Partitioning Causal Effects of Gene Expression on Phenotypes

An MVMR approach was used to dissect the total causal effect of transcript levels on phenotypes ( ${\alpha }_{TP}$ ) into a direct ( ${\alpha }_d$ ) and indirect ( ${\alpha }_i$ ) effect measured through a metabolite. Through inclusion of a metabolite and its associated genetic variants (r² <0.01, p_mQTL<1 × 10^–07), the direct effect of gene expression on a phenotype can be estimated using a multivariable regression model (Burgess et al., 2013 (link)) as the first element of $\hat{\alpha }={\left(B^{\prime }{C}^{-1}B\right)}^{-1}\left(B^{\prime }{C}^{-1}\gamma \right)$
where $B$ is a matrix with two columns containing the standardized effect sizes of n IVs on transcript levels in the first column and on the metabolite levels in the second column, $\gamma$ is a vector of length n containing the standardized effect size of each SNP on the phenotype, and C is the pairwise LD matrix between the n SNPs.
The proportion of direct effect ${\displaystyle \left(\rho \right)}$ is calculated by regressing direct effects ( ${\displaystyle {\alpha }_d}$ ) on total effects ( ${\displaystyle {\alpha }_{TP}}$ ) and then correcting for regression dilution bias: ${\rho }_{corrected}=\frac{\rho }{\sqrt{1-\frac{\sum {\left(SE\left({\alpha }_{TP}\right)\right)}^2}{\sum {\alpha }_{TP}^2}}}$