On the Adaptive Control of the False Discovery Rate in Multiple Testing With Independent Statistics

Abstract

A new approach to problems of multiple significance testing was presented in Benjamini and Hochberg (1995), which calls for controlling the expected ratio of the number of erroneous rejections to the number of rejections–the False Discovery Rate (FDR). The procedure given there was shown to control the FDR for independent test statistics. When some of the hypotheses are in fact false, that procedure is too conservative. We present here an adaptive procedure, where the number of true null hypotheses is estimated first as in Hochberg and Benjamini (1990), and this estimate is used in the procedure of Benjamini and Hochberg (1995). The result is still a simple stepwise procedure, to which we also give a graphical companion. The new procedure is used in several examples drawn from educational and behavioral studies, addressing problems in multi-center studies, subset analysis and meta-analysis. The examples vary in the number of hypotheses tested, and the implication of the new procedure on the conclusions. In a large simulation study of independent test statistics the adaptive procedure is shown to control the FDR and have substantially better power than the previously suggested FDR controlling method, which by itself is more powerful than the traditional family wise error-rate controlling methods. In cases where most of the tested hypotheses are far from being true there is hardly any penalty due to the simultaneous testing of many hypotheses.