Robust multivariable Mendelian randomization based on constrained maximum likelihood

Mendelian randomization (MR) is a powerful tool for causal inference with observational genome-wide association study (GWAS) summary data. Compared to the more commonly used univariable MR (UVMR), multivariable MR (MVMR) not only is more robust to the notorious problem of genetic (horizontal) pleiotropy but also estimates the direct effect of each exposure on the outcome after accounting for possible mediating effects of other exposures. Despite promising applications, there is a lack of studies on MVMR's theoretical properties and robustness in applications. In this work, we propose an efficient and robust MVMR method based on constrained maximum likelihood (cML), called MVMR-cML, with strong theoretical support. Extensive simulations demonstrate that MVMR-cML performs better than other existing MVMR methods while possessing the above two advantages over its univariable counterpart. An application to several large-scale GWAS summary datasets to infer causal relationships between eight cardiometabolic risk factors and coronary artery disease (CAD) highlights the usefulness and some advantages of the proposed method. For example, after accounting for possible pleiotropic and mediating effects, triglyceride (TG), low-density lipoprotein cholesterol (LDL), and systolic blood pressure (SBP) had direct effects on CAD; in contrast, the effects of high-density lipoprotein cholesterol (HDL), diastolic blood pressure (DBP), and body height diminished after accounting for other risk factors.