Why Overlap Is Not Colocalization
Two traits can show association signals in the same genomic region, and their fine-mapping credible sets can contain some of the same variants. That overlap is a useful clue: it identifies candidates worth examining. But it is not, by itself, evidence that the traits share a causal variant. Variants in linkage disequilibrium (LD) can carry similar association signals, even when the causal variant for one trait differs from the causal variant for the other.
The distinction matters when interpreting a molecular QTL alongside a disease or other trait association. A shared causal variant is compatible with a genetic connection between the traits, but overlap between two sets of plausible variants does not establish that sharing. Colocalization addresses a more specific question by comparing models in which neither trait is associated, only one is associated, both have distinct causal variants, or both share one causal variant.
A credible set is not a list of variants that are all causal, nor does membership mean that a variant has a 95% probability of being causal. The set’s target refers to the cumulative posterior probability under the model and its assumptions. In particular, each trait’s set is constructed from evidence for that trait; set overlap does not directly evaluate whether the two traits have the same causal variant.
The Competing Explanations
In the usual five-hypothesis colocalization framework, the hypotheses are mutually exclusive explanations for the two traits within the region: \(H_0\), neither trait is associated; \(H_1\), only trait 1 is associated; \(H_2\), only trait 2 is associated; \(H_3\), both are associated but have distinct causal variants; and \(H_4\), both are associated and share a causal variant. The posterior probabilities depend on the association data, the model, and the prior probabilities assigned to the hypotheses.
Fine-mapping and colocalization are related but not interchangeable. Fine-mapping asks where a causal variant may lie for a trait. Colocalization asks which joint explanation for the two traits is best supported. Under \(H_3\), LD can make variants near each trait’s causal variant look associated with both traits. As a result, the marginal evidence can produce overlapping credible sets even though the causal variants differ. Conversely, a set-overlap rule can miss shared-variant evidence if the sets are sensitive to model uncertainty or how they were constructed.
The appropriate comparison is therefore not simply “Do the sets intersect?” It is whether the joint data and specified priors support \(H_4\) over alternatives, particularly \(H_3\). The colocalization model uses the association patterns across variants and the assumed LD structure to evaluate these alternatives. Its output is still conditional on the model and priors; it is not a guarantee that the true biological mechanism has been identified.
Worked Example: Overlapping Sets Despite Distinct Causal Variants
Suppose a hypothetical region contains variants \(v_1,\ldots,v_4\). Fine-mapping reports the following posterior inclusion probabilities (PIPs), conditional on association for each trait:
| Variant | Trait 1 PIP | Trait 2 PIP |
|---|---|---|
| \(v_1\) | 0.50 | 0.05 |
| \(v_2\) | 0.35 | 0.45 |
| \(v_3\) | 0.10 | 0.40 |
| \(v_4\) | 0.05 | 0.10 |
For trait 1, order the PIPs from largest to smallest and accumulate them: \(0.50+0.35=0.85\), then \(0.85+0.10=0.95\). Its 95% credible set is \(\{v_1,v_2,v_3\}\). For trait 2, the first two PIPs sum to \(0.45+0.40=0.85\); adding \(0.10\) gives \(0.95\). Its 95% credible set is \(\{v_2,v_3,v_4\}\). Thus the sets overlap at \(v_2\) and \(v_3\).
Now suppose the hypothetical data were generated with \(v_1\) causal for trait 1 and \(v_4\) causal for trait 2, and that these variants and nearby markers are in LD. Association evidence at correlated markers can leave posterior probability spread across variants in both sets. The overlap is real as a set operation, but it does not change the stipulated distinct causal variants. To assess the shared-variant explanation, one must compare the colocalization hypotheses; the overlap alone cannot choose between \(H_3\) and \(H_4\).
What the Posterior Comparison Means
A convenient way to see the role of the model and priors is to express posterior odds between \(H_4\) and \(H_3\). Let \(D\) denote the association data for both traits. For each hypothesis \(H_k\), let \(P(D\mid H_k)\) be its marginal likelihood, integrating over the variants and other parameters allowed by that hypothesis. Define the Bayes factor against \(H_0\) as \(BF_k=P(D\mid H_k)/P(D\mid H_0)\), with \(BF_0=1\). Bayes’ rule gives:
This follows because \(P(H_k\mid D)\) is proportional to \(P(H_k)P(D\mid H_k)\); taking the ratio for \(H_4\) and \(H_3\) cancels the common normalizing constant and the denominator \(P(D\mid H_0)\) used in both Bayes factors. Thus the posterior comparison depends on both the relative prior odds and the relative fit of the data under shared and distinct causal variants. A credible-set intersection provides neither quantity by itself.
