Running Single-Signal Colocalization From Summary Statistics

From Regional Summary Statistics to Colocalization

A colocalization analysis asks whether two association signals in a genomic region are consistent with the same causal variant. It does not simply ask whether both traits have small p-values somewhere nearby. The analysis uses the pattern of evidence across variants: a variant that is strongly supported for both traits contributes to the shared-signal hypothesis, while distinct variants supported for each trait contribute to the separate-signals hypothesis.

In a single-signal analysis, each trait is treated as having at most one causal variant in the selected region. The evidence is summarized across a common set of variants, usually using approximate Bayes factors (ABFs) calculated from summary statistics. The H0–H4 hypotheses distinguish no association, association with one trait, distinct causal variants, and a shared causal variant. A posterior probability summarizes support for each hypothesis conditional on the specified model and priors.

Definition: A single-signal colocalization analysis compares two traits over a defined genomic region under the assumption that each trait has at most one causal variant in that region. H4 denotes association with both traits due to the same causal variant; H3 denotes association with both traits due to distinct causal variants.

The regional inputs must be prepared before calculating or interpreting those probabilities. A reliable workflow checks that variants are represented consistently, that the two studies have enough variants in common, and that the association statistics are suitable for the chosen ABF calculation. These checks do not establish that the single-signal assumption is true; they establish that the inputs are coherent enough to apply that model.

Preparing the Two Regional Data Sets

For each trait, retain a defined genomic region and, for every variant, its identifier or genomic position, effect allele, other allele, effect estimate, and standard error (or the variance of the estimate). Allele frequency and sample size are useful supporting information. If the inputs provide only p-values, the method used to obtain ABFs must have enough additional information to reconstruct an association statistic; a p-value alone does not specify the effect estimate and its uncertainty.

The two data sets should refer to the same variants and the same genomic build. Restricting to the intersection of their variant sets makes the comparison explicit. A variant present in only one study can contribute evidence for that study’s association, but it cannot provide a same-variant contribution to H4. If overlap is sparse, an apparent absence of shared evidence can reflect missing coverage rather than evidence against a shared signal.

Allele labels matter because effect estimates are defined relative to an effect allele. Before comparing estimates or frequencies, check that the effect alleles have been made consistent. For example, estimates for the same biological allele should have the same direction; estimates reported relative to opposite alleles have opposite signs. Allele frequencies can help identify inconsistencies, but they must be compared for the same allele and interpreted with the ancestry and sample context in mind. A frequency difference is not automatically an error: study populations can differ. Ambiguous strand variants require particular care when their alleles cannot be resolved from the available information.

The ABF calculation also depends on the statistical scale of the effect estimate and its uncertainty. For an approximately normal estimate, use the estimated effect and its sampling variance. For binary traits or summary statistics reconstructed from p-values, the method’s required inputs and assumptions should be checked rather than treating all reported statistics as interchangeable.

Worked Example: Checking Overlap and Allele Frequencies

Suppose two hypothetical studies report results for variants in the same region. At variant v1, the molecular-trait study reports effect allele A, other allele G, effect estimate \(\hat{\beta}_1=0.20\), standard error \(SE_1=0.05\), and effect-allele frequency 0.30. The disease-trait study reports the same variant and alleles, \(\hat{\beta}_2=0.16\), \(SE_2=0.04\), and effect-allele frequency 0.31. Assume both use the same genome build.

Overlap. Variant v1 occurs in both data sets, so it can contribute to a same-variant comparison. In a full regional analysis, make this check for every retained variant and report how many variants remain in the intersection.

Alleles and frequency. Both studies report A as the effect allele and G as the other allele. The effect-allele frequencies, 0.30 and 0.31, are close. The corresponding minor-allele frequencies are \(\min(0.30,0.70)=0.30\) and \(\min(0.31,0.69)=0.31\). This is a consistency check, not proof that the studies have identical populations.

Association statistics. The standard errors are positive, so the estimates can be converted to z-statistics: \(z_1=0.20/0.05=4.00\) and \(z_2=0.16/0.04=4.00\). The corresponding variances are \(0.05^2=0.0025\) and \(0.04^2=0.0016\). These are the quantities needed for the normal-approximation ABF calculation below.

