On dependence assumption in p-value based multiple test procedures

There are various multiple comparison procedures used in confirmatory clinical studies and exploratory research for multiplicity adjustment. Among them are the Hochberg and Benjamini-Hochberg procedures. A common misconception is that these procedures control the type I error rate properly if the test statistics are independent or positively correlated. In fact, a much stronger positive dependence assumption needs to be satisfied to guarantee the type I error rate control. We give a comprehensive review of the dependence conditions used in multiple testing procedures. We show that a weaker positive dependence assumption may result an inflation of type I error rate by a factor of 2 and discuss the type I error rate control under certain negative dependence conditions.