Source: Ann. Statist.
Volume 29, Number 4
Benjamini and Hochberg suggest that the false discovery rate may
be the appropriate error rate to control in many applied multiple testing
problems. A simple procedure was given there as an FDR controlling procedure
for independent test statistics and was shown to be much more powerful than
comparable procedures which control the traditional familywise error rate. We
prove that this same procedure also controls the false discovery rate when the
test statistics have positive regression dependency on each of the test
statistics corresponding to the true null hypotheses. This condition for
positive dependency is general enough to cover many problems of practical
interest, including the comparisons of many treatments with a single control,
multivariate normal test statistics with positive correlation matrix and
multivariate $t$. Furthermore, the test statistics may be discrete, and the
tested hypotheses composite without posing special difficulties. For all other
forms of dependency, a simple conservative modification of the procedure
controls the false discovery rate. Thus the range of problems for which a
procedure with proven FDR control can be offered is greatly increased.
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