What is the Bonferroni correction and how does it work?

Bonferroni correction is a method for adjusting the significance level (p-value) in multiple testing to control the probability of false positive results (Type I errors).

When multiple statistical tests are conducted simultaneously, the probability of obtaining at least one false positive increases. The Bonferroni correction reduces this risk by dividing the original significance level ( \alpha ) by the number of tests ( m ):

[ \alpha_{adjusted} = \frac{\alpha}{m} ]

Thus, the significance threshold for each individual test becomes stricter. If the p-value of a test is less than ( \alpha_{adjusted} ), the result is considered statistically significant.

Example: If ( \alpha = 0.05 ) and 10 tests are conducted, then the threshold for each test will be 0.005.

This is a simple and conservative method that reduces the likelihood of false positives but may increase the number of missed true effects (Type II errors).