Manifold Asymptotics of Quadratic-Form-Based Inference in Repeated Measures Designs

TL;DR

This paper investigates the asymptotic behavior of quadratic-form-based tests in repeated measures designs, especially in high-dimensional settings where traditional assumptions do not hold, providing new insights and approximations for critical values.

Contribution

It offers a unified analysis of quadratic-form asymptotics across diverse high-dimensional regimes without restrictive assumptions on growth rates.

Findings

Provides asymptotic distributions for quadratic forms in various high-dimensional settings.

Develops an approximation method for critical values in tests based on quadratic forms.

Extensive simulation study validating the proposed asymptotic approximations.

Abstract

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 $\infty$ 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 kind of combinations, where they diverge or are bounded in a unified framework. Studying the limit distributions in detail, we follow Sattler and Pauly (2018) and propose an approximation to obtain critical values. The resulting test together with their approximation approach are investigated in an extensive simulation study with a focus on the exceptional asymptotic frameworks which are the main focus of this work.