Ongoing increases in the size of human genotype and phenotype collections offer the promise of improved understanding of the genetics of complex diseases. In addition to the biological insights that can be gained from the nature of the variants that contribute to the genetic component of complex trait variability, these data bring forward the prospect of predicting complex traits and the risk of complex genetic diseases from genotype data. Here we show that advances in phenotype prediction can be applied to improve the power of genome-wide association studies. We demonstrate a simple and efficient method to model genetic background effects using polygenic scores derived from SNPs that are not on the same chromosome as the target SNP. Using simulated and real data we found that this can result in a substantial increase in the number of variants passing genome-wide significance thresholds. This increase in power to detect trait-associated variants also translates into an increase in the accuracy with which the resulting polygenic score predicts the phenotype from genotype data. Our results suggest that advances in methods for phenotype prediction can be exploited to improve the control of background genetic effects, leading to more accurate GWAS results and further improvements in phenotype prediction.