Solution
The Adjusted p-value That Moved
Show the problem again
Eight hypotheses have p-values 0.041, 0.500, 0.002, 0.034, 0.300, 0.009, 0.030, 0.012. The Benjamini-Hochberg adjusted p-value of the i-th smallest p-value is the smallest value of p(j) × 8 / j over all ranks j ≥ i (capped at 1); it is the smallest FDR level q at which that hypothesis would be rejected. What is the BH-adjusted p-value of the hypothesis with raw p-value 0.009? Give your answer as a decimal or a reduced fraction.
Worked solution
Sorted p-values are 0.002, 0.009, 0.012, 0.030, 0.034, 0.041, 0.300, 0.500. The ratios p(j) × 8 / j for j = 1, ..., 8 are 0.016, 0.036, 0.032, 0.060, 0.0544, 0.0547, 0.3429, 0.5. For the second smallest (0.009), take the minimum over ranks j ≥ 2: min(0.036, 0.032, 0.060, 0.0544, 0.0547, 0.3429, 0.5) = 0.032, attained at rank 3. So the adjusted p-value is 0.032 = 4/125, smaller than its own ratio 0.036: the hypothesis with p = 0.012 would be rejected at q = 0.032, and step-up rejection drags the smaller p-value along with it.
Source: Benjamini-Hochberg adjusted p-values (q-values) with the monotonicity step, as in the standard p.adjust implementation. Statement written for AxiomIQ.