We present a novel method for controlling the k-familywise error rate
(k-FWER) in the linear regression setting using the knockoffs framework first
introduced by Barber and Cand\`es. Our procedure, which we also refer to as
knockoffs, can be applied with any design matrix with at least as many
observations as variables, and does not require knowing the noise variance.
Unlike other multiple testing procedures which act directly on p-values,
knockoffs is specifically tailored to linear regression and implicitly accounts
for the statistical relationships between hypothesis tests of different
coefficients. We prove that knockoffs controls the k-FWER exactly in finite
samples and show in simulations that it provides superior power to alternative
procedures over a range of linear regression problems. We also discuss
extensions to controlling other Type I error rates such as the false exceedance
rate, and use it to identify candidates for mutations conferring
drug-resistance in HIV.Comment: 15 pages, 3 figures. Updated reference
