  The knockoff filter is a recent false discovery rate (FDR) control method for
high-dimensional linear models. We point out that knockoff has three key
components: ranking algorithm, augmented design, and symmetric statistic, and
each component admits multiple choices. By considering various combinations of
the three components, we obtain a collection of variants of knockoff. All these
variants guarantee finite-sample FDR control, and our goal is to compare their
power. We assume a Rare and Weak signal model on regression coefficients and
compare the power of different variants of knockoff by deriving explicit
formulas of false positive rate and false negative rate. Our results provide
new insights on how to improve power when controlling FDR at a targeted level.
We also compare the power of knockoff with its propotype - a method that uses
the same ranking algorithm but has access to an ideal threshold. The comparison
reveals the additional price one pays by finding a data-driven threshold to
control FDR.
