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Abstract This study introduces a novelp-value-based multiple testing approach tailored for generalized linear models. Despite the crucial role of generalized linear models in statistics, existing methodologies face obstacles arising from the heterogeneous variance of response variables and complex dependencies among estimated parameters. Our aim is to address the challenge of controlling the false discovery rate (FDR) amidst arbitrarily dependent test statistics. Through the development of efficient computational algorithms, we present a versatile statistical framework for multiple testing. The proposed framework accommodates a range of tools developed for constructing a new model matrix in regression-type analysis, including random row permutations and Model-X knockoffs. We devise efficient computing techniques to solve the encountered non-trivial quadratic matrix equations, enabling the construction of pairedp-values suitable for the two-step multiple testing procedure proposed by Sarkar and Tang (Biometrika 109(4): 1149–1155, 2022). Theoretical analysis affirms the properties of our approach, demonstrating its capability to control the FDR at a given level. Empirical evaluations further substantiate its promising performance across diverse simulation settings. 

For simultaneous testing of multivariate normal means with known correlation matrix against two-sided alternatives, this paper introduces new methods with proven finite-sample control of false discovery rate. The methods are obtained by shifting each p-value to the left and considering a Benjamini–Hochberg-type linear step-up procedure based on these shifted p-values. The amount of shift for each -value is appropriately determined from the correlation matrix to achieve the desired false discovery rate control. Simulation studies and real-data application show favorable performances of the proposed methods when compared with relevant competitors.