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

Abstract

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.

KEYWORDS:

• Alzheimer’s disease
• Multiple comparison hypothesis test
• Pairwise comparison
• Transformational sparsity
• Variable splitting

Supplementary Materials

The supplementary materials accompanying this paper are comprised of two components. The first component is an online PDF file, which encompasses comprehensive technical proofs and insightful discussions on related works. The second component is a ZIP file, furnished with the programming code and corresponding data that substantiate the findings presented in the paper.

Disclosure Statement

The authors report there are no competing interests to declare.

Notes

1 This particular W statistics is known as $W^{\mathrm{S}}$ in Cao, Sun, and Yao (2023).

Additional information

Funding

The authors gratefully acknowledge National Natural Science Foundation of China/Research Grants Council Joint Research Scheme Grant HKUST635/20, Hong Kong Research Grant Council (HKRGC) Grant 16308321, 16303817, ITF UIM/390, and the State Key Program of National Natural Science Foundation of China under Grant No. 12331009. This research made use of the computing resources of the X-GPU cluster supported by the HKRGC Collaborative Research Fund C6021-19EF.