Valid online inference is an important problem in contemporary multiple testing research,to which various solutions have been proposed recently. It is well-known that these existing methods can suffer from a significant loss of power if the null $p$-values are conservative. In this work, we extend the previously introduced methodology to obtain more powerful procedures for the case of super-uniformly distributed $p$-values. These types of $p$-values arise in important settings, e.g. when discrete hypothesis tests are performed or when the $p$-values are weighted. To this end, we introduce the method of super-uniformity reward (SUR) that incorporates information about the individual null cumulative distribution functions. Our approach yields several new 'rewarded' procedures that offer uniform power improvements over known procedures and come with mathematical guarantees for controlling online error criteria based either on the family-wise error rate (FWER) or the marginal false discovery rate (mFDR). We illustrate the benefit of super-uniform rewarding in real-data analyses and simulation studies. While discrete tests serve as our leading example, we also show how our method can be applied to weighted $p$-values.
翻译:有效的在线推论是当代多种测试研究中的一个重要问题,最近提出了各种解决办法。众所周知,如果无美元价值是保守的,这些现有方法可能会遭受重大权力损失。在这项工作中,我们扩展了以前采用的方法,以获得超统一分布的美元价值的更强有力的程序。这些美元价值类型出现在重要的环境中,例如,进行离散假设测试或加权美元价值时。为此,我们引入了超统一性奖励方法(SUR),该方法将个人无累积分配功能的信息纳入其中。我们的方法产生若干新的“奖励”程序,为已知程序提供统一的权力改进,并配以基于家庭错误率(FWER)或边际错误发现率(MFDR)的数学保证来控制在线错误标准。我们举例说明了在真实数据分析和模拟研究中超级统一性报酬的好处。离散性测试作为我们的主要例子,我们也展示了我们的方法如何应用到加权美元价值。