Rank-Based Procedures in Factorial Designs: Hypotheses about Nonparametric Treatment Effects

TL;DR

This paper introduces a new rank-based nonparametric ANOVA-type statistic for factorial designs that effectively tests hypotheses about treatment effects, especially in heteroscedastic and unbalanced settings.

Contribution

It develops a novel nonparametric ANOVA-type statistic based on ranks suitable for general factorial designs, extending hypothesis testing beyond distribution functions.

Findings

Maintains accurate type-I error rates in unbalanced, heteroscedastic models.

Compares three approximation techniques with theoretical analysis.

Demonstrates effectiveness through extensive simulations.

Abstract

Existing tests for factorial designs in the nonparametric case are based on hypotheses formulated in terms of distribution functions. Typical null hypotheses, however, are formulated in terms of some parameters or effect measures, particularly in heteroscedastic settings. Here this idea is extended to nonparametric models by introducing a novel nonparametric ANOVA-type-statistic based on ranks which is suitable for testing hypotheses formulated in meaningful nonparametric treatment effects in general factorial designs. This is achieved by a careful in-depth study of the common distribution of rank-based estimators for the treatment effects. Since the statistic is asymptotically not a pivotal quantity we propose three different approximation techniques, discuss their theoretic properties and compare them in extensive simulations together with two additionalWald-type tests. An extension of the presented idea to general repeated measures designs is briefly outlined. The proposed rank-based procedures maintain the pre-assigned type-I error rate quite accurately, also in unbalanced and heteroscedastic models.