A Unified Multiple Testing Framework based on rho-values

Multiple testing is an important research area with widespread scientific applications, including in biology and neuroscience. Among popularly adopted multiple testing procedures, many are based on p-values or Local false discovery rate (Lfdr) statistics. However, p-values--often obtained via the probability integral transform of standard test statistics--typically lack information from the alternatives, resulting in suboptimal performance. In contrast, Lfdr-based methods can achieve asymptotic optimality, but their ability to control the false discovery rate (FDR) hinges on accurate estimation of the Lfdr, which can be challenging, especially when incorporating side information. In this article, we introduce a novel and flexible class of statistics, termed rho-values, and develop a corresponding multiple testing framework that integrates the strengths of both p-values and Lfdr, while addressing their respective limitations. Specifically, the rho-value framework unifies these two paradigms through a two-step process: ranking and thresholding. The ranking induced by rho-values closely resembles that of Lfdr-based methods, while the thresholding step aligns with conventional p-value procedures. Therefore, the proposed framework guarantees FDR control under mild assumptions; it maintains the integrity of the structural information encoded by the summary statistics and the auxiliary covariates, and hence can be asymptotically optimal. We demonstrate the advantages of the rho-value framework through comprehensive simulations and analyses of two real datasets: one from microbiome research and another related to attention deficit hyperactivity disorder (ADHD).