Extending the MR-Egger method for multivariable Mendelian randomization to correct for both measured and unmeasured pleiotropy

Abstract

Methods have been developed for Mendelian randomization that can obtain consistent causal estimates while relaxing the instrumental variable assumptions. These include multivariable Mendelian randomization, in which a genetic variant may be associated with multiple risk factors so long as any association with the outcome is via the measured risk factors (measured pleiotropy), and the MR-Egger (Mendelian randomization-Egger) method, in which a genetic variant may be directly associated with the outcome not via the risk factor of interest, so long as the direct effects of the variants on the outcome are uncorrelated with their associations with the risk factor (unmeasured pleiotropy). In this paper, we extend the MR-Egger method to a multivariable setting to correct for both measured and unmeasured pleiotropy. We show, through theoretical arguments and a simulation study, that the multivariable MR-Egger method has advantages over its univariable counterpart in terms of plausibility of the assumption needed for consistent causal estimation and power to detect a causal effect when this assumption is satisfied. The methods are compared in an applied analysis to investigate the causal effect of high-density lipoprotein cholesterol on coronary heart disease risk. The multivariable MR-Egger method will be useful to analyse high-dimensional data in situations where the risk factors are highly related and it is difficult to find genetic variants specifically associated with the risk factor of interest (multivariable by design), and as a sensitivity analysis when the genetic variants are known to have pleiotropic effects on measured risk factors.

Keywords: MR-Egger; Mendelian randomization; invalid instruments; multivariable; pleiotropy.

Figure 1

Causal directed acyclic graph illustrating univariable Mendelian randomization assumptions with potential violation of IV3 by a pleiotropic effect indicated by a dotted line. The genetic effect of G _j on X is ${\beta }_{X_j}$, the direct (pleiotropic) effect of G _j on Y via an independent pathway is α _j (representing the potential violation of the IV3 assumption), and the causal effect of the risk factor X on the outcome Y is θ. U represents the set of variables that confound the association between X and Y

Figure 2

Causal directed acyclic graph illustrating multivariable Mendelian randomization assumptions for a set of genetic variants G _j, 3 risk factors X ₁, X ₂, and X ₃, and outcome Y. The genetic effect of G _j on X _k is ${\beta }_{X_{kj}}$, the direct (pleiotropic) effect of G _j on Y is ${\alpha }_j^{\prime }$, and the causal effect of the risk factor X _k on the outcome Y is θ _k. U _k represents the set of variables that confound the associations between X _k and Y

Figure 3

Causal directed acyclic graph illustrating the causal relationships between the 2 risk factors X ₁ and X ₂, and outcome Y. The causal effect of X ₁ on X ₂ is γ, and the direct causal effect of the risk factor X _k on the outcome Y is θ _k. The total causal effect of X ₁ on Y is θ ₁+γ θ ₂, consisting of the direct effect (θ ₁) and the indirect effect via X ₂ (γ θ ₂). U _k represents the set of variables that confound the associations between X _k and Y