Asymptotic false discovery control of the Benjamini-Hochberg procedure for pairwise comparisons

In a one-way analysis-of-variance (ANOVA) model, the number of pairwise comparisons can become large even with a moderate number of groups. Motivated by this, we consider a regime with a growing number of groups and prove that, when testing pairwise comparisons, the Benjamini-Hochberg (BH) procedure can asymptotically control false discoveries, despite the fact that the involved t-statistics do not exhibit the well-known positive dependence structure required for exact false discovery rate (FDR) control. Following Tukey’s perspective that the difference between the means of any two groups cannot be exactly zero, our main result provides control over the directional false discovery rate and directional false discovery proportion. A key technical contribution of our work is demonstrating that the dependence among the t-statistics is sufficiently weak to establish the convergence result typically required for asymptotic FDR control. Our analysis does not rely on conventional assumptions such as normality, variance homogeneity, or a balanced design, thereby offering a theoretical foundation for applications in more general settings.