Why a Weak eQTL Result Is Not a Colocalization Verdict
A colocalization analysis asks whether genetic association with a trait and genetic association with gene expression are consistent with a shared causal variant. If the eQTL association is weak, however, the data may not distinguish a shared variant from no eQTL signal at all. A nonsignificant eQTL result is therefore not, by itself, evidence that the GWAS and eQTL signals arise from different variants.
The practical task is to diagnose what the weakness means. Consider the eQTL effect estimate and its uncertainty, the sample size and allele frequency that determine precision, and the full posterior distribution over the five hypotheses. The association p-value is useful context, but it does not replace that assessment.
The distinction between \(H_3\) and \(H_4\) is most informative about sharing when there is evidence for association with both traits. In contrast, posterior mass on \(H_0\), \(H_1\), or \(H_2\) may indicate that evidence for one or both associations is absent or weak. A low \(PP.H4\) alone does not tell which situation applies.
What Determines eQTL Precision?
For a simple approximation, suppose expression \(Y_i\) follows an additive linear model \(Y_i=\alpha+\beta G_i+\varepsilon_i\), where \(G_i\) is the number of copies of an allele (0, 1, or 2), \(\beta\) is the per-allele expression effect, and the errors are independent with mean zero and variance \(\sigma^2\). Assume the genotype is measured accurately, individuals are independent, and relevant covariates have been accounted for. If the allele frequency is \(f\), Hardy–Weinberg equilibrium gives genotype variance \(2f(1-f)\). With adequate sample size, the approximate standard error and noncentrality of the association statistic are:
Here \(N\) is the eQTL sample size. The formula shows why a large sample is helpful and why a low-frequency allele can be difficult to assess: either can leave little genotype information. The approximation is less reliable for very rare variants, related individuals, substantial genotype uncertainty, or a model that does not account for important covariates. In those settings, the observed dosage variance and the actual association model matter more than a nominal sample-size calculation.
Worked Example: A Low-Frequency eQTL With Little Information
Suppose a hypothetical study has \(N=120\) individuals, an allele frequency \(f=0.05\), and expression standardized to have residual standard deviation \(\sigma=1\). Under the null, the variance of the genotype-based score is approximately \(N\,2f(1-f)\). Suppose the observed score numerator is \(U=0.40\).
The genotype variance is \(2(0.05)(0.95)=0.095\), so the score variance is \(120(0.095)=11.4\) and its null standard deviation is \(\sqrt{11.4}=3.376\). The standardized score is therefore:
The corresponding two-sided normal-approximation p-value is \(2P(Z\geq 0.1185)\approx 0.9057\), where \(Z\sim N(0,1)\). The score gives essentially no evidence against the null of no eQTL association. This is not evidence that the true eQTL effect is zero: the low allele frequency and small sample provide little information. For scale, the model-based standard error for a per-allele effect is \(1/\sqrt{11.4}=0.2962\) expression standard deviations, which is large. A colocalization result in this setting should not be described as evidence for distinct causal variants merely because the eQTL test is nonsignificant.
Reading Association Strength Alongside the Posterior
An eQTL p-value describes compatibility of an association statistic with a null model; it is not a direct measure of the probability that an eQTL exists. Colocalization instead compares the likelihood of the observed association patterns under its competing hypotheses, combined with the specified prior probabilities. Thus, a weak eQTL can leave the analysis uncertain among hypotheses rather than specifically favoring \(H_3\).
When both traits have meaningful association support, the relative posterior support for sharing versus distinct signals is often summarized by \(PP.H4/(PP.H3+PP.H4)\). This is a conditional comparison within the two hypotheses that posit association for both traits. It must be read alongside \(PP.H0\), \(PP.H1\), and \(PP.H2\); it is not a substitute for checking whether the data support associations in the first place.
Worked Example: A Precisely Measured eQTL
Consider a hypothetical eQTL study with \(N=4{,}000\), allele frequency \(f=0.30\), residual expression standard deviation \(\sigma=1\), and estimated per-allele effect \(\hat{\beta}=0.20\) expression standard deviations. Assume the additive model and other conditions above are reasonable.
The genotype variance is \(2(0.30)(0.70)=0.42\), giving \(N\,2f(1-f)=4{,}000(0.42)=1{,}680\). The approximate standard error is \(1/\sqrt{1{,}680}=0.02440\) expression standard deviations per allele. Hence:
The two-sided normal-approximation p-value is about \(2.4\times10^{-16}\). This is strong evidence of an eQTL association in this hypothetical setting. It is not, on its own, evidence that the eQTL shares a causal variant with the GWAS signal; that conclusion depends on the joint pattern of association across variants and on the colocalization posterior.
For comparison, if the same effect and allele frequency were observed in \(N=120\), the standard error would be \(1/\sqrt{120(0.42)}=0.1409\), and \(z=0.20/0.1409=1.419\), with a two-sided p-value of about 0.156. The effect is unchanged in this comparison, but the smaller sample yields much weaker evidence. A nonsignificant result at that sample size would not be surprising even if the effect were real.
