A Two-Sample Robust Bayesian Mendelian Randomization Method Accounting for Linkage Disequilibrium and Idiosyncratic Pleiotropy with Applications to the COVID-19 Outcome

Mendelian randomization (MR) is a statistical method exploiting genetic variants as instrumental variables to estimate the causal effect of modifiable risk factors on an outcome of interest. Despite wide uses of various popular two-sample MR methods based on genome-wide association study summary level data, however, those methods could suffer from potential power loss or/and biased inference when the chosen genetic variants are in linkage disequilibrium (LD), and have relatively large direct effects on the outcome whose distribution might be heavy-tailed which is commonly referred to as the idiosyncratic pleiotropy. To resolve those two issues, we propose a novel Robust Bayesian Mendelian Randomization (RBMR) model that uses the more robust multivariate generalized $t$-distribution to model such direct effects in a probabilistic model framework which can also incorporate the LD structure explicitly. The generalized $t$-distribution can be represented as a Gaussian scaled mixture so that our model parameters can be estimated by the EM-type algorithms. We compute the standard errors by calibrating the evidence lower bound (ELBO) using the likelihood ratio test. Through extensive simulation studies, we show that our RBMR has robust performance compared to other competing methods. We also apply our RBMR method to two benchmark data sets and find that RBMR has smaller bias and standard errors. Using our proposed RBMR method, we found that coronary artery disease (CAD) is associated with increased risk of coronavirus disease 2019 (COVID-19). We also develop a user-friendly R package RBMR for public use.