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