Worked Single-Signal Colocalization At A Locus

From Summary Statistics to a Shared-Signal Question

At a genomic locus, a GWAS association and an eQTL association can occur in the same region without being driven by the same variant. Linkage disequilibrium can make different causal variants appear associated with overlapping sets of nearby variants. Single-signal colocalization compares these explanations by combining variant-level evidence from the two traits under explicit hypotheses.

The calculation below uses approximate Bayes factors (ABFs) for each candidate variant. It illustrates the single-signal model and the five hypothesis posteriors, not a claim that an eQTL mediates a GWAS association. The earlier material on reading PP.H0 through PP.H4 gives the hypothesis labels; here the focus is constructing and interpreting them from a small numerical locus.

Definition: Let \(A_i\) and \(B_i\) denote the Bayes factors for association at variant \(i\) for the GWAS trait and the eQTL trait, respectively. A Bayes factor compares the marginal likelihood under an association model with that under no association. The five hypotheses are: \(H_0\), neither trait is associated at the locus; \(H_1\), only the GWAS trait is associated; \(H_2\), only the eQTL trait is associated; \(H_3\), both traits are associated but have distinct causal variants; and \(H_4\), both traits share a causal variant.

The single-signal model assumes that each trait has either no causal variant in the region or exactly one. Under \(H_3\), the two causal variants must be different; under \(H_4\), they are the same. The candidate variants must refer to the same genomic locus, and the summary statistics must be harmonized so that variant identity, alleles, and effect directions are comparable. Each ABF must also be computed under an appropriate sampling model and effect-size prior. These assumptions define the calculation; they are not verified just because the output contains five posterior probabilities.

Computing Variant-Level Evidence

For a normally distributed effect estimate \(\hat{\beta}\) with standard error \(s_\beta\), a common Wakefield ABF uses \(V=s_\beta^2\) and a prior variance \(W\) for the effect under association. With \(z=\hat{\beta}/s_\beta\), the ABF is

\operatorname{ABF} = \sqrt{\frac{V}{V+W}} \exp\left(\frac{z^2W}{2(V+W)}\right).

This is the association-model marginal likelihood divided by the null-model likelihood. An ABF greater than 1 favors association over the null for that variant under the specified prior; it is not a posterior probability. Changing the effect prior can change the ABF, so the prior scale and the trait’s effect units need to be scientifically coherent.

Worked Example: An ABF for One Hypothetical GWAS Variant

Suppose a hypothetical variant has estimated effect \(\hat{\beta}=0.8\) with standard error \(s_\beta=0.2\), on a specified trait scale. Use a prior effect variance \(W=0.16\). Then \(z=0.8/0.2=4\), and \(V=0.2^2=0.04\). The normal approximation and the chosen prior are assumed appropriate for this illustration.

Substituting into the ABF formula gives

\operatorname{ABF} = \sqrt{\frac{0.04}{0.04+0.16}} \exp\left(\frac{4^2(0.16)}{2(0.20)}\right) = \sqrt{0.2}\exp(6.4) \approx 269.2.

Thus, under these assumptions, the observed summary statistic is about 269.2 times as likely under the association model, integrated over its effect prior, as under the null model. This is strong evidence for this variant considered by itself; the number is not the probability that the variant is causal.

Hypothesis Weights and Posterior Probabilities

Let \(p_1\) be the prior probability assigned to association of the GWAS trait at a particular variant, \(p_2\) the corresponding prior for the eQTL trait, and \(p_{12}\) the prior for a shared causal variant. In the usual rare-association-prior calculation, the unnormalized weights across candidate variants are

\begin{aligned} w_0 &= 1,\\ w_1 &= p_1\sum_i A_i,\\ w_2 &= p_2\sum_i B_i,\\ w_3 &= p_1p_2\sum_{i\ne j} A_iB_j,\\ w_4 &= p_{12}\sum_i A_iB_i. \end{aligned}

The off-diagonal sum in \(w_3\) pairs the GWAS evidence at one variant with the eQTL evidence at a different variant. The diagonal sum in \(w_4\) pairs the two traits’ evidence at the same variant. The products \(p_1p_2\) and \(p_{12}\) encode the prior weights for distinct and shared causal configurations, respectively. These are unnormalized weights, not yet posterior probabilities. Their use corresponds to the standard rare-prior approximation in which configurations with multiple causal variants per trait are outside the model.

