Colocalization-Informed Molecular Mediation

Why Shared Genetic Signals Need Careful Interpretation

Suppose a genetic variant is associated with both a molecular trait, such as gene expression, and a health outcome. This pattern can be consistent with mediation: the variant changes the molecular trait, which in turn changes the outcome. But it can also arise when the molecular-trait association and outcome association are driven by different variants that are correlated through linkage disequilibrium (LD). Even a convincing association for each trait does not, by itself, tell us whether the same causal variant underlies both.

Colocalization analysis addresses this locus-level question. It compares evidence for a shared causal variant with evidence for distinct causal variants in a genomic region. Its contribution to molecular mediation is important but limited: support for a shared variant strengthens the case that the two association signals belong together, while the causal pathway from the molecular trait to the outcome still requires separate assumptions and evidence.

Definition: Colocalization is an analysis of genetic association data for two traits in a region that evaluates whether their association signals are most consistent with a shared causal variant or with distinct causal variants. In molecular mediation, the two traits are typically a molecular trait and an outcome. Colocalization is evidence about shared genetic architecture at the locus; it does not, on its own, estimate or establish the causal effect of the molecular trait on the outcome.

Competing Hypotheses and Their Posterior Support

A common colocalization framework considers five hypotheses. Let trait 1 be the molecular trait and trait 2 the outcome. “Causal variant” here means a variant responsible for an association signal under the model for that trait; it does not mean that the variant acts through the molecular trait to change the outcome.

Hypothesis	Interpretation
H₀	Neither trait has an associated causal variant in the region.
H₁	Only the molecular trait has an associated causal variant.
H₂	Only the outcome has an associated causal variant.
H₃	Both traits have an associated causal variant, but the variants are distinct.
H₄	Both traits have an associated causal variant, and the causal variant is shared.

The analysis combines association evidence across variants in the region with prior probabilities for the hypotheses. Its output is commonly reported as posterior probabilities \(PP_0,\ldots,PP_4\), which sum to 1. \(PP_4\) is the posterior probability assigned to the shared-variant hypothesis under the specified data model and priors. It is not the probability that molecular mediation is true.

For clarity, one standard approximate calculation uses a single-causal-variant model for each trait. Suppose there are \(M\) candidate variants, and let \(ABF_{1i}\) and \(ABF_{2i}\) be approximate Bayes factors for association of variant \(i\) with traits 1 and 2, respectively. An approximate Bayes factor measures how much the data favor an association model over a no-association model. Let \(p_1\) and \(p_2\) be small per-variant prior probabilities of association for each trait, and \(p_{12}\) the per-variant prior probability of a shared association. Unnormalized posterior weights can be written as:

\begin{aligned} W_0 &= 1,\\ W_1 &= p_1\sum_{i=1}^{M} ABF_{1i},\\ W_2 &= p_2\sum_{i=1}^{M} ABF_{2i},\\ W_3 &= p_1p_2\sum_{i\ne j} ABF_{1i}ABF_{2j},\\ W_4 &= p_{12}\sum_{i=1}^{M} ABF_{1i}ABF_{2i},\\ PP_k &= \frac{W_k}{\sum_{\ell=0}^{4}W_\ell}. \end{aligned}

The shared-variant weight \(W_4\) combines evidence for both traits at the same variant. In contrast, \(W_3\) combines molecular-trait evidence at one variant with outcome evidence at a different variant. The calculation therefore compares more than whether each trait has a small association p-value: it assesses whether the regional pattern is better explained by the same or by distinct causal variants.

Model assumptions: In this basic framework, the variants being compared are in the same region and can be aligned to the same alleles; association evidence and LD information are suitable for that region; and each trait has at most one causal variant in the region. The result also depends on the chosen association priors. If a region contains multiple causal signals for either trait, a single-signal analysis may misrepresent the evidence; a method that accounts for multiple signals is needed.

