymptotically valid in the first two cases and provides an accurate approximation for the latter. We motivate and illustrate the application with a sleep lab data set and also discuss the practical meaning of $d\to \mathrm{\infty }$ in case of repeated measures designs. It turns out that the limit behaviour depends on how the number of repeated measures is increased which is crucial for application.

Keywords:

• approximations
• high-dimensional data
• quadratic forms
• repeated measures

Acknowledgements

The authors would like to thank Marius Placzek for providing the graphs displayed in this paper. Moreover, the authors are grateful to an Associate Editor and two expert referees for helpful comments which considerably improved the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Supplemental data

In the supplement to the current paper (cf. Pauly et al. [21]) we provide additional technical results as well as supporting simulations for the investigated approximation. Supplemental data for this article can be accessed at doi:10.1080/02331888.2015.1050022.

Additional information

Funding

This work was supported by the German Research Foundation project DFG-PA 2409/3-1.