Sum-based global tests are highly popular in multiple hypothesis testing. In this paper we propose a general closed testing procedure for sum tests, which provides confidence lower bounds for the proportion of true discoveries (TDP), simultaneously over all subsets of hypotheses. Our method allows for an exploratory approach, as simultaneity ensures control of the TDP even when the subset of interest is selected post hoc. It adapts to the unknown joint distribution of the data through permutation testing. Any sum test may be employed, depending on the desired power properties. We present an iterative shortcut for the closed testing procedure, based on the branch and bound algorithm, which converges to the full closed testing results, often after few iterations. Even if it is stopped early, it controls the TDP. The feasibility of the method for high dimensional data is illustrated on brain imaging data. We compare the properties of different choices for the sum test through simulations.

翻译：在多个假设测试中,基于总和的全球测试非常普遍。在本文中,我们提议对总测试采用一般封闭测试程序,该程序为真实发现的比例提供了信任度较低的下限,同时覆盖所有假设子集。我们的方法允许一种探索性方法,因为同时性确保了对TDP的控制,即使有兴趣的子集是临时选定的。它适应了通过变换测试对数据进行不为人知的联合分配。任何总测试都可以使用,视想要的功率特性而定。我们根据分支和约束算法为封闭测试程序提供了一个迭接捷径,该算法往往在很少迭代之后与完全封闭测试结果汇合。即使它被早期停止,它也控制TDP。高维数据方法的可行性在脑成像数据上加以说明。我们比较了通过模拟进行总和测试的不同选择的特性。