Unbiased causal inference with Mendelian randomization and covariate-adjusted GWAS data

Mendelian randomization (MR) facilitates causal inference with observational data using publicly available genome-wide association study (GWAS) results. In a GWAS, one or more heritable covariates may be adjusted for to estimate the direct effects of SNPs on a focal trait or to improve statistical power, which may introduce collider bias in SNP-trait association estimates, thus affecting downstream MR analyses. Numerical studies suggested that using covariate-adjusted GWAS summary data might introduce bias in univariable Mendelian randomization (UVMR), which can be mitigated by multivariable Mendelian randomization (MVMR). However, it remains unclear and even mysterious why/how MVMR works; a rigorous theory is needed to explain and substantiate the above empirical observation. In this paper, we derive some analytical results when multiple covariates are adjusted for in the GWAS of exposure and/or the GWAS of outcome, thus supporting and explaining the empirical results. Our analytical results offer insights to how bias arises in UVMR and how it is avoided in MVMR, regardless of whether collider bias is present. We also consider applying UVMR or MVMR methods after collider-bias correction. We conducted extensive simulations to demonstrate that with covariate-adjusted GWAS summary data, MVMR had an advantage over UVMR by producing nearly unbiased causal estimates; however, in some situations it is advantageous to apply UVMR after bias correction. In real data analyses of the GWAS data with body mass index (BMI) being adjusted for metabolomic principal components, we examined the causal effect of BMI on blood pressure, confirming the above points.