Why a Strong Colocalization Result Needs Diagnosis
A colocalization analysis can report strong support for a shared causal variant even when the evidence is fragile. The posterior is conditional on the variants analyzed, the model used to represent association signals, and the priors supplied to the analysis. A high posterior is therefore not a substitute for checking whether those conditions are plausible.
This tutorial diagnoses four routes to false or overstated support: unmodeled signals, linkage disequilibrium (LD), prior choice, and incomplete variant coverage. The aim is not to re-run a full colocalization analysis, but to identify what a result does and does not establish.
The single-signal model used in basic colocalization assumes at most one causal variant per trait in the region. When multiple signals are plausible, the multi-signal approaches covered in earlier material are more appropriate. Even with those methods, a result depends on the quality of the LD information and on which variants are available for analysis.
Diagnostic Case 1: An Unmodeled Signal
If a trait has more than one causal signal but the analysis represents it with only one, the fitted signal can summarize a mixture. The resulting variant-level evidence may overlap with another trait's signal even when the underlying causal variants do not. The problem is not simply that the estimated effect is imprecise: the fitted model does not represent the data-generating pattern it is being used to interpret.
Worked Example: A Secondary Signal Hidden by a Single-Signal Fit
Suppose a hypothetical region contains variants \(v_1\), \(v_2\), and \(v_3\). Trait X has causal signals at \(v_1\) and \(v_3\); trait Y has a causal signal at \(v_2\). Assume \(v_1\) and \(v_2\) are in strong LD, with correlation \(r=0.95\), while \(v_3\) is approximately independent of them. A single-signal fit selects \(v_1\) as the leading signal for X and \(v_2\) for Y, and fails to represent X's second signal.
For a standardized single-causal signal, the expected marginal association statistic at a correlated tag is approximately the causal-variant statistic multiplied by the LD correlation. If the causal-variant \(Z\)-statistic is 8 at \(v_1\), then the expected tag statistic at \(v_2\) is approximately
Likewise, if trait Y has a \(Z\)-statistic of 8 at \(v_2\), its expected tag statistic at \(v_1\) is approximately 7.6. Thus, the marginal patterns at \(v_1\) and \(v_2\) can look similar for the two traits even though their causal variants differ. Trait X's additional signal at \(v_3\) is another warning that the single-signal representation is incomplete.
A single-signal analysis might consequently give a high \(PP.H4\). That posterior would summarize support under a model that omits X's second signal; it would not demonstrate that the model's one-signal description is adequate. The diagnostic response is to examine regional association patterns and assess whether each trait has multiple signals, then use an appropriate multi-signal analysis and inspect signal-level colocalization. The numerical \(Z\)-statistic calculation illustrates LD tagging; it does not, by itself, calculate a colocalization posterior.
A useful warning sign is a regional pattern that cannot be described by one association peak per trait, such as distinct peaks or shoulders, or a leading variant that changes after conditioning on another signal. A high single-signal posterior should not be treated as a resolution of those patterns.
Diagnostic Case 2: LD and the Reference Panel
LD makes a variant's marginal association statistic reflect not only its own causal status but also its correlation with causal variants. LD-aware fine-mapping and multi-signal methods use an LD matrix to separate signals. If that matrix does not represent the analyzed study population, the separation can be wrong. This can make distinct signals look more alike, or make a genuinely shared signal appear less alike.
Worked Example: A Reference That Overstates Tagging
Suppose two different traits have distinct causal variants, \(a\) and \(b\). In the study population, the correlation between them is \(r=0.20\). An LD reference panel instead gives \(r=0.90\). For a simplified illustration, let a causal-variant \(Z\)-statistic be 10. The expected marginal statistic at the other variant due to tagging is approximately \(r\) times 10.
Using the study LD: the expected tag statistic is \(0.20 \times 10=2.0\). Using the mismatched reference: the corresponding prediction is \(0.90 \times 10=9.0\). The reference therefore represents the variants as far more strongly correlated than they are in the study.
A method that uses the mismatched LD can have difficulty distinguishing which variant drives each trait's association. If it assigns substantial support to the same variant for both traits, that support may reflect the incorrect correlation structure rather than a shared causal variant. This example does not imply that every LD mismatch produces a false \(PP.H4\); the direction and size of the distortion depend on the signal configuration and method.
Check that the LD reference matches the study's ancestry and variant set, and compare conclusions under plausible alternative LD inputs when mismatch is a concern. In particular, do not interpret a shared-variant assignment as secure simply because the same high-LD group contains the leading variants.
LD also matters when interpreting overlapping credible sets. Two credible sets can overlap because several variants are correlated, even if the traits' causal variants differ. Overlap is not itself a test of shared causality; the relevant evidence is how the specified colocalization model assigns support across its hypotheses.
Diagnostic Case 3: Prior Choice and Posterior Support
A Bayesian colocalization posterior combines the data's relative support for the hypotheses with prior probabilities. In the standard approximate Bayes factor formulation, let \(BF_{1i}\) and \(BF_{2i}\) be the Bayes factors for association at variant \(i\) for traits 1 and 2. With small per-variant priors \(p_1\), \(p_2\), and \(p_{12}\) for association with trait 1, association with trait 2, and a shared association, respectively, the unnormalized hypothesis weights are approximately
The posterior for a hypothesis is its weight divided by the sum of all five weights. This calculation explains why strong association evidence alone does not fix \(PP.H4\): the relative weights for shared and distinct causal variants also depend on \(p_{12}\) relative to \(p_1p_2\).
