For 3 randomly chosen data sets from simulation 2, comprising strong, moderate, and weak IV scenarios, we compared 5 strategies for calculating standard errors and 95% confidence intervals for the MR estimate. First, we used a sequential regression approach, where linear regression was used to generate a coefficient for the G-X association; this coefficient was then used to generate predicted values of X for all persons with data on Y (nY). The association between the predicted X and Y was assessed using linear regression with robust standard errors to mitigate the failure of the method to account for the uncertainty in the predicted X. We also used the SUR/delta method described above; Fieller's theorem, which is a method for calculating confidence intervals for a ratio of 2 normally distributed variables (12); and a Bayesian method using weakly informative prior distributions (13). Finally, we used a bootstrap method for confidence interval estimation in which 1,000 random samples equal in size to the original sample were drawn, with replacement, from each of the samples used to generate the first-stage and reduced-form estimates.
