A Bonferroni threshold

Topic: High-dimensional analysis · more questions on this topic
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With m=5000 tests and \alpha=0.05, find the Bonferroni threshold and the required |z| for a two-sided normal test. Use \Phi^{-1}(1-10^{-5}/2)=4.42. What would the Benjamini–Hochberg thresholds be for the 5th and the 100th smallest p-values at q^*=0.05?

Worked answer (click to show)

The threshold is 0.05/5000=10^{-5}. For a two-sided normal test this requires

|z|\ge\Phi^{-1}(1-5\times 10^{-6})=4.42.

The Benjamini–Hochberg thresholds at q^*=0.05 are 5(0.05)/5000=5\times 10^{-5} for the fifth smallest and 100(0.05)/5000=10^{-3} for the hundredth. The threshold relaxes linearly with rank, so the more discoveries are made, the easier it is to make the next.