We performed simulation studies using real genetic data from the UK Biobank and the 1KG European sample (N = 503) as an external LD reference panel. SNP effect sizes were simulated using (1) a point-normal model as specified in Eq. (12) with different numbers of causal variants (100, 1000, 10,000, and 100,000), which represent extremely sparse to highly polygenic genetic architectures; and (2) a normal mixture model comprised 10 group-one SNPs, 1000 group-two SNPs and 10,000 group-three SNPs, and the three effect size groups explained 10%, 20%, and 70% of the total heritability, respectively. The simulated trait was generated by the sum of all genetic markers, weighted by their simulated effect sizes, and adding a normally distributed noise term which fixed the heritability at 0.5. We then conducted GWAS to produce a marginal least squares effect size estimate for each SNP, and applied each polygenic prediction method to the GWAS summary statistics. For P+T, LDpred, and PRS-CS, tuning parameters were selected in a validation data set of 3000 individuals that are unrelated to the training sample. The predictive performance of
