risk of incorrectly rejecting the null hypothesis. However, if 100 tests are each conducted at the 5% level and all corresponding null hypotheses are true, the expected number of incorrect rejections (also known as false positives or Type I errors) is 5. If the tests are statistically independent from each other, the probability of at least one incorrect rejection is approximately 99.4%. The multiple comparisons problem also applies to confidence intervals. A single confidence interval with a 95% coverage probability level will contain the true value of the parameter in 95% of samples. However, if one considers 100 confidence intervals simultaneously, each with 95% coverage probability, the expected number of non-covering intervals is 5. If the intervals are statistically independent from each other, the probability that at least one interval does not contain the population parameter is 99.4%. Techniques have been developed to prevent the inflation of false positive rates and non-coverage rates that occur with multiple statistical tests. Classification of multiple hypothesis tests Controlling procedures If m independent comparisons are performed, the family-wise error rate (FWER), is given by Hence, unless the tests are perfectly positively dependent (i.e., identical), increases as the number of comparisons increases. If we do not assume that the comparisons are independent, then we can still say: which follows from Boole's inequality. Example: There are different ways to assure that the family-wise error rate is at most . The most conservative method, which is free of dependence and distributional assumptions, is the Bonferroni correction . A marginally less conservative correction can be obtained by solving the equation for the family-wise error rate of independent comparisons for . This yields , which is known as the Šidák correction. Another procedure is the Holm–Bonferroni method, which uniformly delivers more power than the simple Bonferroni correction, by testing only the lowest p-value () against the strictest criterion, and the higher p-values () against progressively less strict criteria. . For continuous problems, one can employ Bayesian logic to compute from the prior-to-posterior volume ratio. Continuous generalizations of the Bonferroni and Šidák correction are presented in. Multiple testing correction Multiple testing correction refers to making statistical tests more stringent in order to counteract the problem of multiple testing. The best known such adjustment is the Bonferroni correction, but other methods have been developed. Such methods are typically designed to control the family-wise error rate or the false discovery rate. Large-scale