On the power of some binomial modifications of the Bonferroni multiple test

Widely used in testing statistical hypotheses, the Bonferroni multiple test has a rather low power that entails a high risk to accept falsely the overall null hypothesis and therefore to not detect really existing effects. We suggest that when the partial test statistics are statistically independent, it is possible to reduce this risk by using binomial modifications of the Bonferroni test. Instead of rejecting the null hypothesis when at least one of n partial null hypotheses is rejected at a very high level of significance (say, 0.005 in the case of n = 10), as it is prescribed by the Bonferroni test, the binomial tests recommend to reject the null hypothesis when at least k partial null hypotheses (say, k = [n/2]) are rejected at much lower level (up to 30-50%). We show that the power of such binomial tests is essentially higher as compared with the power of the original Bonferroni and some modified Bonferroni tests. In addition, such an approach allows us to combine tests for which the results are known only for a fixed significance level. The paper contains tables and a computer program which allow to determine (retrieve from a table or to compute) the necessary binomial test parameters, i.e. either the partial significance level (when k is fixed) or the value of k (when the partial significance level is fixed).