Online testing procedures assume that hypotheses are observed in sequence, and allow the significance thresholds for upcoming tests to depend on the test statistics observed so far. Some of the most popular online methods include alpha investing, LORD++ (hereafter, LORD), and SAFFRON. These three methods have been shown to provide online control of the "modified" false discovery rate (mFDR) under a condition known as conditional superuniformity. However, to our knowledge, LORD & SAFFRON have only been shown to control the traditional false discovery rate (FDR) under an independence condition on the test statistics. Our work bolsters these results by showing that SAFFRON and LORD additionally ensure online control of the FDR under a "local" form of nonnegative dependence. Further, FDR control is maintained under certain types of adaptive stopping rules, such as stopping after a certain number of rejections have been observed. Because alpha investing can be recovered as a special case of the SAFFRON framework, our results immediately apply to alpha investing as well. In the process of deriving these results, we also formally characterize how the conditional superuniformity assumption implicitly limits the allowed p-value dependencies. This implicit limitation is important not only to our proposed FDR result, but also to many existing mFDR results.

翻译：在线测试程序假定,假设按顺序观察,并允许即将进行的测试的重要阈值取决于迄今所观察到的测试统计数据。一些最受欢迎的在线方法包括阿尔法投资、 耶和华++(以下称 耶和华+)和SAFFRON。这三种方法已经证明对“修改”假发现率(MFDR)提供在线控制,其条件被称为有条件的超统一。然而,据我们所知,根据测试统计的独立条件,耶和华 & SAFFFRRON只能控制传统的虚假发现率(FDR ) 。我们的工作支持了这些结果,表明SAFFRRON和耶和耶和华还确保了FDR在“局部”形式的非负面依赖下对FDR的在线控制。此外,FDR控制在某些类型的适应性停止规则下得以维持,例如在观察到了一定数量的拒绝后停止。然而,ADR投资作为SAF框架的一个特殊案例,我们的结果也立即适用于ADR投资。在得出这些结果的过程中,我们还正式说明,有条件的SUforfornformlity假设如何默认地限制了我们现有的结果。