Figure 1.

Principle of MRBEE

(A) Traditional MR methods are vulnerable to weak instrument bias arising from the estimation errors in GWAS associations for the exposure(s) and outcome. The direction of the bias is influenced by the degree of sample overlap between the studies where the red and blue points refer to two simulated data with 0% and 100% sample overlap. The shadow regions represent the 95% confidence interval regions.

(B) MRBEE corrects for weak instrument bias using bias-correction terms which are calculated from the matrix of correlations between measurement errors for all exposures and the outcome. In this example with myopia and its four exposures, the numbers in the lower triangle of the table are the correlations estimated using LD score regression and that in the upper triangle of the table are the correlations estimated using non-significant SNPs.

(C) MRBEE uses an iterative estimation procedure, where horizontally pleiotropic IVs are removed at each iteration until convergence. The y axis in panels (2) and (4) reflect the SNP association with the outcome not mediated by the exposures. The numbers under the red vertical lines represent p values.

(D) After estimating causal effects, MRBEE performs genome-wide horizontal pleiotropy testing to find loci associated with the outcome (e.g., myopia) that were not detected in the original GWAS.