Collider bias correction for multiple covariates in GWAS using robust multivariable Mendelian randomization

Genome-wide association studies (GWAS) have identified many genetic loci associated with complex traits and diseases in the past 20 years. Multiple heritable covariates may be added into GWAS regression models to estimate direct effects of genetic variants on a focal trait, or to improve the power by accounting for environmental effects and other sources of trait variations. When one or more covariates are causally affected by both genetic variants and hidden confounders, adjusting for them in GWAS will produce biased estimation of SNP effects, known as collider bias. Several approaches have been developed to correct collider bias through estimating the bias by Mendelian randomization (MR). However, these methods work for only one covariate, some of which utilize MR methods with relatively strong assumptions, both of which may not hold in practice. In this paper, we extend the bias-correction approaches in two aspects: first we derive an analytical expression for the collider bias in the presence of multiple covariates, then we propose estimating the bias using a robust multivariable MR (MVMR) method based on constrained maximum likelihood (called MVMR-cML), allowing the presence of invalid instrumental variables (IVs) and correlated pleiotropy. We also established the estimation consistency and asymptotic normality of the new bias-corrected estimator. We conducted simulations to show that all methods mitigated collider bias under various scenarios. In real data analyses, we applied the methods to two GWAS examples, the first a GWAS of waist-hip ratio with adjustment for only one covariate, body-mass index (BMI), and the second a GWAS of BMI adjusting metabolomic principle components as multiple covariates, illustrating the effectiveness of bias correction.Author summary: Genome-wide association studies (GWAS) are powerful in identifying genetic variants influencing complex traits and diseases. However, adjusting for heritable covariates in GWAS may introduce collider bias when both genetic variants and confounders may causally influence these covariates. In this study, for the first time we derived the analytical form of the bias term in GWAS with multiple covariates, enabling bias estimation and correction using any MVMR method. On the other hand, many existing MVMR methods may not be robust to invalid IVs and are designed for independent samples. Since GWAS data of multiple traits are needed, overlapping samples become inevitable. Hence, while investigating the performance of many MVMR methods, we mainly adopt MVMR-cML, a novel MVMR approach robust to invalid IVs and sample overlap. Our simulations underscore that most MVMR methods effectively reduce collider bias across various scenarios. Furthermore, by accounting for correlations among GWAS statistics, as well as the linkage disequilibrium (LD) between the target SNP and IVs, we establish the consistency and asymptotic normality of the bias-corrected estimator based on MVMR-cML. The application of our bias-correction approach to two published GWAS data examples illustrates its utility and efficacy.