The control of the false discovery rate in multiple testing under dependency

Benjamini and Hochberg suggest that the false discovery rate may\nbe the appropriate error rate to control in many applied multiple testing\nproblems. A simple procedure was given there as an FDR controlling procedure\nfor independent test statistics and was shown to be much more powerful than\ncomparable procedures which control the traditional familywise error rate. We\nprove that this same procedure also controls the false discovery rate when the\ntest statistics have positive regression dependency on each of the test\nstatistics corresponding to the true null hypotheses. This condition for\npositive dependency is general enough to cover many problems of practical\ninterest, including the comparisons of many treatments with a single control,\nmultivariate normal test statistics with positive correlation matrix and\nmultivariate $t$. Furthermore, the test statistics may be discrete, and the\ntested hypotheses composite without posing special difficulties. For all other\nforms of dependency, a simple conservative modification of the procedure\ncontrols the false discovery rate. Thus the range of problems for which a\nprocedure with proven FDR control can be offered is greatly increased.