The key arithmetic identity is that the sum over all variant pairs is the product of the marginal sums. Removing the diagonal leaves the distinct-variant evidence:

\sum_{i\ne j} A_iB_j = \left(\sum_i A_i\right)\left(\sum_j B_j\right) - \sum_i A_iB_i.

Each \(A_iB_j\) occurs once in the product of sums. The terms with \(i=j\) are exactly the same-variant terms; subtracting them therefore leaves precisely the terms with \(i\ne j\). Once all five weights are calculated, normalize them using \(T=\sum_{k=0}^4 w_k\), so that \(PP.H_k=w_k/T\). The posteriors sum to 1, apart from rounding.

Worked Example: All Five Hypothesis Posteriors at a Three-Variant Locus

Consider a hypothetical locus with three candidate variants. The GWAS and eQTL ABFs are as follows; they are assumed to have been calculated using compatible summary-statistic models and to refer to the same harmonized variants.

Variant	GWAS ABF \(A_i\)	eQTL ABF \(B_i\)
1	269.2	60.06
2	16.37	1.100
3	0.6672	0.4942

Use hypothetical priors \(p_1=p_2=0.0001\) and \(p_{12}=0.00001\). First sum the ABFs:

\sum_i A_i=269.2+16.37+0.6672=286.2372, \qquad \sum_i B_i=60.06+1.100+0.4942=61.6542.

The product of these sums is \(286.2372\times61.6542=17647.725576\), or approximately \(17647.73\). The same-variant sum is

\begin{aligned} \sum_i A_iB_i &=(269.2)(60.06)+(16.37)(1.100)+(0.6672)(0.4942)\\ &=16168.152+18.007+0.32973024\\ &=16186.48873024. \end{aligned}

Therefore the distinct-variant sum is \(17647.725576-16186.48873024=1461.23684576\). Using these values, the five weights are

\begin{aligned} w_0&=1,\\ w_1&=0.0001(286.2372)=0.02862372,\\ w_2&=0.0001(61.6542)=0.00616542,\\ w_3&=0.0001^2(1461.23684576)\approx0.0000146124,\\ w_4&=0.00001(16186.48873024)\approx0.1618649. \end{aligned}

Their total is \(T\approx1.1966686\). Dividing each weight by this total gives

\begin{aligned} PP.H0&\approx0.8357,& PP.H1&\approx0.02392,& PP.H2&\approx0.005152,\\ PP.H3&\approx0.00001221,& PP.H4&\approx0.1353. \end{aligned}

These values sum to approximately 1. Under this model and these priors, \(H_0\) has the largest posterior probability, about 0.8357. \(H_4\) has posterior probability about 0.1353, while \(H_3\) has very little posterior probability. Thus the calculation does not establish a shared signal: the posterior probability assigned to no association is still much larger than that assigned to a shared association.

What the Posterior Pattern Says

The posterior for \(H_4\) is the probability assigned to the shared-variant hypothesis among the five specified hypotheses, conditional on the data, priors, and model. It is not the probability that the GWAS trait is caused by altered gene expression, nor does it establish that the gene mediates a biological effect. Colocalization addresses whether the two association signals are consistent with a shared causal variant; causal interpretation requires additional evidence and assumptions.

It is useful to separate two comparisons. First, \(PP.H4\) versus \(PP.H3\) concerns whether the two traits’ signals are more consistent with a shared variant than distinct variants, conditional on the model’s hypothesis set. In the example, the posterior odds are

\frac{PP.H4}{PP.H3} = \frac{w_4}{w_3} \approx \frac{0.1618649}{0.0000146124} \approx 1.108\times10^4.

This is strong relative support for \(H_4\) over \(H_3\). But it does not mean \(H_4\) is highly probable overall: \(H_0\), \(H_1\), and \(H_2\) also retain posterior mass, especially \(H_0\). Relative evidence among two alternatives and absolute posterior probability across all alternatives answer different questions.

Worked Example: Identifying Which Variant Drives the Shared-Signal Weight

For \(H_4\), each variant contributes \(A_iB_i\). At variant 1 the contribution is \(269.2(60.06)=16168.152\); the total same-variant sum is \(16186.48873024\). Thus variant 1’s share of the unnormalized \(H_4\) evidence is

\frac{16168.152}{16186.48873024}\approx0.9989.

