Split Knockoffs for Multiple Comparisons: Controlling the Directional False Discovery Rate

Multiple comparisons in hypothesis testing often encounter structural constraints in various applications. For instance, in structural Magnetic Resonance Imaging for Alzheimer’s Disease, the focus extends beyond examining atrophic brain regions to include comparisons of anatomically adjacent regions. These constraints can be modeled as linear transformations of parameters, where the sign patterns play a crucial role in estimating directional effects. This class of problems, encompassing total variations, wavelet transforms, fused LASSO, trend filtering, and more, presents an open challenge in effectively controlling the directional false discovery rate. In this article, we propose an extended Split Knockoff method specifically designed to address the control of directional false discovery rate under linear transformations. Our proposed approach relaxes the stringent linear manifold constraint to its neighborhood, employing a variable splitting technique commonly used in optimization. This methodology yields an orthogonal design that benefits both power and directional false discovery rate control. By incorporating a sample splitting scheme, we achieve effective control of the directional false discovery rate, with a notable reduction to zero as the relaxed neighborhood expands. To demonstrate the efficacy of our method, we conduct simulation experiments and apply it to two real-world scenarios: Alzheimer’s Disease analysis and human age comparisons. Supplementary materials for this article are available online.

假设检验中的多重比较问题在诸多实际应用场景中常面临结构性约束。例如，针对阿尔茨海默病（Alzheimer’s Disease）的结构磁共振成像（Magnetic Resonance Imaging, MRI）研究中，分析重点不仅限于探查脑萎缩区域，还需对解剖学上相邻的脑区开展比较。这类约束可被建模为参数的线性变换，其中符号模式在方向性效应的估计过程中发挥着关键作用。此类问题涵盖总变差（total variations）、小波变换（wavelet transforms）、融合套索（fused LASSO）、趋势滤波等多个范畴，在有效控制方向性错误发现率（directional false discovery rate）方面仍存在开放性挑战。本文提出一种扩展版分裂柯普（Split Knockoff）方法，专门用于解决线性变换约束下的方向性错误发现率控制问题。所提方法将严苛的线性流形约束松弛至其邻域，并采用了优化领域中常用的变量拆分技术。该方法构建了正交设计框架，可同时提升检验效力并优化方向性错误发现率的控制效果。通过引入样本拆分策略，我们实现了方向性错误发现率的有效控制，且随着松弛邻域的扩大，该错误发现率可显著降至零。为验证所提方法的有效性，我们开展了仿真实验，并将其应用于两个实际场景：阿尔茨海默病分析与人类年龄比较研究。本文的补充材料可在线获取。