RE: “MULTIVARIABLE MENDELIAN RANDOMIZATION: THE USE OF PLEIOTROPIC GENETIC VARIANTS TO ESTIMATE CAUSAL EFFECTS” In the manuscript “Multivariable Mendelian Randomiza- tion: the Use of Pleiotropic Genetic Variants to Estimate

In the manuscript “Multivariable Mendelian Randomization: the Use of Pleiotropic Genetic Variants to Estimate Causal Effects” (1), we commented on an analysis method recently used in the literature (2), which we referred to as a “regression-based method.” We presented simulation results showing that the method had some weaknesses—in particular, not estimating the same parameter as that from a 2-stage least-squares analysis of individual-level data, and giving widely varying results when there were causal effects between the risk factors. A specific criticism of the method is that the uncertainty in the summarized associations (beta coefficients) used in the method is ignored. We recently noticed a simple modification which can be made to the regression-based method that results in estimates which closely approximate those from a 2-stage least-squares analysis and remain stablewhen there are causal effects between the risk factors. In this letter, we describe this modification of the regression-based method (we refer to the modified method as a “weighted regression-based method”) and repeat the simulations from the main paper using this modified method. We assume that data are available on the associations of J uncorrelated genetic variants with each ofK risk factors, such that the association estimate of variant j with risk factor k is Xkj, with standard error σXkj, and the association estimate of variant j with the outcome is Yj, with standard error σYj. This notation matches that of our original article (1). Theweighted regressionbased method is performed by regression of the association estimatesYjon each of theXkj in amultivariableweighted regressionmodel, using the σ 2 Yj as weights (inverse-varianceweights) (3). The regression model should have no intercept. When there is only 1 risk factor, the estimate from theweighted regression-based method is P j XjYjσ 2 Yj = P j X 2 j σ 2 Yj ;