Worked Example: Changing the Shared-Variant Prior
Consider two variants with hypothetical Bayes factors \(BF_1=(10{,}000,100)\) and \(BF_2=(100,10{,}000)\). Let \(p_1=p_2=10^{-4}\). The sums for the single-trait hypotheses are both \(10{,}100\), so
The shared-variant sum is \(10{,}000\times100+100\times10{,}000=2{,}000{,}000\). The distinct-variant sum is \(10{,}000\times10{,}000+100\times100=100{,}010{,}000\). Thus \(W_4=p_{12}\times2{,}000{,}000\), while \(W_3=10^{-8}\times100{,}010{,}000=1.0001\).
First set \(p_{12}=10^{-6}\). Then \(W_4=2\), and the total weight is \(1+1.01+1.01+1.0001+2=6.0201\), using \(W_0\approx1\). Therefore \(PP.H4\approx2/6.0201=0.3322\). Now change only the shared prior to \(p_{12}=10^{-8}\). Then \(W_4=0.02\), the total is \(4.0401\), and \(PP.H4\approx0.02/4.0401=0.00495\).
The data Bayes factors are unchanged, but the posterior support for sharing changes substantially because the prior changes. These numbers are a demonstration of prior sensitivity, not a recommendation for a particular prior. Priors should be chosen and justified in light of the analysis context, and sensitivity to plausible choices should be reported when conclusions depend on them.
The ratio of shared to distinct weights is especially informative:
This is a comparison of posterior weights for \(H_4\) and \(H_3\), not the complete \(PP.H4\), because \(H_0\), \(H_1\), and \(H_2\) also contribute to the posterior denominator. It makes explicit that an apparently strong preference for sharing can reflect the shared prior as well as the relative evidence at matching versus distinct variants.
Diagnostic Case 4: Incomplete Variant Coverage
A colocalization result concerns the variants that were available and analyzed. If one or both causal variants are missing, the remaining variants may be imperfect proxies. When both traits have a well-measured proxy in common, distinct causal variants can appear to point to the same analyzed variant. Conversely, missing coverage can conceal evidence for a genuinely shared signal.
Worked Example: A Shared Proxy for Two Missing Causal Variants
Suppose trait X is caused by variant \(c_1\), while trait Y is caused by a different variant \(c_2\). Neither causal variant is present in the common variant set used for colocalization, but proxy variant \(q\) is present in both datasets. Assume \(r(c_1,q)=0.90\) and \(r(c_2,q)=0.92\), and suppose each causal variant has a \(Z\)-statistic of 10 in its own trait.
Under the same approximate tagging relationship, the expected marginal \(Z\)-statistics at \(q\) are \(0.90\times10=9.0\) for trait X and \(0.92\times10=9.2\) for trait Y. The analysis sees strong association at the same observed proxy for both traits, while the two distinct causal variants are absent. That observed overlap can overstate the evidence that the causal variant is shared.
The diagnostic is to audit the analyzed variant set against each study's available coverage and ask whether plausible causal variants were omitted. Where possible, expand coverage, harmonize the variants and alleles, and assess whether the conclusion is robust to including additional well-measured variants. If the causal variants remain unobserved, describe the result as support among the analyzed variants rather than proof that the underlying causal variants are identical.
A Practical Diagnostic Sequence
When a result seems to support colocalization strongly, use a sequence of checks before interpreting it:
Ask whether each trait's regional pattern is consistent with the number of signals the analysis allows. If multiple signals are plausible, examine a suitable multi-signal analysis rather than relying on a single-signal result.
Confirm that LD information represents the study population and analyzed variants. Look for distinct causal variants that could create similar marginal patterns through tagging.
Record the priors used and assess whether a reasonable alternative changes the balance between \(H_4\) and \(H_3\). Do not select a prior simply because it produces the preferred conclusion.
Verify that relevant variants were measured or imputed, and that variants and alleles are aligned across traits. Missing or unmatched variants can change which hypotheses are distinguishable.
Report what the posterior supports under the specified model, priors, LD information, and variant set. Do not turn that conditional result into a stronger claim than the analysis justifies.
Common Mistake and Key Takeaway
Wrong reasoning: “The posterior probability for \(H_4\) is high, so the traits must share a causal variant.” This ignores that the posterior is conditional on model assumptions, priors, LD inputs, and analyzed coverage. It also confuses evidence for a shared variant with a broader causal explanation of how the traits are related.
Correction: A high \(PP.H4\) is evidence favoring a shared causal variant over the alternatives represented in the fitted model, given the specified data and priors. It should be interpreted alongside checks for multiple signals, LD mismatch, prior sensitivity, and missing variants. If those checks reveal limitations, report them as qualifications rather than treating the posterior as definitive.
Check Your Understanding
For each question, identify what the result can support and what additional diagnostic check is needed.
- A single-signal analysis reports high \(PP.H4\), but one trait has two clear regional association peaks. What assumption should be questioned?
- Why can two distinct causal variants produce similar marginal association patterns when they are in strong LD?
- In the prior-sensitivity example, why did \(PP.H4\) change even though all Bayes factors stayed fixed?
- Both traits have strong association at the same observed proxy, but their likely causal variants are absent from the shared variant set. What limitation should be reported?
- What does a high \(PP.H4\) establish under the fitted model, and what does it not establish by itself?