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, 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.

A comprehensive treatment of quadratic-form-based inference in repeated measures designs under diverse asymptotics

“…Here (•) − denotes some generalized inverse of the matrix and H 0 (H) can equivalently be written as H 0 (T ) : T μ = 0. As discussed in [18], T has the form T = T W ⊗ T S for projection matrices T W and T S . Now hypotheses of interest are for example given by (a) No group effect:…”

“…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.…”

Small sample sizes: A big data problem in high-dimensional data analysis

“…As explained above, the vector of t-test type statistics follows, asymptotically, a multivariate normal distribution with expectation 0 and correlation matrix R in low-dimensional settings (d fixed). This means that a proper resampling algorithm must be designed in such a way that the resampling distribution of T, say T Ã , converges to the Nð0; RÞ distribution, respectively, where the correlation matrix must be identical to the one defined in Equation (8). Moreover, in high-dimensional settings (with d !…”

“…These may be tackled by applying global testing procedures, which have been developed for different high-dimensional repeated measures and multivariate ANOVA models by several authors. 1 – 8 Furthermore, multivariate tests based on interpoint distances have been proposed. 9 – 12 Testing global null hypotheses and herewith answering the question whether any difference among the repeated measurements per or across endpoints exists, however, does usually not answer the main question of the practitioners—that is the specific localization of the responsible experimental conditions that lead to the overall significance conclusion.…”

HRM: An R Package for Analysing High-dimensional Multi-factor Repeated Measures

“…Although it generally performs well, it is not asymptotically exact. There are asymptotic tests such as those proposed by or Sattler and Pauly (2017), converging under some assumptions such as min{n i , i = 1, . .…”