The increase in the mean χ2-statistic for each trait from the GWAS results to the MTAG results can be used to calculate a “GWAS-equivalent sample size” for MTAG. Under the assumptions of LD score regression,14 the expected χ2-statistic for some SNP with LD score lj is E(χj2|lj)=Njh2ljM+Nja+1, where Nj is the sample size for SNP j; h2 is the SNP heritability of the trait; M is the number of SNPs for which we define the SNP heritability; and a is the variance due to biases (e.g., due to population stratification). Note that E(χj2|lj)−1 scales linearly with Nj as long as M and lj are held constant in the additional samples.24–26 Since the individuals included in all GWASs are of European ancestry, M and lj are indeed expected to be approximately constant.24–26 Thus, we can use the mean χ2-statistic from the GWAS and the MTAG results to calculate how much larger the GWAS sample size would have to be to give a mean χ2-statistic equal to that attained by MTAG: NGWAS-equiv,j=NGWAS,jχMTAG2¯−1χGWAS2¯−1, where χGWAS2¯ is the mean χ2-statistic in the GWAS results
