Multivariate analysis of variance (MANOVA) is a powerful and versatile method to infer and quantify main and interaction effects in metric multivariate multi-factor data. It is, however, neither robust against change in units nor meaningful for ordinal data. Thus, we propose a novel nonparametric MANOVA. Contrary to existing rank-based procedures, we infer hypotheses formulated in terms of meaningful Mann–Whitney-type effects in lieu of distribution functions. The tests are based on a quadratic form in multivariate rank effect estimators, and critical values are obtained by bootstrap techniques. The newly developed procedures provide asymptotically exact and consistent inference for general models such as the nonparametric Behrens–Fisher problem and multivariate one-, two-, and higher-way crossed layouts. Computer simulations in small samples confirm the reliability of the developed method for ordinal and metric data with covariance heterogeneity. Finally, an analysis of a real data example illustrates the applicability and correct interpretation of the results.

中文翻译：

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有意义效应中的非参数 MANOVA

多元方差分析 (MANOVA) 是一种强大且通用的方法，可用于推断和量化度量多元多因素数据中的主要效应和交互效应。然而，它对单位的变化既不稳健，对有序数据也没有意义。因此，我们提出了一种新的非参数 MANOVA。与现有的基于等级的程序相反，我们推断出根据有意义的 Mann-Whitney 型效应而不是分布函数制定的假设。检验基于多元秩效应估计器中的二次形式，并且临界值通过自举技术获得。新开发的程序为一般模型（例如非参数 Behrens-Fisher 问题和多变量单、双和更高的交叉布局）提供了渐近精确和一致的推理。小样本中的计算机模拟证实了所开发的具有协方差异质性的序数和度量数据方法的可靠性。最后，对真实数据示例的分析说明了结果的适用性和正确解释。