  Genome-wide association studies (GWAS) have been widely used to examine the
association between single nucleotide polymorphisms (SNPs) and complex traits,
where both the sample size n and the number of SNPs p can be very large.
Recently, cross-trait polygenic risk score (PRS) method has gained extremely
popular for assessing genetic correlation of complex traits based on GWAS
summary statistics (e.g., SNP effect size). However, empirical evidence has
shown a common bias phenomenon that even highly significant cross-trait PRS can
only account for a very small amount of genetic variance (R^2 often <1%). The
aim of this paper is to develop a novel and powerful method to address the bias
phenomenon of cross-trait PRS. We theoretically show that the estimated genetic
correlation is asymptotically biased towards zero when complex traits are
highly polygenic/omnigenic. When all p SNPs are used to construct PRS, we show
that the asymptotic bias of PRS estimator is independent of the unknown number
of causal SNPs m. We propose a consistent PRS estimator to correct such
asymptotic bias. We also develop a novel estimator of genetic correlation which
is solely based on two sets of GWAS summary statistics. In addition, we
investigate whether or not SNP screening by GWAS p-values can lead to improved
estimation and show the effect of overlapping samples among GWAS. Our results
may help demystify and tackle the puzzling "missing genetic overlap" phenomenon
of cross-trait PRS for dissecting the genetic similarity of closely related
heritable traits. We illustrate the finite sample performance of our
bias-corrected PRS estimator by using both numerical experiments and the UK
Biobank data, in which we assess the genetic correlation between brain white
matter tracts and neuropsychiatric disorders.
