## What is Bonferroni Correction?

The Bonferroni Correction is one of the earliest procedures for controlling the Family-Wise Error Rate (FWER) which is the error rate across a set of tightly related significance tests. The goal of the correction is to maintain the overall type I error rate which is computed under the null hypothesis that all nulls tested in the significance tests are false. The correction provides a conservative bound on alpha_{m}.

Bonferroni derived the calculation that the overall α of performing **m** significance tests is equal to **1 - (1 - α _{per test})^{m}** which is the probability that one of them will result in a statistically significant outcome. The simple Bonferroni correction would suggest performing each test at level α/m to maintain the Family-Wise Error Rate (FWER) fixed at α, but this is a conservative adjustment when the comparisons are not independent.

The Bonferroni correction can be applied when there is more than one primary KPI in an A/B test and finding any of them to be statistically significant would result in deciding against the control. Still, it is a bit conservative in the presence of positive dependence so the Sidak Correction is usually slightly more powerful and thus preferred.

While the Bonferroni correction can also be applied to a multivariate test (A/B/n test) it is not the best choice as it takes no account of the dependency present between the tests due to the fact that they are all against a common control. In such cases the Dunnett's correction provides a significantly more powerful method that also control FWER as defined above.