Why Sample Overlap Matters
A GWAS association estimate and an eQTL association estimate can be correlated even when neither association is caused by the other. If some people contribute to both analyses, their genotype and their phenotype residuals contribute to both estimates. When the two phenotypes are correlated after accounting for genotype and other covariates, random sampling fluctuations can move the two estimates together or in opposite directions. A colocalization analysis that treats the estimates as independent may then misstate how much evidence supports a shared causal variant.
This is a question about dependence between association estimates, not about linkage disequilibrium (LD) among variants. LD concerns how genotypes at different variants co-vary; overlap concerns how estimates for two traits co-vary because some individuals are shared. Both can matter in a locus analysis, but they are distinct sources of dependence.
A Working Model and an Overlap Approximation
Consider association analyses for the same allele-coded variant, with genotype \(G\), in samples of sizes \(n_Y\) and \(n_X\). Let \(n_O\) be the number of participants included in both analyses. For a simple derivation, assume the genotype has variance \(v_G\) in both samples; the genotype is independent of the phenotype residuals; and the residuals have standard deviations \(\sigma_Y\) and \(\sigma_X\), with correlation \(\rho_e\), among overlapping participants. Assume observations are independent across individuals, the overlap is correctly identified, and the association models use compatible covariates. After residualizing for those covariates, write the two models as
Here \(\varepsilon_{Yi}\) and \(\varepsilon_{Xi}\) are residuals after accounting for the modeled genotype and covariates. Under these assumptions, the ordinary least-squares estimates have approximate standard errors \(\sigma_Y/\sqrt{n_Yv_G}\) and \(\sigma_X/\sqrt{n_Xv_G}\). Their sampling covariance is approximately
The shared individuals supply the terms that contribute to both estimate numerators. For each overlapping person, the expected product of the genotype-weighted residuals is approximately \(v_G\sigma_{YX}\); non-overlapping people contribute no cross-product between the two numerators. Dividing the sum of these \(n_O\) expected cross-products by the two approximate regression denominators, \(n_Yv_G\) and \(n_Xv_G\), gives the covariance above. Dividing that covariance by the product of the two standard errors gives the useful Z-statistic approximation
The approximation uses the same allele orientation in both analyses. Reversing the effect allele in only one analysis reverses the sign of its estimates and Z-statistics. With covariate adjustment, unequal genotype variances, or meta-analysis across cohorts, the simple count-based expression may be inadequate: the relevant overlap and covariance are effectively weighted by the score contributions and analysis design. The formula is a diagnostic starting point, not a universal correction.
Worked Example: Approximating Correlation Between Z-Statistics
Suppose a hypothetical GWAS has \(n_Y=10{,}000\) participants and a hypothetical eQTL analysis has \(n_X=5{,}000\), with \(n_O=3{,}000\) participants in both. Suppose the residual correlation between the two phenotypes in the overlap is estimated to be \(\rho_e=0.40\). Assume the model conditions above are reasonable.
The overlap fraction in the correlation formula is
Therefore,
The estimated sampling correlation is about \(0.170\), not \(0.40\): the residual correlation is scaled by the amount of overlap relative to the two sample sizes. This is a moderate dependence under the stated approximation, not proof that the observed associations share a causal variant. If the residual correlation were near zero, the same overlap counts alone would imply little correlation by this formula.
When Colocalization Assumptions Are Affected
Standard Bayesian colocalization combines evidence from two association analyses to compare hypotheses such as no association, association with one trait only, distinct causal variants, or a shared causal variant. In common implementations, the likelihood contribution from the GWAS and the eQTL analysis is factorized as if the two sets of association evidence were independent, conditional on the candidate causal configuration. Sample overlap coupled with correlated residual phenotypes violates that independence approximation: at a given variant, the two association estimates can share sampling noise.
Positive estimate correlation can make evidence for two traits appear concordant more often than an independent-trait model predicts; negative correlation can make it appear less concordant. Neither direction has a guaranteed effect on a particular posterior probability. The outcome depends on the observed association patterns, LD, priors, and model details. Thus overlap is a reason to assess calibration and sensitivity, not a reason to apply a fixed adjustment to a reported posterior probability.
The distinction matters especially when the evidence for a shared variant is modest and the two traits have appreciable residual correlation. Correlated noise at a variant can contribute to similar-looking associations. Conversely, dependence does not mean a shared causal signal is absent: it means that a method relying on independence may overstate or understate the evidential strength. The single-causal-variant or multiple-signal assumptions are separate modeling choices; overlap does not by itself resolve them.
Worked Example: Covariance of Two Effect Estimates
Suppose hypothetical analyses have \(n_Y=8{,}000\), \(n_X=6{,}000\), and \(n_O=2{,}000\). Let the genotype variance be \(v_G=0.50\), the GWAS residual standard deviation be \(\sigma_Y=2.0\) outcome units, and the eQTL residual standard deviation be \(\sigma_X=1.0\) expression units. Suppose the residual correlation is \(\rho_e=-0.30\). Assume the same genotype scale and compatible covariate adjustment in both analyses.
First, the residual covariance is \(\sigma_{YX}=\rho_e\sigma_Y\sigma_X=(-0.30)(2.0)(1.0)=-0.60\) outcome-expression units. The approximate covariance of the effect estimates is
The standard errors under the same simple model are
Their product is approximately \(0.0005774\), so the implied correlation is \(-0.0000500/0.0005774\approx-0.0866\). This agrees with the Z-statistic formula: \((-0.30)\times2{,}000/\sqrt{8{,}000\times6{,}000}\approx-0.0866\). The negative covariance means that, under these assumptions, sampling errors tend to move the two estimates in opposite directions. The effect estimates have different units, so their covariance also has outcome-expression units; their correlation is unitless.
