A General Framework for the Multiple Nonparametric Behrens–Fisher Problem With Dependent Replicates

In many trials and experiments, subjects are not only observed once but multiple times, resulting in a cluster of possibly correlated observations (e.g., brain regions per patient). Observations often do not fulfill model assumptions of mixed models and require the use of nonparametric methods. In this article, we develop and present a purely nonparametric rank-based procedure that flexibly allows the unbiased and consistent estimation of the Wilcoxon-Mann-Whitney effect <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>P</mml:mi> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo><</mml:mo> <mml:mi>Y</mml:mi> <mml:mo>)</mml:mo> <mml:mo>+</mml:mo> <mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow> <mml:mrow><mml:mn>2</mml:mn></mml:mrow> </mml:mfrac> <mml:mi>P</mml:mi> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>=</mml:mo> <mml:mi>Y</mml:mi> <mml:mo>)</mml:mo></mml:mrow> <mml:annotation>$$ P\left(X<Y\right)+\frac{1}{2}P\left(X=Y\right) $$</mml:annotation></mml:semantics> </mml:math> in clustered data designs. Compared with existing methods, we allow flexible weights to be used in effect estimation. Additionally, we develop global and multiple contrast test procedures to test null hypotheses formulated regarding the generalized Mann-Whitney effects and for the computation of range-preserving simultaneous confidence intervals in a unified way. Extensive simulation studies show that these methods control the type-I error rate well and have reasonable power to detect alternatives in various situations.