Causal model selection

Mediation analysis

The GTEx study identified trans-eQTLs that are also cis-eQTLs and asked if the cis-eGene could be the cause of the trans-eQTL association (see fig above), that is, if the following model is supported by the data:

where $Z$ is a SNP that is a cis-eQTL for gene $X$ and trans-eQTL for gene $Y$. This model implies that $X$ blocks the path between $Z$ and $Y$, and hence that $Z$ and $Y$ are independent conditional on $X$, in mathematical notation

$$ Z \perp Y \mid X $$

The principle for testing whether the model $Z\to X \to Y$ is true using the conditional independence criterion is illustrated in the figure below. Assuming linear relations between all variables, three conditions must be met:

1. The expression levels of $X$ differ significantly between the genotype groups of $Z$ (to confirm the $Z\to X$ association).
2. The expression levels of $Y$ differ significantly between the genotype groups of $Z$ (to confirm the $Z\to Y$ association).
3. The residuals of $Y$ after regression on $X$ do not differ differ significantly between the genotype groups of $Z$ (to confirm that the $Z\to Y$ association is mediated by $X$, and hence that $Z\to X \to Y$ is true).

Instrumental variable analysis / Mendelian randomization

The mediation method fails if $X$ and $Y$ are affected by common cause $U$ (which may be an unknown or hidden variable):

In this case, conditioning on $X$ opens the collider $Z \to X \leftarrow U$, creating a path $Z\; — \; U \to Y$, such that the residuals of $Y$ will still show a difference between the genotype groups of $Z$, and the mediation method concludes (wrongly!) that the $Z\to Y$ association must be due to another factor than $X$ (no causal $X\to Y$ relation).

Instrumental variable, known as Mendelian randomization (MR), is an alternative causal inference approach that is not affected by hidden confounders $U$, but with subtly different underlying assumptions.

Specifically, in MR we assume that the $Z\to Y$ association must be due to $X$ (for instance because $X$ is the only cis-eGene of $Z$, and trans-eQTL associations must be mediated by some initial cis effects), and we seek to estimate the magnitude of the causal effect of $X$ on $Y$.

The diagram above can be written as a structural equation model

$$ \begin{aligned} X &= a Z + c_X U + E_X\ Y &= b X + c_Y U + E_Y \end{aligned} $$

where $E_X$ and $E_Y$ are error terms, mutually independent and independent of $Z$ and $U$. Since $Z$ and $U$ are assumed to be independent (no arrows in the diagram), it follows that

$$ \begin{aligned} \mathrm{cov}(Y,Z) &= b\; \mathrm{cov}(X,Z) \end{aligned} $$

and hence the causal effect of $X$ on $Y$ is estimated by the ratio of covariances:

$$ b = \frac{\mathrm{cov}(Y,Z)}{\mathrm{cov}(X,Z)} $$

Assignment