Towards "simultaneous selective inference": post-hoc bounds on the false discovery proportion

19 Mar 2018  ·  Eugene Katsevich, Aaditya Ramdas ·

The false discovery rate (FDR) is a popular error criterion for multiple testing, but it has been criticized for lacking flexibility. A target FDR level $q$ must be set in advance, and the resulting rejection set cannot be contracted or expanded without invalidating FDR control. In exploratory settings, it is desirable to allow the experimenter more freedom to choose a rejection set, while still preserving some Type-I error guarantees. In this paper, we show that the entire path of rejection sets considered by a variety of existing FDR procedures (like BH, knockoffs, and many others) can be endowed with simultaneous high-probability bounds on FDP. The path can be defined based on either the p-values themselves, side information, or a combination of the two. FDR procedures maintain an estimate of FDP for each rejection set on the path, stopping when this estimate exceeds $q$. We show that inflating this FDP estimate by a small, explicit, multiplicative constant bounds the FDP with high probability across the entire path. These results allow the scientist to $\textit{spot}$ one or more suitable rejection sets (Select Post hoc On the algorithm's Trajectory) as they wish, for example by picking a data-dependent size or error level, after examining the FDP bounds for the whole path, and still get a valid high probability FDP bound on the chosen set. Bounding the FDP simultaneously for a selected path of rejection sets (simultaneous selective inference) can be a fruitful middle ground between fully simultaneous inference (guarantees for all possible rejection sets), and fully selective inference (guarantees only for one rejection set).

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