False Discovery Control for Pairwise Comparisons - An Asymptotic Solution to Williams, Jones and Tukeys Conjecture

Under weak moment and asymptotic conditions, we offer an affirmative answer to whether the BH procedure (Benjamini and Hochberg, 1995) can control the false discovery rate in testing pairwise comparisons of means under a one-way ANOVA layout. Specifically, despite the fact that the two sample t-statistics do not exhibit positive regression dependency (Benjamini and Yekutieli, 2001), our result shows that the BH procedure can asymptotically control the directional false discovery rate as conjectured by Williams, Jones, and Tukey (1999). Such a result is useful for most general situations when the number of variables is moderately large and/or when idealistic assumptions such as normality and a balanced design are violated.