Statisticians increasingly face the problem to reconsider the adaptability of classical inference techniques. In particular, divers 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 its limit distributions in a general asymptotic framework and additionally propose improved small sample approximations. Finally its small sample performance is investigated in simulations and the applicability is illustrated by a real data analysis.

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Clustered data arise frequently in many practical applications whenever units are repeatedly observed under a certain condition. One typical example for clustered data are animal experiments, where several animals share the same cage and should not be assumed to be completely independent. Standard methods for the analysis of such data are Linear Mixed Models and Generalized Estimating Equations—however, checking their assumptions is not easy, especially in scenarios with small sample sizes, highly skewed, count, and ordinal or binary data. In such situations, Wilcoxon–Mann–Whitney type effects are suitable alternatives to mean-based or other distributional approaches. Hence, no specific data distribution, symmetric or asymmetric, is required. Within this work, we will present different estimation techniques of such effects in clustered factorial designs and discuss quadratic- and multiple contrast type-testing procedures for hypotheses formulated in terms of Wilcoxon–Mann–Whitney effects. Additionally, the framework allows for the occurrence of missing data: estimation and testing hypotheses are based on all-available data instead of complete-cases. An extensive simulation study investigates the precision of the estimators and the behavior of the test procedures in terms of their type-I error control. One real world dataset exemplifies the applicability of the newly proposed procedures.

Split-Plot or Repeated Measures Designs with multiple groups occur naturally in sciences. Their analysis is usually based on the classical Repeated Measures ANOVA. Roughly speaking, the latter can be shown to be asymptotically valid for large sample sizes n i assuming a fixed number of groups a and time points d. However, for high-dimensional settings with d > n i , this argument breaks down and statistical tests are often based on (standardized) quadratic forms. Furthermore, analysis of their limit behaviour is usually based on certain assumptions on how d converges to ∞ with respect to n i . As this may be hard to argue in practice, we do not want to make such restrictions. Moreover, sometimes also the number of groups a may be large compared to d or n i . To also have an impression about the behaviour of (standardized) quadratic forms as test statistic, we analyze their asymptotics under diverse settings on a, d and n i . In fact, we combine all kinds of combinations, where they diverge or are bounded in a unified framework. To this aim, we assume equal covariance matrices between all groups. Studying the limit distributions in detail, we follow Sattler and Pauly (2018) and propose an approximation to obtain critical values. The resulting test and its approximation approach are investigated in an extensive simulation study focusing on the exceptional asymptotic frameworks that are the main focus of this work.

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