Practical Checks and Alternatives
A useful assessment combines study-design information with the association results. No one check establishes that the independence assumption is harmless, and a small apparent overlap count does not guarantee negligible dependence if residual correlation is strong.
Check cohort membership, recruitment sites, controls reused across analyses, and whether each GWAS or eQTL meta-analysis contains the same component studies. Count overlap at the participant or cohort level when possible; do not infer it only from total sample sizes.
Among overlapping individuals, consider the correlation between the analyzed phenotypes after accounting for relevant covariates and genotype. A raw phenotype correlation is not automatically the residual correlation required by the approximation.
Compare ancestry, variant coverage, allele coding, covariate adjustment, and the way estimates were combined across cohorts. Inconsistent designs or strongly unequal weights make a simple overlap-count calculation less reliable.
If participant-level data are available, estimate the joint sampling covariance using a suitable joint model or a robust covariance estimator. With summary data, use a method that explicitly represents cross-trait covariance if available, or report sensitivity to plausible dependence and treat results cautiously when the assumption cannot be checked.
A null-marker check can provide supporting evidence: after excluding the locus of interest and controlling the analysis of correlated markers, compare GWAS and eQTL Z-statistics at markers expected not to carry effects for either trait. But LD, widespread polygenicity, population structure, and selection of markers can all induce or distort correlation. Such a check is not a direct estimate of overlap-induced covariance unless those other sources are handled. Study records and a model aligned to the design are usually more informative.
| Approach | What it addresses | Main limitation |
|---|---|---|
| Independent-trait colocalization likelihood | Combines two association evidence sets under an independence approximation. | Can be miscalibrated when overlapping samples produce non-negligible cross-trait covariance. |
| Joint or covariance-aware analysis | Represents dependence between the trait association estimates in the likelihood or test. | Requires suitable covariance information and assumptions matching the cohort design. |
| Sensitivity analysis | Shows whether conclusions change over plausible dependence values or analysis choices. | Does not identify the true dependence when it is unknown, and cannot substitute for missing design information. |
Common Pitfalls and Limitations
- “The same number of participants means the estimates are highly correlated.” Similar sample sizes do not establish overlap. Correlation depends on who is shared and on residual covariance, as well as the analysis design.
- “Any overlap invalidates colocalization.” Not necessarily. If the overlap is small or residual correlation is negligible, induced dependence may be weak. The concern is whether the independence approximation is consequential for the data and conclusion.
- “Positive phenotype correlation always raises the shared-variant posterior.” It can increase apparent concordance, but posterior changes depend on all observed evidence and model assumptions. There is no universal direction or scalar correction.
- “Cross-trait estimate correlation is just LD.” It is not. LD links variant estimates within a trait analysis; sample overlap can correlate the two traits’ estimates at the same variant. Both may be present simultaneously.
- “A posterior probability solves the dependence problem.” A posterior is conditional on its likelihood and prior assumptions. If the likelihood treats dependent evidence as independent, a precise-looking posterior does not repair that misspecification.
The simple formulas also rely on an approximate linear model and comparable genotype and covariate treatment. Meta-analysis weights, related participants, heterogeneous residual variances, imputation differences, or sample selection can change the covariance. When those features are important, use design-specific covariance estimates or methods that model the joint data rather than treating the count-based approximation as exact.
Worked Example: A Sensitivity Check for a Difference in Z-Statistics
Suppose a hypothetical locus has \(Z_Y=3.1\) and \(Z_X=2.7\). Their difference is \(0.4\). For illustration, suppose the overlap calculation suggests a cross-trait Z-statistic correlation of \(0.17\). Under a bivariate standard normal approximation, the variance of the difference is
If the Z-statistics were independent, the standard deviation of the difference would be \(\sqrt{2}=1.414\), giving \(0.4/1.414=0.283\). With \(\rho_Z=0.17\), it would instead be \(\sqrt{2(1-0.17)}=\sqrt{1.66}=1.288\), giving \(0.4/1.288=0.311\). This calculation is not a colocalization test and does not determine whether the locus shares a causal variant. It illustrates that dependence changes joint uncertainty even for a simple comparison, so an independence-based interpretation should not be assumed without assessment.
Check Your Understanding
Answer each question, then compare with the solutions below.
- A GWAS has 12,000 participants, an eQTL analysis has 8,000, and 4,000 are shared. If the residual phenotype correlation is \(0.25\), what is the approximate correlation between their Z-statistics under the simple model?
- If the samples overlap but the two phenotype residuals are uncorrelated under the model, what does the overlap approximation imply about cross-trait estimate correlation? Name one reason the real analyses might still depart from this result.
- Why can positive residual correlation complicate evidence for a shared variant even when the marginal association estimates from each study are individually unbiased?
- Does a cross-trait correlation between association estimates mean the two variants are in LD? Explain the distinction.
Solutions. (1) The scale factor is \(4{,}000/\sqrt{12{,}000\times8{,}000}=4{,}000/9{,}798.0\approx0.4082\). Thus the approximate Z correlation is \(0.4082\times0.25=0.1021\). (2) The formula gives zero because \((n_O/\sqrt{n_Yn_X})\rho_e=0\). Real estimates may still be dependent because of related participants, population structure, incompatible covariate adjustment, or other design features outside the simple model. (3) Unbiasedness concerns the expectation of each estimate separately; it does not imply that their sampling errors are independent. Correlated errors can make the joint evidence differ from the product likelihood used under independence. (4) No. LD is correlation between genotypes at different variants; the cross-trait estimate correlation here is dependence between two traits’ estimates, potentially because the same people contribute to both analyses.