In statistics, the Bonferroni correction is a method used to counteract the problem of multiple comparisons. It was developed and introduced by Italian mathematician Carlo Emilio Bonferroni. The correction is based on the idea 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. So, if it is desired that the significance level for the whole family of tests should be (at most) α, then the Bonferroni correction would be to test each of the individual tests at a significance level of α/n. Statistically significant simply means that a given result is unlikely to have occurred by chance assuming the null hypothesis is actually correct (i.e., no difference among groups, no effect of treatment, no relation among variables).
The Bonferroni correction is derived by observing Boole's inequality. If n tests are performed, each of them significant with probability β, (where β is unknown) then the probability that at least one of them comes out significant is (by Boole's inequality) ≤ nβ. Our intention is for this probability to equal α, the significance level for the entire series of tests. By solving for β, we get β = α/n. This result does not require that the tests be independent.
当有k个均数需作两两比较时,比较的次数共有c= = k!/(2!(k-2)!)=k(k-1)/2
例如,设α=0.05,c=3(即k=3),其累积Ⅰ类错误的概率为α'=1-(1-0.05)3 =1-(0.95)3 = 0.143