Why LD Can Mimic a Shared Signal

A variant that is not causal can still show association because it is correlated with a causal variant. This is the key reason that overlapping association peaks do not by themselves establish colocalization. A short derivation makes the point. Let \(Z_c\) be the standardized genotype at a causal variant and \(Z_k\) the standardized genotype at a nearby marker, with correlation \(r_{kc}\). Suppose the outcome follows \(Y=\beta_c Z_c+\varepsilon\), where the residual \(\varepsilon\) is uncorrelated with the genotypes. The marginal regression coefficient of \(Y\) on \(Z_k\) is:

\frac{\operatorname{Cov}(Z_k,Y)}{\operatorname{Var}(Z_k)} = \frac{\operatorname{Cov}(Z_k,\beta_c Z_c+\varepsilon)}{1} = \beta_c r_{kc}.

Thus a marker correlated with the causal variant can inherit a nonzero marginal association. If the molecular trait is caused by variant \(c_1\) and the outcome by a distinct variant \(c_2\), their marginal association patterns can still overlap when both are correlated with the same nearby markers. Colocalization uses the pattern across variants and the LD structure to compare this possibility with a shared causal variant.

Worked Example: Strong Evidence for a Shared Variant

Consider a hypothetical region with two traits and a colocalization analysis that reports unnormalized hypothesis weights \(W_0=1\), \(W_1=2\), \(W_2=1\), \(W_3=4\), and \(W_4=32\). These are illustrative weights from the model, not measurements from a real study. Normalize them to interpret the posterior evidence.

The total weight is \(1+2+1+4+32=40\). Therefore \(PP_4=32/40=0.800\), while \(PP_3=4/40=0.100\). The posterior probability assigned to a shared causal variant is 0.800 under this model and these priors. Among the two hypotheses in which both traits are associated, the shared-variant fraction is \(32/(32+4)=0.889\).

This pattern favors a shared variant over distinct variants, but it is not a posterior probability of 0.800 that the molecular trait mediates the outcome. A shared variant could influence both traits through a pathway other than the proposed molecular mediation pathway, or the model assumptions could be inadequate. The result supports a locus-level interpretation, not a complete causal mechanism.

Worked Example: Two Associations but Evidence for Distinct Variants

In a second hypothetical region, suppose both traits have strong association evidence, but the posterior weights are \(W_0=0.1\), \(W_1=0.2\), \(W_2=0.1\), \(W_3=18\), and \(W_4=1.8\). The total is \(0.1+0.2+0.1+18+1.8=20.2\).

The posterior for distinct causal variants is \(PP_3=18/20.2=0.891\), and for a shared causal variant it is \(PP_4=1.8/20.2=0.0891\). Among the hypotheses where both traits are associated, the shared-variant fraction is \(1.8/(18+1.8)=0.0909\).

The analysis therefore favors distinct causal variants, despite evidence that each trait is associated in the region. This weakens the claim that the two association signals reflect one shared genetic perturbation. It does not prove that the molecular trait has no causal effect on the outcome: mediation could involve other variants or loci, and the regional data could be insufficient or poorly represented by the model.

Connecting Colocalization to a Mediation Estimate

In a genetic mediation analysis, let \(\beta_{GM}\) denote the genetic effect on a molecular mediator \(M\), and let \(\beta_{MY}\) denote the causal effect of the mediator on an outcome \(Y\), under the assumptions of the mediation model. The product \(\beta_{GM}\beta_{MY}\) is an indirect genetic effect through \(M\) in the product-estimator framework. Colocalization can help assess whether the molecular and outcome association signals used in that interpretation plausibly arise from the same variant. It does not supply \(\beta_{MY}\), and it does not establish the assumptions that identify it.

For a mediated interpretation, relevant conditions include a well-defined mediator and outcome, appropriate alignment of the genetic effects, and the identification assumptions for genetic mediation. Those assumptions include the absence of unaccounted pathways by which the instrument affects the outcome when interpreting an effect as mediated. Colocalization addresses a different question: whether the molecular and outcome signals at a locus are consistent with a common causal variant. Both kinds of evidence matter, but neither replaces the other.

