  In many scientific problems, researchers try to relate a response variable
$Y$ to a set of potential explanatory variables $X = (X_1,\dots,X_p)$, and
start by trying to identify variables that contribute to this relationship. In
statistical terms, this goal can be posed as trying to identify $X_j$'s upon
which $Y$ is conditionally dependent. Sometimes it is of value to
simultaneously test for each $j$, which is more commonly known as variable
selection. The conditional randomization test (CRT) and model-X knockoffs are
two recently proposed methods that respectively perform conditional
independence testing and variable selection by, for each $X_j$, computing any
test statistic on the data and assessing that test statistic's significance by
comparing it to test statistics computed on synthetic variables generated using
knowledge of $X$'s distribution. Our main contribution is to analyze their
power in a high-dimensional linear model where the ratio of the dimension $p$
and the sample size $n$ converge to a positive constant. We give explicit
expressions of the asymptotic power of the CRT, variable selection with CRT
$p$-values, and model-X knockoffs, each with a test statistic based on either
the marginal covariance, the least squares coefficient, or the lasso. One
useful application of our analysis is the direct theoretical comparison of the
asymptotic powers of variable selection with CRT $p$-values and model-X
knockoffs; in the instances with independent covariates that we consider, the
CRT provably dominates knockoffs. We also analyze the power gain from using
unlabeled data in the CRT when limited knowledge of $X$'s distribution is
available, and the power of the CRT when samples are collected retrospectively.
