Six strategies produce p-values 0.001, 0.008, 0.021, 0.030, 0.041 and 0.400. Apply Bonferroni at a family-wise error rate of 5% and Benjamini-Hochberg at a false discovery rate of 5%, and state exactly which strategies each rule keeps.

Six strategies produce p-values 0.001, 0.008, 0.021, 0.030, 0.041 and 0.400. Apply Bonferroni at a family-wise error rate of 5% and Benjamini-Hochberg at a false discovery rate of 5%, and state exactly which strategies each rule keeps.

Approach: Compute the single Bonferroni threshold, then walk the sorted p-values against the step-up boundary and take the largest index that clears it.

Bonferroni keeps the first two and Benjamini-Hochberg keeps the first five. Bonferroni tests each hypothesis at 0.05/6 = 0.00833, so only 0.001 and 0.008 survive, and the family-wise error rate, the chance of even one false keep, is held at 5%. Benjamini-Hochberg sorts the p-values ascending and finds the largest k with p_(k) <= 0.05*k/6: the comparisons are 0.001 against 0.00833, 0.008 against 0.01667, 0.021 against 0.025, 0.030 against 0.0333, 0.041 against 0.0417 and 0.400 against 0.05, so k = 5 and everything up to 0.041 is kept, including entries that individually fail Bonferroni. The false discovery rate is the expected proportion of false keeps among the keeps, so with 5 kept at 5% you expect about 0.25 of them to be spurious.

Follow-up: Change the largest p-value from 0.400 to 0.045. Which strategies does each rule keep now, and why does one p-value change the fate of another?

Key concepts: Bonferroni correction, Benjamini-Hochberg, false discovery rate, family-wise error rate.