Robust multivariable Mendelian randomization based on constrained maximum likelihood

Abstract

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.

Keywords: GWAS summary data; IV; direct causal effect; instrumental variable; mediation analysis; pleiotropy.

Figure 1

True causal models for multivariable MR (A and B) A general (A) and a specific (B) causal graph showing the relationships among one IV $\left(G_i\right)$, multiple exposures $\left(X_1,\dots ,X_L\right)$, an unmeasured confounder $\left(U\right)$, and the outcome $\left(Y\right)$.

Figure 2

True causal models in simulations (A–C) Three scenarios of simulated genetic instruments for $i=1,\dots ,K$ (A); $i=K+1,\dots ,K+m_1$ (B); and $i=K+m_1+1,\dots ,20$ (C). ${\gamma }_{X_{1}i},{\gamma }_{X_{2}i}\stackrel{iid.}{\sim }\mathcal{U}\left(\mathrm{0,0.22}\right)$, ${\alpha }_i\sim \mathcal{N}\left(\mathrm{0.1,0}.2^2\right)$, and $\left({\theta }_1,{\theta }_2\right)=\left(\mathrm{0.1,0.2}\right)$.

Figure 3

The estimated effects (and 95% confidence intervals) of each of the eight risk factors on CAD by various UVMR and MVMR methods The conditional F-statistic is given in the parentheses following each exposure name.

Figure 4

The estimated effects (and 95% confidence intervals) of DBP on CAD with various sets of exposures by MVMR-cML-DP Left corresponds to the sets of two exposures (DBP plus one of the other seven risk factors). Right corresponds to the sets of the six exposures after excluding one of seven risk factors marked out in the left. Results from UVMR-cML-DP and MVMR-cML-DP in Figure 3 are also added at bottom for comparison.