Bonferroni cutoff for 25 tests

Question

You run 25 hypothesis tests and want to keep the family-wise error rate (the chance of any false positive across all 25) at 5%.

Using the Bonferroni correction, what per-test p-value cutoff must each individual test beat to be called significant?

Accepted Answer

The Bonferroni idea. If you split your total 5% false-alarm budget evenly across the tests, each test gets only a tiny slice. With m tests and a target family-wise rate \alpha, the per-test cutoff is \frac{\alpha}{m} = \frac{0.05}{25} = 0.002. So a test counts as significant only if its p-value is at most 0.002, not the usual 0.05. Why this controls the family. Each test now has at most a 0.2% chance of a false alarm. Add up the worst case across all 25 (this is the union bound: the chance of any alarm is at most the sum of the individual chances), and you get at most 25 	imes 0.002 = 0.05. …

Bonferroni cutoff for 25 tests

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