Meta data of reading

• Journal: bioRxiv
• Year: 2017
• DOI: http://dx.doi.org/10.1101/219428

Idea

This paper is to perform the mediation analysis using GWAS summary statistic and eQTL data. The goal is to infer whether gene expression mediates the genotype-phenotype association. The idea is to incorperate multiple genes and multiple SNPs to take LD into account. Also, the unmediated effect is also modelled explicitly.

Model

\[\begin{align*} y &\sim N(G\theta, \sigma^2 I) \\ \theta &= \sum_{k = 1}^K \beta_k \vec\alpha_k + \vec\gamma \end{align*}\]

, where \(\beta_k\) is the effect of gene \(k\) expression on phenotype. \(\vec\gamma\) is unmediated effect of genotype on phenotype. \(\vec{\alpha}_k\) is the effect of genotype on gene \(k\).

Here, the idea comes from the fact that \(\hat\theta\) coming from single-effect model (GWAS summary statistic) follows \(\hat\theta \sim N(SRS^{-1}\theta, SRS)\) where \(\theta\) is the effect of true multi-site model. Following this idea, the paper proposed a likelihood function for \(\hat\theta\) given \(\hat\alpha\). Specifically, \(\hat\theta \sim N(SRS^{-1} \sum_{k = 1}^K \hat{\vec\alpha_k} \beta_k + \vec\gamma, SRS)\).

The ultimate goal is to infer \(\beta_k\) with the consideration of \(\vec\gamma\). So, a spike-and-slab prior is assigned to \(\beta_k\) and a normal prior is assigned to \(\vec\gamma\). One more techical note is that variational Bayes was used in inference where LD matrix \(R\) is replaced by a low rank approximation to ensure efficient/stable computaton.

In practice

The paper defined only considered genes with at least on eQTL with p-value > 0.05. All SNPs were included to allow unmediated effect explain away mediated effect. It is unclear what is computed for prioritizing genes, but it is likely to be PIP of \(\beta_k\) for each gene \(k\).