Empirical process theory for robust inference in multivariate analyses and multiple testing

Motivation: This cumulative dissertation is about multivariate and multiple testing methods which are applicable for data beyond normality. In medicine, psychology, and biology, there is a need for testing procedures that are applicable in complex factorial designs and have robust properties at the same time. Most established approaches are less robust in the sense that they rely on strong assumptions such as normality and homogeneity, which cannot be justified in many quantitative research studies. Apart from single testing problems, which underlie anova and manova, multiple testing problems are equally relevant. The more complex the underlying factorial design, the more realistic it is that a multiple testing problem is of interest. Therefore, easy-interpretable and powerful solutions for single and multiple testing problems are of interest. To have robust methods at hand, there is a recent trend to resampling-based alternatives to classical analysis-of-variance (ANOVA) and multivariate analysis-of-variance (MANOVA), which rely on fewer assumptions. While there already exist robust alternatives for well-known and even complexer mean-based factorial designs, the situation is different for estimands beyond the mean. Inference regarding quantiles, especially regarding the median, is a potentially robust and powerful alternative to mean-based inference, especially in context of skewed and heavy-tailed data. Another issue, where the use of resampling methods can be feasible, is the presence of covariates. For example in medicine and psychology, there is often is the desire or even a requirement to adjust for covariates. In this situation, the covariate-adjusted means can be an adequate estimand. All research questions are motivated by real data examples from psychology or biology. They exemplify the high applicability of and the need for the respective methods. Methods: This dissertation fills some gaps in the methodology for robust inference in multivariate analyses and multiple testing. The contributions are described as follows: The first article introduces a manova with quantiles as estimands for general factorial designs by implementing resampling tests based on quadratic form-type statistics. The asymptotic theory is based on empirical processes. The second article includes a general comparison of one- and two-sided quantile-based simultaneous multiple testing procedures and extends some existing methods for quantile-based multiple testing to one-sided testing problems. An extensive Monte Carlo simulation was carried out, which provides a detailed view of the behaviour of the testing procedures under the null hypothesis and under the alternative. In the third article, simultaneous multiple testing on covariate-adjusted means was made available through the implementation of multiple contrast test procedures in a semiparametric model for multivariate analysis-of-covariance (mancova). Results: In all three articles, it has been shown that the new methods are asymptotically valid and consistent resampling tests. To reach this conclusion, aspects of asymptotic statistics and empirical process theory have been applied. Apart from that, Monte Carlo simulations have been used to analyse the performance of the methods under the null hypothesis and under the alternative. This allows an insight into the behaviour of the new methods, especially on small sample sizes. For the first and the third article, the simulations allow an initial evaluation of the performance of the new methods. In the second article, the comparative simulation studies enable a deeper insight in which scenarios the behaviour of the methods is satisfactory and in which it is less good. In that context, some surprising results have been obtained.