Worked Example: Combining Shared-Variant Evidence With an Indirect Effect

Suppose a hypothetical analysis of a molecular trait and a continuous outcome gives \(PP_4=0.96\). At the candidate variant, the estimated effect on the molecular trait is \(\beta_{GM}=0.25\) standard deviations of the molecular trait per allele. A separate mediation analysis estimates \(\beta_{MY}=0.40\) standard deviations of the outcome per one-standard-deviation increase in the mediator. Under the product-estimator assumptions, calculate the indirect genetic effect.

The product is \(\beta_{GM}\beta_{MY}=(0.25\ \text{mediator SD per allele})(0.40\ \text{outcome SD per mediator SD})=0.100\ \text{outcome SD per allele}\). The mediator-SD units cancel, leaving an indirect effect of 0.100 outcome standard deviations per allele through the specified mediator.

The high \(PP_4\) supports the interpretation that the molecular and outcome association signals share a causal variant in this region. Together with the product estimate, this is stronger evidence for the proposed mediation account than either result alone. It still does not show that the indirect effect is causally identified unless the mediation assumptions hold. Nor does \(PP_4=0.96\) mean there is a 96% probability that the indirect effect is correct; it is posterior support for the shared-variant hypothesis within the colocalization model.

Interpreting Results and Checking Robustness

A useful report gives the posterior support across the competing hypotheses, not just a label such as “colocalized.” In particular, reporting both \(PP_4\) and \(PP_3\) helps distinguish strong shared-variant evidence from a result in which both traits are associated but distinct variants remain plausible. Some analyses also report \(PP_4/(PP_3+PP_4)\), the shared-variant fraction conditional on both traits being associated under these two hypotheses. That conditional quantity answers a narrower question than \(PP_4\), because it excludes the hypotheses that one or both traits have no association in the region.

Prior choices matter because the posterior compares hypotheses using both the data and prior probabilities. In the weights above, \(p_{12}\) directly contributes to the shared-variant weight, while \(p_1p_2\) contributes to the distinct-variant weight. A careful analysis should state the priors and consider whether conclusions change under reasonable alternatives. A posterior close to a decision threshold should not be treated as a categorical biological fact; no single threshold is universally appropriate for every study.

The basic model also assumes at most one causal variant per trait in the region. If either trait has multiple independent signals, a single-signal analysis may assign misleading support to \(H_3\) or \(H_4\). The analyst should examine whether the model suits the data and use an approach that accounts for multiple signals when necessary. Finally, the molecular and outcome association data must refer to compatible alleles and genomic variants; otherwise, apparent disagreement or agreement can be an artifact of mismatched data rather than biological evidence.

Common Mistake: Treating Colocalization as Mediation Proof

Wrong reasoning: “The molecular trait and outcome have a high \(PP_4\), so the molecular trait causes the outcome.” This conclusion is too strong. \(PP_4\) supports a shared causal variant for the two association signals; it does not determine the direction of an effect, establish that the molecular trait lies on the causal pathway, or rule out other routes from the variant to the outcome.

Correction: State the result at the level it supports: “The regional data favor a shared causal variant for the molecular trait and outcome.” Then assess that finding alongside a mediation estimate and its identification assumptions. Keep the shared-variant conclusion separate from the causal pathway conclusion.

Key takeaway: Colocalization is a locus-level test of whether two association signals are consistent with the same causal variant rather than distinct variants in LD. It can strengthen the genetic coherence of a molecular mediation hypothesis, but shared-variant evidence is not, by itself, proof that the molecular trait mediates the genetic effect on the outcome.

Check Your Understanding

Answer in terms of the hypotheses, the model, and the causal interpretation.

What distinction does \(H_4\) make from \(H_3\) in a colocalization analysis?
Why can two distinct causal variants produce overlapping marginal association patterns in a region?
If \(PP_4=0.70\), does that mean there is a 70% probability that the molecular trait mediates the outcome? Explain.
Why should \(PP_4\) be interpreted alongside \(PP_3\), and what question does \(PP_4/(PP_3+PP_4)\) answer?
Name one model assumption whose failure could make a basic single-signal colocalization result misleading.