Approximate Bayes Factors for Each Variant

For a single trait and variant, let \(\hat{\beta}\) be the estimated effect, \(V=SE^2\) its estimated sampling variance, and \(W\) the prior variance for the effect under the association alternative. The null model sets the true effect to zero. The alternative assigns the true effect a mean-zero normal prior with variance \(W\). Define \(z=\hat{\beta}/\sqrt{V}\) and \(r=W/(V+W)\). Under these assumptions, the Wakefield ABF is the marginal likelihood under the alternative divided by the likelihood under the null:

\mathrm{ABF} = \sqrt{1-r}\exp\left(\frac{r z^2}{2}\right), \qquad r=\frac{W}{V+W}.

An ABF greater than 1 favors the association alternative over the null for that variant, with the size of the evidence depending on both the z-statistic and the prior variance. An ABF is not a p-value and is not itself a posterior probability. The prior variance \(W\) is in squared effect units, so it must be chosen on the same scale as \(V\).

The formula follows by integrating over the alternative’s effect prior. Under the null, \(\hat{\beta}\) has a normal sampling distribution with mean zero and variance \(V\). Under the alternative, integrating \(\hat{\beta}\mid\beta\sim N(\beta,V)\) over \(\beta\sim N(0,W)\) gives \(\hat{\beta}\sim N(0,V+W)\). Taking the ratio of these two normal densities gives:

\frac{f(\hat{\beta}\mid\text{alternative})} {f(\hat{\beta}\mid\text{null})} = \sqrt{\frac{V}{V+W}} \exp\left\{ \frac{\hat{\beta}^{\,2}}{2} \left(\frac{1}{V}-\frac{1}{V+W}\right) \right\} = \sqrt{1-r}\exp\left(\frac{r z^2}{2}\right).

This derivation makes the assumptions visible: the estimate is treated as approximately normal with known or suitably estimated variance, and the alternative prior is normal with variance \(W\). In practice, the ABFs from both traits are computed for each retained variant using declared, trait-appropriate prior effect variances.

Worked Example: Calculating an ABF

For a hypothetical molecular-trait association, let \(\hat{\beta}=0.20\) trait units per effect allele, \(SE=0.05\) trait units per allele, and prior variance \(W=0.04\) squared trait units per allele. Assume the normal approximation above is appropriate.

First, \(V=SE^2=0.05^2=0.0025\), and \(z=0.20/0.05=4.00\). Then

r=\frac{0.04}{0.0025+0.04} =\frac{0.04}{0.0425} \approx 0.941176.

Substituting into the ABF formula gives

\log(\mathrm{ABF}) = \frac{1}{2}\log(1-0.941176) +\frac{0.941176(4.00)^2}{2} \approx -1.41661+7.52941 =6.11280.

Therefore, \(\mathrm{ABF}=\exp(6.11280)\approx 451.6\). Under the stated normal-prior model, the data at this variant are about 452 times as likely under the association alternative as under the null. This is a likelihood ratio for the two models, not a 452-fold posterior probability and not a statement that this variant is certainly causal.

Combining Variant Evidence Into H0–H4

Let \(\mathrm{ABF}_{1i}\) and \(\mathrm{ABF}_{2i}\) denote the ABFs for trait 1 and trait 2 at variant \(i\). With per-variant prior probabilities \(p_1\) and \(p_2\) for association with each trait, and \(p_{12}\) for a shared causal variant, the unnormalized hypothesis weights are conventionally written:

\begin{aligned} W_0 &= 1,\\ W_1 &= p_1\sum_i \mathrm{ABF}_{1i},\\ W_2 &= p_2\sum_i \mathrm{ABF}_{2i},\\ W_3 &= p_1p_2\sum_{i\ne j}\mathrm{ABF}_{1i}\mathrm{ABF}_{2j},\\ W_4 &= p_{12}\sum_i \mathrm{ABF}_{1i}\mathrm{ABF}_{2i}. \end{aligned}

Here H3 sums over pairs of distinct variants, whereas H4 sums products for the same variant. The posterior probability for hypothesis \(H_k\) is \(W_k/\sum_{\ell=0}^{4}W_\ell\). The prior terms are essential: posterior outputs depend on the specified priors as well as on the association data. These expressions use the usual rare-prior approximation in which the prior mass for multiple causal configurations is represented by the displayed weights.

The previous derivation of H0–H4 posterior probabilities motivates these weights; in an applied analysis, the key task is to supply coherent variant-level evidence and understand what the returned probabilities condition on. The following hypothetical calculation makes the distinction between H3 and H4 explicit.

