Bonferroni cutoff for 40 tests at a 10% family level

Question

An analyst runs 40 hypothesis tests and is willing to tolerate a 10% family-wise error rate (a 10% chance of any false positive across the 40).

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

Accepted Answer

Split the chosen budget. Here the total false-alarm budget is 10%, not 5%. Bonferroni divides that budget evenly over the tests. With m tests and target \alpha, \frac{\alpha}{m} = \frac{0.10}{40} = 0.0025. So each test is significant only if its p-value is at most 0.0025. Why it holds. Every test risks at most 0.25% of a false alarm. Summing the worst case across all 40 gives at most 40 	imes 0.0025 = 0.10, matching the 10% family target. Note the flexibility. The numerator is whatever family-wise rate you choose, 5%, 10%, or anything else. Bonferroni just takes that number and divides by ho…

Bonferroni cutoff for 40 tests at a 10% family level

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