  Mediation analysis is difficult when the number of potential mediators is
larger than the sample size. In this paper we propose new inference procedures
for the indirect effect in the presence of high-dimensional mediators for
linear mediation models. We develop methods for both incomplete mediation,
where a direct effect may exist, as well as complete mediation, where the
direct effect is known to be absent. We prove consistency and asymptotic
normality of our indirect effect estimators. Under complete mediation, where
the indirect effect is equivalent to the total effect, we further prove that
our approach gives a more powerful test compared to directly testing for the
total effect. We confirm our theoretical results in simulations, as well as in
an integrative analysis of gene expression and genotype data from a
pharmacogenomic study of drug response. We present a novel analysis of gene
sets to understand the molecular mechanisms of drug response, and also identify
a genome-wide significant noncoding genetic variant that cannot be detected
using standard analysis methods.
