LD among SNPs. It is important to note that in accordance with one of the basic assumptions for MR9, only the SNPs that are strongly associated with the risk factor should be used as the instruments for MR analyses including GSMR. We demonstrate using simulations (Supplementary Note 1) that if we use independent SNPs that are associated with the exposure at P < 5×10−8, there is no inflation in the GSMR test-statistics under the null hypothesis that bxy = 0 (Supplementary Fig. 1a), that the estimate of bxy by GSMR is unbiased under the alternative hypothesis that bxy≠0 (Supplementary Table 1), and that bxy approximately equals to logOR (where OR is the effect of risk factor on disease in observational study without confounding) (Supplementary Fig. 2). GSMR accounts for LD if the SNP instruments are not fully independent. This is demonstrated by the simulation that in the presence of LD the test-statistic is well calibrated under the null (Supplementary Fig. 1b) and that the estimate of bxy is unbiased under the alternative (Supplementary Table 1). In comparison with the existing methods that use summary data to make causal inference12,13,16,18, GSMR is more powerful as demonstrated by simulation (Supplementary Fig. 3)
