While there exists several inferential methods for analyzing functional data in factorial designs, there is a lack of statistical tests that are valid (i) in general designs, (ii) under non-restrictive assumptions on the data generating process and (iii) allow for coherent post-hoc analyses. In particular, most existing methods assume Gaussianity or equal covariance functions across groups (homoscedasticity) and are only applicable for specific study designs that do not allow for evaluation of interactions. Moreover, all available strategies are only designed for testing global hypotheses and do not directly allow a more in-depth analysis of multiple local hypotheses. To address the first two problems (i)-(ii), we propose flexible integral-type test statistics that are applicable in general factorial designs under minimal assumptions on the data generating process. In particular, we neither postulate homoscedasticity nor Gaussianity. To approximate the statistics' null distribution, we adopt a resampling approach and validate it methodologically. Finally, we use our flexible testing framework to (iii) infer several local null hypotheses simultaneously. To allow for powerful data analysis, we thereby take the complex dependencies of the different local test statistics into account. In extensive simulations we confirm that the new methods are flexibly applicable. Two illustrate data analyses complete our study. The new testing procedures are implemented in the R package multiFANOVA, which will be available on CRAN soon.

翻译：尽管存在多种推理方法可用于分析因子设计中的函数型数据，但目前仍缺乏满足以下条件的统计检验方法：(i)适用于一般设计，(ii)对数据生成过程不作严格假设，(iii)支持一致的后续分析。特别地，现有大多数方法假设数据服从高斯分布或组间协方差函数相等（同方差性），且仅适用于无法评估交互效应的特定研究设计。此外，现有策略均仅用于检验全局假设，无法直接对多个局部假设进行深度分析。为解决前两个问题(i)-(ii)，我们提出了灵活积分型检验统计量，该统计量在数据生成过程的极简假设下适用于一般因子设计，特别地，既不要求同方差性也不要求高斯性。为近似统计量的零分布，我们采用重抽样方法并对其进行方法论验证。最后，利用该灵活检验框架(iii)同时推断多个局部零假设。为实现高效数据分析，我们考虑了不同局部检验统计量之间的复杂依赖性。通过大量模拟实验验证了新方法的灵活性，并通过两个实例数据分析完善研究。新检验程序已在R包multiFANOVA中实现，即将于CRAN上线。