Rank‐based permutation approaches for non‐parametric factorial designs

Abstract

Inference methods for null hypotheses formulated in terms of distribution functions in general non-parametric factorial designs are studied. The methods can be applied to continuous, ordinal or even ordered categorical data in a unified way, and are based only on ranks. In this set-up Wald-type statistics and ANOVA-type statistics are the current state of the art. The first method is asymptotically exact but a rather liberal statistical testing procedure for small to moderate sample size, while the latter is only an approximation which does not possess the correct asymptotic α level under the null. To bridge these gaps, a novel permutation approach is proposed which can be seen as a flexible generalization of the Kruskal-Wallis test to all kinds of factorial designs with independent observations. It is proven that the permutation principle is asymptotically correct while keeping its finite exactness property when data are exchangeable. The results of extensive simulation studies foster these theoretical findings. A real data set exemplifies its applicability.

Randomization Tests for Weak Null Hypotheses in Randomized Experiments

“…Rank statistics are attractive for the FRT because their distributions do not depend on the outcome values under Fisher's sharp null hypothesis (Lehmann 1975;Rosenbaum 2002a). They are useful for testing some weak null hypotheses (Brunner and Puri 2001;Chung and Romano 2016a;Umlauft et al 2017). However, it is not entirely clear what causal effects they are testing under the potential outcomes framework.…”

Rank-Based Procedures in Factorial Designs: Hypotheses About Non-Parametric Treatment Effects

“…() and Umlauft et al . (). In our situation this becomes even worse because of the complicated structure of the covariance matrix V involving more unknown quantities which must be estimated.…”

Factorial analyses of treatment effects under independent right-censoring

“…On the other hand, nonparametric estimation and the inference of adequate effects in such designs can be rather involved. In particular, most existing inference procedures have focused on testing hypotheses formulated in terms of distribution functions Brunner and Puri, 2001;Alvo, 2005, 2008;Akritas, 2011;Dutta and Datta, 2016;Friedrich et al, 2017;Umlauft et al, 2017a) which cannot be inverted to obtain confidence intervals or regions for meaningful effects. Only recently, nonparametric methods for inferring adequate effects in general factorial designs with independent and dependent observations have been established (Konietschke et al, 2016;Brunner et al, 2017;Umlauft et al, 2017b;.…”

“…In particular, most existing inference procedures have focused on testing hypotheses formulated in terms of distribution functions. 7–15 But those cannot be inverted to obtain confidence intervals or regions for meaningful effects. Only recently, nonparametric methods for inferring adequate effects in general factorial designs with independent and dependent observations have been established.…”