Inference for high-dimensional split-plot-designs: A unified approach for small to large numbers of factor levels

Abstract

Statisticians increasingly face the problem to reconsider the adaptability of classical inference techniques. In particular, diverse types of high-dimensional data structures are observed in various research areas; disclosing the boundaries of conventional multivariate data analysis. Such situations occur, e.g., frequently in life sciences whenever it is easier or cheaper to repeatedly generate a large number $d$ of observations per subject than recruiting many, say $N$, subjects. In this paper, we discuss inference procedures for such situations in general heteroscedastic split-plot designs with $a$ independent groups of repeated measurements. These will, e.g., be able to answer questions about the occurrence of certain time, group and interactions effects or about particular profiles. The test procedures are based on standardized quadratic forms involving suitably symmetrized U-statistics-type estimators which are robust against an increasing number of dimensions $d$ and/or groups $a$. We then discuss their limit distributions in a general asymptotic framework and additionally propose improved small sample approximations. Finally, the small sample performance is investigated in simulations and applicability is illustrated by a real data analysis.