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，通过工具变量（IV）方法推断暴露与结局之间因果关系的统计方法。然而，作为大多数MR方法基础的多变量逆方差加权（IVW）方法，在存在大量弱IV时无法得到无偏的因果效应估计。为解决此问题，我们提出了基于偏差校正估计方程的MR方法（MRBEE），该方法能在弱IV众多的条件下推断无偏的因果关系，并同时校正水平多效性。虽然MRBEE的实际意义已在我们的并行工作（Lorincz-Comi, 2023）中得到验证，但本文建立了多变量IVW和MRBEE在弱IV众多情况下的统计理论。首先，我们证明多变量IVW估计的偏差是由变量误差偏差引起的，其幅度和方向分别受弱工具偏差以及暴露与结局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在因果估计的无偏性和因果推断的渐近有效性上均优于多变量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，而室内活动则不是。