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在诸多实际应用中常遭遇结构约束。以针对阿尔茨海默病（Alzheimer’s Disease）的结构磁共振成像（Magnetic Resonance Imaging）研究为例，其分析范畴不仅涵盖脑萎缩区域的检测，还需包含解剖学相邻区域的比较。此类约束可被建模为参数的线性变换，其中符号模式在方向效应的估计过程中发挥着关键作用。涵盖总变分、小波变换、融合套索（fused LASSO）、趋势滤波等方法的这类问题，在有效控制定向错误发现率（directional false discovery rate）方面仍面临公开挑战。本文提出一种扩展版的分裂拷贝（Split Knockoff）方法，专门用于解决线性变换约束下的定向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，并借助优化领域常用的变量分裂技术实现这一过程。该方法论可生成正交设计，从而同时兼顾检验效能与定向错误发现率的控制。通过引入样本拆分方案，本文实现了定向错误发现率的有效控制，且随着松弛邻域的扩大，该错误率可显著降至零。为验证所提方法的有效性，本文开展了模拟实验，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。本文的补充材料可在线获取。