reflects the true sampling distribution of the IV estimate with a weak IV, which has long tails and is asymmetrical and is modeled poorly by a normal distribution. In the complete-data MR setting, the reliance on normality assumptions for constructing confidence intervals has been shown to lead to poor coverage properties with weak IVs (13). In our work, coverage under the null was not underestimated when IVs were strong, but it was overestimated, with increasingly conservative confidence intervals, as IV strength decreased (Web Table 1). Table 1.A Comparison of Different Methods of Estimating 95% Confidence Intervals for Selected Simulated Data SetsaStrong IV (R2 = 0.025) (Theoretical F = 50)bModerate IV (R2 = 0.005) (Theoretical F = 10)Weak IV (R2 = 0.002) (Theoretical F = 5)βSECIβSECIβSECISubsample IV approach Delta method0.1480.0570.037, 0.2590.1520.132−0.108, 0.4110.0810.161−0.234, 0.397 Sequential regressionc0.0550.039, 0.2560.128−0.099, 0.4030.159−0.231, 0.394 Fieller's theoremN/A0.040, 0.272N/A−0.108, 0.562N/A−0.291, 0.602 Bootstrapd0.0680.014, 0.2800.137−0.117, 0.4210.551−0.999, 1.162 Bayesian0.1430.0560.040, 0.2580.1740.289−0.161, 0.7780.0890.443−0.563, 0.9752-sample IV approach Delta method0.1170.068−0.015, 0.2500.0510.119−0.182, 0.284−0.0860.163−0.405, 0.232 Sequential regressionc0.065−0.011, 0.2450.118−0.181, 0.2820.160−0.440, 0.227 Fieller's theoremN/A−0.012, 0.267N/A−0.201, 0.336N/A−0.610, 0.280 Bootstrapd0.071−0.023, 0.2570.138−0.221, 0.3220.997−2.041, 1.868 Bayesian0.1190.072−0.013, 0.2730.0550.169−0.232, 0.390−0.1000.456−1.012, 0.554Abbreviations: CI, confidence interval; IV, instrumental variable; N/A, not applicable; SE, standard error.a The simulated data sets consisted of 10,000 persons with data on G and
