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

Mendelian randomization (MR) is a popular epidemiological approach that utilizes genetic variants as instrumental variables (IVs) to infer the causal relationships between exposures and an outcome in the genome-wide association studies (GWAS) era. It is well-known that the inverse-variance weighted (IVW) estimate of causal effect suffers from bias caused by the violation of valid IV conditions, however, the quantitative degree of this bias has not been well characterized and understood. This paper contributes to the theoretical investigation and practical solution of the causal effect estimation in multivariable MR. First, we prove that the bias of IVW estimate is a product of the weak instrument and estimation error biases, where the estimation error bias is caused linearly by measurement error and confounder biases with a trade-off due to the sample overlap among exposure and outcome GWAS cohorts. Second, we demonstrate that our novel multivariable MR approach, MR using Bias-corrected Estimating Equation (MRBEE), can estimate the causal effect unbiasedly in the presence of many weak IVs. Asymptotic behaviors of IVW and MRBEE are investigated under moderate conditions, where MRBEE is shown superior to IVW in terms of unbiasedness and asymptotic validity. Simulations exhibit that only MRBEE can provide a strongly asymptotically unbiased estimate of causal effect in comparison with existing MR methods. Applied to data from the UK Biobank, MRBEE can eliminate weak instrument and estimation error biases and provide valid causal inferences. R package MRBEE is available online.