Mendelian randomization (MR) is an instrumental variable (IV) approach to infer causal relationships between exposures and outcomes with genome-wide association studies (GWAS) summary data. However, the multivariable inverse-variance weighting (IVW) approach, which serves as the foundation for most MR approaches, cannot yield unbiased causal effect estimates in the presence of many weak IVs. To address this problem, we proposed the MR using Bias-corrected Estimating Equation (MRBEE) that can infer unbiased causal relationships with many weak IVs and account for horizontal pleiotropy simultaneously. While the practical significance of MRBEE was demonstrated in our parallel work (Lorincz-Comi (2023)), this paper established the statistical theories of multivariable IVW and MRBEE with many weak IVs. First, we showed that the bias of the multivariable IVW estimate is caused by the error-in-variable bias, whose scale and direction are inflated and influenced by weak instrument bias and sample overlaps of exposures and outcome GWAS cohorts, respectively. Second, we investigated the asymptotic properties of multivariable IVW and MRBEE, showing that MRBEE outperforms multivariable IVW regarding unbiasedness of causal effect estimation and asymptotic validity of causal inference. Finally, we applied MRBEE to examine myopia and revealed that education and outdoor activity are causal to myopia whereas indoor activity is not.

翻译：孟德尔随机化（MR）是一种利用全基因组关联研究（GWAS）汇总数据推断暴露与结局因果关系的工具变量（IV）方法。然而，作为大多数MR方法基础的多变量逆方差加权（IVW）方法，在存在多个弱工具变量时无法获得无偏的因果效应估计。为解决这一问题，我们提出了基于偏倚校正估计方程的MR方法（MRBEE），该方法能在存在多个弱工具变量的情况下推断无偏因果关系，同时校正水平多效性。虽然MRBEE的实践意义已在我们的平行研究（Lorincz-Comi (2023)）中得到验证，但本文建立了多变量IVW和MRBEE在存在多个弱工具变量时的统计理论。首先，我们证明多变量IVW估计的偏倚源于变量误差偏倚，其幅度和方向分别受弱工具变量偏倚以及暴露与结局GWAS队列样本重叠的影响。其次，我们研究了多变量IVW和MRBEE的渐近性质，表明MRBEE在因果估计无偏性和因果推断渐近有效性方面均优于多变量IVW。最后，我们将MRBEE应用于近视研究，发现教育和户外活动是近视的因果因素，而室内活动则非。