Worked Example: Overlap With Stronger Support for Distinct Variants
Suppose the two 95% credible sets overlap, but a colocalization analysis gives the following illustrative priors and Bayes factors. These values are hypothetical and summarize evidence from the full association patterns, not from counting shared set members.
| Hypothesis | Prior probability | Bayes factor versus \(H_0\) |
|---|---|---|
| \(H_0\) | 0.900 | 1 |
| \(H_1\) | 0.040 | 2 |
| \(H_2\) | 0.040 | 1.5 |
| \(H_3\) | 0.019 | 24 |
| \(H_4\) | 0.001 | 3 |
The unnormalized posterior weights are prior times Bayes factor: \(0.900\), \(0.080\), \(0.060\), \(0.456\), and \(0.003\), respectively. Their sum is \(1.499\). Therefore:
Although the credible sets overlap, this analysis gives much more posterior support to \(H_3\) than to \(H_4\). The posterior odds \(H_4:H_3\) are \(0.003/0.456=0.00658\), or about 1 to 152. The conclusion is not that overlap is irrelevant; rather, it is not enough to override the evidence favoring distinct causal variants in this model.
How to Read Evidence for a Shared Variant
The reverse situation is also possible: overlapping sets may accompany strong evidence for \(H_4\), but it is the model comparison—not the overlap—that supplies that evidence. For instance, if both traits’ association patterns are best explained by the same variant, the data can favor \(H_4\) over \(H_3\). That conclusion depends on how well the alternatives explain the complete pattern, including uncertainty about which variant is responsible.
Worked Example: Strong Posterior Support for Sharing
Consider another hypothetical analysis with the same prior probabilities as above. Suppose its Bayes factors are \(BF_0=1\), \(BF_1=1\), \(BF_2=1\), \(BF_3=2\), and \(BF_4=5000\). The unnormalized posterior weights are \(0.900\), \(0.040\), \(0.040\), \(0.038\), and \(5.000\). The total is \(6.018\), so:
Here the posterior probability of \(H_4\) is about 0.831 under the stated model and priors, and the posterior odds of \(H_4\) to \(H_3\) are \(5/0.038=131.6\). If the credible sets also overlap, that overlap is consistent with the colocalization result, but it is not the calculation that produced it. The result supports a shared causal variant within the region and model; it does not by itself establish a molecular mechanism or a causal effect of one trait on the other.
A Practical Interpretation
A useful interpretation keeps three statements separate. First, the credible sets overlap: the sets contain at least one common variant. Second, the colocalization analysis favors one joint hypothesis over another: for example, \(H_4\) over \(H_3\). Third, any biological interpretation concerns what a shared variant might imply, and requires evidence beyond this statistical comparison. Combining these statements into “the overlap proves the traits share a causal variant” skips the central model comparison.
The comparison is meaningful only for a coherent analysis. The variants must refer to the same genomic region and be represented consistently across the two trait analyses. The colocalization model must also be appropriate to the signal structure: when there are multiple signals, a single-signal comparison can conflate distinct signals or miss sharing, so the relevant signal-level comparisons should be used. The result remains conditional on the chosen priors and assumptions about the data and LD.
It is often more informative to report posterior support for \(H_3\) and \(H_4\) together than to report only a high \(P(H_4\mid D)\). A small \(P(H_4\mid D)\) can occur because the evidence favors \(H_3\), but it can also occur because the data support \(H_0\), \(H_1\), or \(H_2\). Reporting the relevant competing posterior probabilities makes the interpretation clearer. As with any posterior probability, the value is conditional on the model and priors used; it is not a frequentist probability that the hypothesis is true.
Common Mistake
Wrong reasoning: “The 95% credible sets share two variants, so the traits must have the same causal variant.” The sets summarize marginal fine-mapping uncertainty for the traits separately; LD can make variants associated with one trait plausible candidates for the other even when the true causal variants differ.
Correction: Treat set overlap as a descriptive clue, then use a colocalization model that compares shared and distinct causal-variant explanations. Interpret \(P(H_4\mid D)\) in relation to \(P(H_3\mid D)\), the other hypotheses, and the analysis assumptions. Neither set intersection nor a single posterior value should be turned into a causal claim about the traits’ relationship.
Check Your Understanding
Answer each question by distinguishing set membership from evidence under a joint model.
- What does a 95% fine-mapping credible set summarize, and what does its 95% target not mean?
- How can two traits have overlapping credible sets if their causal variants differ?
- In the posterior-odds expression for \(H_4\) versus \(H_3\), what roles do the prior odds and Bayes-factor ratio play?
- If \(P(H_4\mid D)\) is small, why is it useful to examine \(P(H_3\mid D)\) and the other hypothesis probabilities?
- Why does evidence for a shared causal variant not, by itself, establish a mechanism or a causal effect of one trait on another?