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). However, to our knowledge, they 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 nonnegative dependence. Because alpha investing can be recovered as a special case of the SAFFRON framework, the same result applies to this method as well. Our result also allows for certain forms of adaptive stopping times, for example, stopping after a certain number of rejections have been observed.
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SAFFFRON。这三种方法已经证明可以在线控制“修改”假发现率(MFDR ) 。然而,据我们所知,这些方法只显示在测试统计数据的独立条件下控制了传统的虚假发现率(FDR ) 。 我们的工作支持了这些结果,显示SAFFRON和耶和华在非负面依赖下进一步确保了FDR的在线控制。由于阿尔法投资可以作为SAFRON框架的一个特殊案例被回收,同样的结果也适用于这种方法。我们的结果还允许某些适应性停止时间,例如,在观察到一定数量的拒绝后停止。
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