AdaPT: an interactive procedure for multiple testing with side information

We consider the problem of multiple-hypothesis testing with generic side information: for each hypothesis H-i we observe both a p-value p(i) and some predictor x(i) encoding contextual information about the hypothesis. For large-scale problems, adaptively focusing power on the more promising hypotheses (those more likely to yield discoveries) can lead to much more powerful multiple-testing procedures. We propose a general iterative framework for this problem, the adaptive p-value thresholding procedure which we call AdaPT, which adaptively estimates a Bayes optimal p-value rejection threshold and controls the false discovery rate in finite samples. At each iteration of the procedure, the analyst proposes a rejection threshold and observes partially censored p-values, estimates the false discovery proportion below the threshold and proposes another threshold, until the estimated false discovery proportion is below alpha. Our procedure is adaptive in an unusually strong sense, permitting the analyst to use any statistical or machine learning method she chooses to estimate the optimal threshold, and to switch between different models at each iteration as information accrues. We demonstrate the favourable performance of AdaPT by comparing it with state of the art methods in five real applications and two simulation studies.