So approximately 99.89% of the same-variant evidence comes from variant 1. This says that variant 1 dominates the \(H_4\) sum in this three-variant calculation. It does not mean there is a 99.89% posterior probability that variant 1 is causal: the denominator here is only the sum of the \(H_4\) variant contributions, not the total probability across all hypotheses and configurations.

Interpretation rule: Report the posterior pattern across all five hypotheses, and distinguish \(PP.H4\) from the ratio \(PP.H4/PP.H3\). State the priors and single-signal model. Describe a high \(H_4\) posterior as support for a shared association signal under that model, not proof of mediation or biological causation.

Diagnostics, Comparisons, and Limitations

Before interpreting a result, check that variant identifiers, genomic build, alleles, and effect directions are harmonized and that the two analyses cover an adequately overlapping set of variants. Inspect whether the locus contains usable association evidence for each trait, whether ABFs are finite and calculated on sensible effect scales, and whether the posterior weights normalize as expected. A useful arithmetic check is to verify that the \(H_3\) off-diagonal sum plus the \(H_4\) diagonal sum equals \((\sum_i A_i)(\sum_i B_i)\). Rounding can cause small discrepancies, but not large ones.

A small \(PP.H4\) should not automatically be interpreted as evidence for distinct causal variants. It can arise because neither trait has convincing locus-level evidence, because only one trait does, or because the available variants do not provide strong shared-variant evidence. Compare all five posteriors before deciding which explanation is most consistent with the calculation. Conversely, a large \(PP.H4\) is conditional on the assumed candidate set, priors, ABFs, and single-signal model.

The single-signal approach is computationally simple and makes the shared-versus-distinct comparison explicit. Its key limitation is that it allows at most one causal variant per trait in the locus. If a region contains multiple independent signals, the model can misrepresent their evidence. The per-variant ABFs and LD structure also require care: correlated variants are not independent biological replications, and a calculation based on marginal summary statistics does not itself establish that the single-signal assumption is adequate. The model is therefore most defensible when one signal per trait is a plausible representation of the locus and the inputs are well harmonized. More complex signal structures require methods beyond this calculation.

Calculation pitfall: Using the product \((\sum_i A_i)(\sum_i B_i)\) as the \(H_3\) sum counts same-variant pairs too. Subtract \(\sum_i A_iB_i\) to retain only distinct-variant pairs.
Interpretation pitfall: Saying “\(PP.H4/PP.H3\) is enormous, so \(H_4\) is almost certain” ignores posterior mass on \(H_0\), \(H_1\), and \(H_2\). Report the full posterior distribution.
Model pitfall: Treating \(H_4\) as proof that expression mediates the GWAS trait confuses a shared association signal with a causal pathway. Colocalization alone does not identify mediation.

Check Your Understanding

Answer each question using the definitions and calculations above. Full solutions follow the questions.

For two variants, \(A=(5,2)\) and \(B=(3,4)\), calculate the total pairwise product, the same-variant sum, and the distinct-variant sum.
In the three-variant example, why is \(PP.H4\approx0.1353\) not inconsistent with the very large ratio \(PP.H4/PP.H3\)?
Suppose a locus has strong GWAS evidence but negligible eQTL evidence. Which hypotheses may retain posterior support, and why is a low \(PP.H4\) not by itself evidence for \(H_3\)?
What does the approximately 0.9989 share for variant 1 in the worked \(H_4\) calculation represent, and what does it not represent?

Solutions. (1) The product of sums is \((5+2)(3+4)=49\). The same-variant sum is \(5(3)+2(4)=23\), so the distinct-variant sum is \(49-23=26\), also \(5(4)+2(3)=26\). (2) The ratio compares only \(H_4\) with \(H_3\); it says the shared hypothesis outweighs the distinct-signal hypothesis. But \(H_0\) has weight 1 and dominates the total in the example, so the posterior probability of \(H_4\) across all five hypotheses remains about 0.1353. (3) \(H_1\) may retain support because the GWAS alone is associated, while \(H_0\) may also retain support depending on the strength of the evidence and the priors. With little eQTL evidence, the data need not favor two associated traits with distinct variants, so low \(PP.H4\) does not imply high \(PP.H3\). (4) It is the fraction of the \(H_4\) same-variant evidence sum contributed by variant 1. It is not the posterior probability that variant 1 is causal, because it conditions only on the \(H_4\) contributions and does not account for other hypotheses.