Multiple Comparisons and Bonferroni

= − ,

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Example: a researcher tests 20 hypotheses at 5% significance, and none of them has a real effect. What is the chance that at least one result comes out “significant” anyway? Change the values or draw again.

Number of tests k Significance α (%) Simulation seed

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What each term means

Chance of At Least One False Positive: The probability that, among k tests with no real effect, at least one comes out “significant”. Also called the family-wise error rate.
Complement (1 −): Easier than counting the cases with one, two or more false positives is computing the chance of none and subtracting it from 1.
No False Positive ((1 − α)ᵏ): Each test with no effect gets it right with chance 1 − α; with k independent tests, all get it right with chance (1 − α)ᵏ, which shrinks fast.
Bonferroni Correction (α′ = α/k): Requiring a p-value below α/k in each test keeps the chance of any false positive below α. The price is losing power to find real effects.