Unbiased estimation and asymptotically valid inference in multivariable Mendelian randomization with many weak instrumental variables

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. In this paper, we prove that the bias of the multivariable IVW estimate is a product of weak instrument and estimation error biases, where the latter is linearly composed of measurement error and confounder biases with a trade-off due to sample overlap among multiple GWAS cohorts. To address this problem, we propose a novel multivariable MR approach, MR using Bias-corrected Estimating Equation (MRBEE), which can infer unbiased causal relationships with many weak IVs. Asymptotic behaviors of multivariable IVW and MRBEE are investigated under moderate conditions, showing that MRBEE outperforms multivariable IVW in terms of unbiasedness and asymptotic validity. We apply MRBEE to examine myopia and confirm that schooling and driving time are causal factors for myopia. A novel locus of myopia is identified in the subsequent whole-genome pleiotropy test.