This figure illustrates the mechanism in summary cis-eQTL GWAS data that leads to missing data in eQTL-MVMR and how this missing data can be addressed using imputation. (A) Only SNP-gene pairs within a defined distance have association estimates present in cis-eQTL summary data. This figure demonstrates this by displaying the available data for SNPs and genes ordered by their chromosomal position using data from the eQTLGen Consortium (Vosa et al. 2018). (B) (left) Visual display of the pattern of missing in the design matrix $B^(Ω)$ used in eQTL-MVMR. Imputation can be performed by setting missing values to be 0 (“Zero imp.”) or by applying the low-rank approximation (“MV imp.”) to $B^(Ω)$ described in Algorithm 1. “Soft impute” is the soft imputation method of Hastie et al. (2015) and “Normal imp.” is a gene-pairwise imputation method based on the multivariate normal distribution, more fully described in the Supplementary material. $|Ω|$ is the total number of missing values in a simulation of 1000 replicates performed using real data in the CCDC163 gene region. These data were GWAS summary statistics of gene expression in blood tissue measured in 236 unrelated non-Hispanic White individuals. Full details of this simulation are presented in the Supplementary material. (right) An example of the MV imp. method applied to summary data for nine genes on chromosome 22 using cis-eQTL data from the eQTLGen Consortium (Vosa et al. 2018).