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We tackle the problem of selecting from among a large number of variablesthose that are 'important' for an outcome. We consider situations where groupsof variables are also of interest in their own right. For example, eachvariable might be a genetic polymorphism and we might want to study how a traitdepends on variability in genes, segments of DNA that typically containmultiple such polymorphisms. Or, variables might quantify various aspects ofthe functioning of individual internet servers owned by a company, and we mightbe interested in assessing the importance of each server as a whole on theaverage download speed for the company's customers. In this context, todiscover that a variable is relevant for the outcome implies discovering thatthe larger entity it represents is also important. To guarantee meaningful andreproducible results, we suggest controlling the rate of false discoveries forfindings at the level of individual variables and at the level of groups.Building on the knockoff construction of Barber and Candes (2015) and themultilayer testing framework of Barber and Ramdas (2016), we introduce themultilayer knockoff filter (MKF). We prove that MKF simultaneously controls theFDR at each resolution and use simulations to show that it incurs little powerloss compared to methods that provide guarantees only for the discoveries ofindividual variables. We apply MKF to analyze a genetic dataset and find thatit successfully reduces the number of false gene discoveries without asignificant reduction in power.

Multilayer Knockoff Filter: Controlled variable selection at multiple resolutions