  The family-wise error rate (FWER) has been widely used in genome-wide
association studies. With the increasing availability of functional genomics
data, it is possible to increase the detection power by leveraging these
genomic functional annotations. Previous efforts to accommodate covariates in
multiple testing focus on the false discovery rate control while
covariate-adaptive FWER-controlling procedures remain under-developed. Here we
propose a novel covariate-adaptive FWER-controlling procedure that incorporates
external covariates which are potentially informative of either the statistical
power or the prior null probability. An efficient algorithm is developed to
implement the proposed method. We prove its asymptotic validity and obtain the
rate of convergence through a perturbation-type argument. Our numerical studies
show that the new procedure is more powerful than competing methods and
maintains robustness across different settings. We apply the proposed approach
to the UK Biobank data and analyze 27 traits with 9 million single-nucleotide
polymorphisms tested for associations. Seventy-five genomic annotations are
used as covariates. Our approach detects more genome-wide significant loci than
other methods in 21 out of the 27 traits.