Worked Example: Reading Posterior Probabilities From Two Variants

Suppose the shared variant set contains only variants A and B. Their ABFs are \(\mathrm{ABF}_{1A}=1000\), \(\mathrm{ABF}_{1B}=1\), \(\mathrm{ABF}_{2A}=800\), and \(\mathrm{ABF}_{2B}=1\). For illustration, set \(p_1=p_2=0.0001\) and \(p_{12}=0.00001\).

The unnormalized H1 and H2 weights are

W_1=0.0001(1000+1)=0.1001,\qquad W_2=0.0001(800+1)=0.0801.

For H3, the distinct-variant pairs are trait 1 at A with trait 2 at B, and trait 1 at B with trait 2 at A:

W_3=10^{-8}\{(1000)(1)+(1)(800)\}=0.000018.

For H4, multiply the two traits’ ABFs at the same variant and sum:

W_4=0.00001\{(1000)(800)+(1)(1)\}=8.00001.

Since \(W_0=1\), the total weight is \(1+0.1001+0.0801+0.000018+8.00001=9.180228\). Thus

\begin{aligned} PP(H0)&=1/9.180228\approx 0.1089,\\ PP(H1)&=0.1001/9.180228\approx 0.01090,\\ PP(H2)&=0.0801/9.180228\approx 0.008726,\\ PP(H3)&=0.000018/9.180228\approx 0.00000196,\\ PP(H4)&=8.00001/9.180228\approx 0.8714. \end{aligned}

In this toy analysis, H4 receives the largest posterior probability, about 87.1%, because the strongest evidence for both traits is at variant A. H0 still has about 10.9% posterior probability under these priors and this limited variant set. The result supports a shared causal variant within the modeled region and assumptions; it does not prove that variant A is causal or that the molecular trait mediates the disease association.

Interpreting Outputs and Avoiding Overclaiming

A colocalization report commonly gives posterior probabilities for H0 through H4. These probabilities sum to one, up to rounding, and describe relative support among the specified hypotheses given the regional data, the ABF model, and the priors. A high PP(H4) is evidence favoring a shared causal variant over the other hypotheses considered. It is not the probability that a gene is biologically causal, and it does not establish a mechanism such as mediation.

Interpret the full pattern, not just the largest number. High PP(H3) indicates support for both traits being associated but with distinct causal variants. High PP(H1) or PP(H2) indicates that the evidence favors association with only one trait in the region. High PP(H0) indicates that the modeled data provide little support for association with either trait. When probabilities are diffuse, the appropriate conclusion is that the analysis does not sharply distinguish the hypotheses.

A useful practical record includes the genomic interval, number of variants in each study and in their intersection, the ABF inputs and prior variances, and the priors used for H0–H4. This makes it possible to distinguish a weak colocalization result from one driven by limited overlap or particular prior choices. The single-signal assumption is also central: if a region contains multiple causal signals, the single-signal posterior may not adequately represent the underlying evidence.

Common Mistakes

Treating nearby association peaks as colocalization. Two small p-values in the same region do not show that the same variant drives both associations. Colocalization compares evidence across variants, including the same-variant evidence for H4 and distinct-variant evidence for H3.
Ignoring allele orientation. Comparing effect estimates as though they refer to the same allele can reverse signs and conceal input problems. Align effect alleles before interpreting effects or using frequencies as a check.
Calling an ABF a posterior probability. An ABF compares likelihoods under two models. Posterior probabilities additionally depend on prior probabilities and on the evidence across the region.
Reading PP(H4) as proof of mediation. A shared association signal does not establish which trait causes the other, or that one mediates the other. It supports shared variant evidence under the model.
Ignoring variant coverage. A small shared variant set can limit the ability to recognize a shared signal. Report overlap and interpret absence of H4 evidence in light of what each study actually measured.

Key takeaway: A single-signal colocalization result is only as interpretable as its regional inputs. Check the common variant set and allele definitions, calculate ABFs under stated assumptions, and read the H0–H4 posterior probabilities as conditional evidence—not as causal proof.

Check Your Understanding

Answer these questions using the definitions and workflow above.

Why must a variant’s effect allele be checked before comparing effect estimates from two studies?
For an effect estimate of 0.12 and standard error 0.03, what are \(V\) and \(z\)?
In the H3 weight, why are pairs with \(i=j\) excluded?
If PP(H4) is large, what conclusion is supported, and what conclusion about mediation is not established?
Why should an analysis report the number of variants shared between the two regional data sets?