Bonferroni correction

In statistics, the Bonferroni correction states that if an experimenter is testing n dependent or independent hypotheses on a set of data, then one way of maintaining the familywise error rate is to test each individual hypothesis at a statistical significance level of 1/n times what it would be if only one hypothesis were tested. Statistically significant simply means that a given result is unlikely to have occurred by chance.

1 - (1 - α)^1/n = α (corrected for n comparisons)

For example, to test two independent hypotheses on the same data at 0.05 significance level, instead of using a p value threshold of 0.05, one would use a stricter threshold equal to the square root of 0.05. Notably one can derive valid confidence intervals matching the test decision using the Bonferroni correction by using 100*(1-α^1/n)% confidence intervals.

The Bonferroni correction is a safeguard against multiple tests of statistical significance on the same data falsely giving the appearance of significance, as 1 out of every 20 hypothesis-tests is expected to be significant at the α = 0.05 level purely due to chance. Furthermore, the probability of getting a significant result with n tests at this level of significance is 1-0.95ⁿ (1-probability of not getting a significant result with n tests).

It was developed by Italian mathematician Carlo Emilio Bonferroni.

A uniformly more powerful test procedure (i.e. more powerful regardless of the values of the unobservable parameters) is the Holm-Bonferroni method, however current methods for obtaining confidence intervals for the Holm-Bonferroni method do not guarantee confidence intervals that are contained within those obtained using the Bonferroni correction. A less restrictive criterion that does not control the familywise error rate is the rough false discovery rate giving (3/4)0.05 = 0.0375 for n = 2 and (21/40)0.05 = 0.02625 for n = 20.