Bonferroni cutoff for eight tests

Question

A researcher runs 8 hypothesis tests and wants the family-wise error rate (the chance of any false positive across the 8) held at 5%.

Using the Bonferroni correction, what per-test p-value cutoff must each test beat?

Accepted Answer

Divide the budget. Bonferroni spreads the 5% total false-alarm budget evenly over the tests. With m tests and target \alpha, each test must clear \frac{\alpha}{m} = \frac{0.05}{8} = 0.00625. So a result is significant only if its p-value is at most 0.00625. Why it works. Each test now risks at most 0.625% of a false alarm. Summing the worst case over all 8 (the union bound) gives at most 8 	imes 0.00625 = 0.05, so the family-wide false-alarm chance stays within 5%. Sanity check. A p-value of 0.02 would pass the naive 0.05 test but fails the Bonferroni bar of 0.00625, so it is not declared si…

Bonferroni cutoff for eight tests

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