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Abstract:
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In many multiple testing problems, the extremely large number of questions being tested means that we may be unable to find many signals after correcting for multiple testing, even when using false discovery rate (FDR) for more flexible error control. At the same time, prior information can indicate that the signals are more likely to be found in some hypotheses than others, or that they may tend to cluster together spatially. If we use our data to fit weights that reveal where the signals are most likely to be found, can we then reuse the same data to perform weighted multiple testing and identify discoveries? Our method, the structure adaptive Benjamini-Hochberg algorithm (SABHA), uses the data twice in this way in order to boost power. We find that as long as the first step is constrained so as to not overfit to the data, we maintain FDR control at nearly the target level for the overall two-stage procedure. Interestingly, the excess FDR can be related to the Rademacher complexity or Gaussian width of the class from which we choose the adaptive weights. We validate our results on simulated data and real fMRI data, testing our method over a range of different data structures.
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