Different statistical functionals are omnipresent in nonparametric statistics. Functionals like the Mann-Whitney functional (relative effect) P(X<Y) measure a location or stochastic tendency effect for two independent samples. We extend this by additionally incorporating an overlap index (nonparametric dispersion measure). We develop the joint asymptotic distribution of their rank-based estimators and construct confidence regions based on a resampling approach.

Extending the two-sample rank sum test (Wilcoxon, Mann and Whitney), we propose a new test based on these functionals for the hypothesis of distribution equality. As we simultaneously test different functionals we get a much larger consistency region as in the classical test and in most cases a substantial improvement in the power to the Kolmogorow Smirnov Test, as our simulation shows. We additionally show the usability of our approach by applying the test to different real data sets.

Evaluating the effect of a group variable on a response variable, while controlling for nuisance variables (covariates), is referred to as analysis of covariance (ANCOVA). The idea is to reduce response variance by accounting some of the total variance to the covariates, and thus, increase power of statistical tests for group effects. Besides theory for parametric ANCOVA, which is restricted to normally distributed data, there exist also some semi- and nonparametric approaches. However, none of the existing approaches can be reliably applied to small sample studies, since various simulations have indicated poor type-I error control in these situations. We close this gap by enhancing existing methods with newly developed small sample tests. In other words, we build a robust ANCOVA that is applicable when classical parametric assumptions are violated or when available data is limited. The performance of our tests is evaluated in extensive simulations that demonstrate an improved type-I error control, while maintaining competitive power.

The determination of alert concentrations, where a pre-specified threshold of the response variable is exceeded, is an important goal of concentration–response studies. The traditional approach is based on investigating the measured concentrations and attaining statistical significance of the alert concentration by using a multiple t-test procedure.

In this talk, we propose a new model-based method to identify alert concentrations, based on first fitting a concentration–response curve and second constructing a simultaneous confidence band for the difference of the response of a concentration compared to the control. In order to obtain these confidence bands, we use a bootstrap approach which can be applied to any functional form of the concentration–response curve. This particularly offers the possibility to investigate also those situations where the concentration–response relationship is not monotone and, moreover, allows to detect alerts at concentrations which were not measured during the study, providing a highly flexible framework for the problem at hand.

We demonstrate the validity of the method by means of a simulation study and present an application to a real dataset investigating the effect of different concentrations of the compound VPA on the development of hESC to neuroectoderm.

A common problem in clinical trials is to test whether an effect of an explanatory variable on the response, e.g. the effect of the dose of a compound on efficacy, is similar between two groups. In this context, similarity is defined as equivalence up to a pre-specified threshold specifying the accepted deviation between the groups. Such question is usually assessed by testing whether the (marginal) effects of the explanatory variable on the response are similar, based on, for example, confidence intervals for differences, or, to mention another example, the distance between two parametric models. These approaches typically assume a univariate continuous or binary outcome variable. An approach for associated bivariate binary response variables, based on the Gumbel model, has been recently introduced (Möllenhoff et al., 2021).

In this talk, we propose a flexible extension of such methodology that builds on a generalized joint regression framework with Gaussian copula. Compared to existing approaches, this allows for various scales of the outcome variables (e.g. continuous, binary, categorical, ordinal) including mixed outcomes as well as responses with more than two dimensions. We demonstrate the validity of our approach by means of a simulation study. An efficacy-toxicity case study demonstrates the practical relevance of the approach.

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Möllenhoff, K., Dette, H., and Bretz, F. (2021). Testing for similarity of binary efficacy–toxicity responses. Biostatistics, 23(3), 949–966.