LAWS: A Locally Adaptive Weighting and Screening Approach to Spatial Multiple Testing

Abstract

Exploiting spatial patterns in large-scale multiple testing promises to improve both power and interpretability of false discovery rate (FDR) analyses. This article develops a new class of locally adaptive weighting and screening (LAWS) rules that directly incorporates useful local patterns into inference. The idea involves constructing robust and structure-adaptive weights according to the estimated local sparsity levels. LAWS provides a unified framework for a broad range of spatial problems and is fully data-driven. It is shown that LAWS controls the FDR asymptotically under mild conditions on dependence. The finite sample performance is investigated using simulated data, which demonstrates that LAWS controls the FDR and outperforms existing methods in power. The efficiency gain is substantial in many settings. We further illustrate the merits of LAWS through applications to the analysis of two-dimensional and three-dimensional images. Supplementary materials for this article are available online.

Keywords:

• Adjusted p-value
• Covariate-assisted inference
• Dependent tests
• False discovery rate
• Structured multiple testing

Supplementary Materials

This supplement contains the proofs of the main results (Section A) and some additional numerical and theoretical explanations (Sections B and C).

Notes

1 In other applications such as climate change analysis, one observes incomplete data points at irregular locations (e.g., weather monitoring stations) but needs to make inference at every point in the whole spatial domain. This setting goes beyond the scope of our work; see Sun et al. (2015) for related discussions.

2 The actual order would not affect the methodology or theory as the weights are fully determined by the spatial structure. We only need an ordering for characterizing the dependence structure between all pairs of p-values.

Additional information

Funding

The research of Tony Cai was supported in part by NSF grants DMS-1712735 and DMS-2015259 and NIH funding: grants R01-GM129781 and R01-GM123056. The research of Wenguang Sun was supported in part by NSF grants DMS-CAREER 1255406 and DMS-1712983. The research of Yin Xia was supported in part by NSFC grants 12022103, 11771094, and 11690013.