Worked Example: Distinct-Signal Support Versus Insufficient eQTL Evidence
Suppose a hypothetical colocalization analysis reports \(PP.H0=0.05\), \(PP.H1=0.10\), \(PP.H2=0.05\), \(PP.H3=0.70\), and \(PP.H4=0.10\). These probabilities sum to 1. The posterior support that both traits are associated is \(PP.H3+PP.H4=0.80\). Within those two hypotheses, the conditional share support is:
In this hypothetical result, most posterior support lies on both traits being associated with distinct causal variants, rather than sharing one. That interpretation is more informative than saying only that \(PP.H4=0.10\), because the high \(PP.H3\) indicates a competing distinct-signal explanation. It still depends on the assumed priors, variant set, and adequacy of the colocalization model.
Now consider a different hypothetical result: \(PP.H0=0.45\), \(PP.H1=0.05\), \(PP.H2=0.40\), \(PP.H3=0.08\), and \(PP.H4=0.02\). The conditional share support among \(H_3\) and \(H_4\) is \(0.02/(0.08+0.02)=0.20\), but only \(0.10\) of the posterior supports both traits being associated. Much of the posterior instead supports no association or an expression-only association. This is weak evidence for colocalization, but it does not establish that two detected signals are distinct: the data provide little posterior support for two signals to compare.
Diagnostics and Common Failure Modes
Before interpreting a weak eQTL result, check whether the apparent lack of evidence has a plausible measurement or design explanation. The following checks help distinguish limited information from a pattern that genuinely favors distinct signals.
- Check sample size and precision. Report the eQTL effect estimate and standard error, not just whether its p-value crosses a threshold. A wide interval indicates that many effect sizes remain compatible with the data.
- Check allele frequency and genotype quality. Low-frequency alleles provide less genotype variation. Imputation uncertainty or filtering can reduce usable information further; the nominal sample size alone may overstate precision.
- Check the locus and variant coverage. Confirm that the relevant variants are represented and that GWAS and eQTL variants have been harmonized. Poor coverage or allele mismatches can weaken or distort apparent cross-trait agreement.
- Check the posterior pattern, not just \(PP.H4\). High \(PP.H3\) alongside \(PP.H4\) can support distinct signals; high \(PP.H0\), \(PP.H1\), or \(PP.H2\) may instead reflect absent or weak evidence for one or both traits.
- Check model adequacy. The five-hypothesis comparison depends on assumptions about the candidate variants, priors, and causal-signal structure. If the region has multiple signals, an analysis that does not account for them can produce misleading summaries. The earlier material on conditional and fine-mapping-based colocalization addresses that issue.
Sample overlap is another consideration: when GWAS and eQTL estimates share participants, their errors may be correlated. This affects interpretation of joint evidence, so use an analysis appropriate to the study design rather than treating the two estimates as automatically independent.
| Observed pattern | Reasonable interpretation | What not to conclude |
|---|---|---|
| Weak eQTL statistic with a large standard error | The eQTL evidence is imprecise; sharing remains difficult to assess. | That the eQTL effect is zero or that signals are distinct. |
| Strong association support for both traits and high \(PP.H3\) relative to \(PP.H4\) | The fitted model favors distinct variants over a shared variant. | That the result is independent of priors, variant coverage, or model assumptions. |
| Large posterior support on \(H_0\), \(H_1\), or \(H_2\) | One or both association signals may be unsupported in the analyzed data. | That a low \(PP.H4\) alone demonstrates distinct signals. |
A common mistake is to reason, “The eQTL p-value is above 0.05, so the GWAS and expression signals cannot share a variant.” This confuses a failure to reject a no-association null with evidence for a distinct causal variant. The correction is to examine precision and posterior support: distinct-signal evidence requires support for both associations and, within the relevant model, stronger support for \(H_3\) than \(H_4\). If the eQTL is underpowered, the appropriate conclusion is that sharing is unresolved, not that it has been disproved.
Check Your Understanding
Answer each question, then compare with the solutions below.
- In the simple additive model, what happens to the approximate standard error when sample size increases while allele frequency and residual variance stay fixed?
- A study has a nonsignificant eQTL p-value and a very large standard error. Does this alone support \(H_3\) over \(H_4\)? Explain.
- If \(PP.H3=0.60\) and \(PP.H4=0.20\), calculate the conditional share support among these two hypotheses.
- Why can a low-frequency allele lead to weak eQTL evidence even when the number of participants seems substantial?
Solutions. (1) Since \(SE(\hat{\beta})\approx\sigma/\sqrt{N\,2f(1-f)}\), increasing \(N\) decreases the standard error in proportion to \(1/\sqrt{N}\). (2) No. A large standard error means the eQTL estimate is imprecise; the result does not by itself favor distinct variants over a shared one. Examine the full posterior and whether both associations are supported. (3) \(PP.H4/(PP.H3+PP.H4)=0.20/(0.60+0.20)=0.25\), or 25% conditional support for sharing within those two hypotheses. (4) The genotype variance \(2f(1-f)\) is small when \(f\) is near zero, so the study has less genetic information and typically a larger standard error.