Cross-Cohort Heterogeneity In Genetic Correlations

Why Compare Genetic Correlations Across Cohorts?

A genetic correlation between two traits describes the alignment of their additive genetic influences in a specified population. Comparing that quantity across cohorts can help assess whether the relationship is similar across study settings or whether the data are inconsistent with a shared value. But two estimates that look different are not necessarily evidence of different genetic architecture: each estimate has sampling uncertainty, and each cohort must be estimating a sufficiently comparable quantity.

The comparison therefore has two parts. First, quantify uncertainty in the difference between estimates, allowing for dependence if the cohorts share participants. Second, consider whether differences in trait definitions, scales, or measurement could make the estimates target different quantities. A statistical test addresses compatibility with a specified common-value model; by itself, it does not identify why estimates differ.

Estimands and Cross-Cohort Contrasts

The covariance and correlation answer related but different questions. Covariance retains the units of the traits and can change when either trait’s scale changes. Correlation standardizes covariance by genetic standard deviations, so it is unitless; it can nevertheless differ across cohorts when genetic variances or the relationship between traits differ. A comparison of correlations is not interchangeable with a comparison of covariances.

A contrast is meaningful only when the cohorts’ trait definitions and analysis targets are sufficiently comparable. Harmonizing variant and trait labels is important, but does not by itself ensure that a trait was measured in the same way or represents the same construct. Differences in scales, instruments, thresholds, or ascertainment can complicate interpretation. In particular, a difference in an estimate is not automatically evidence that underlying biological effects differ.

Uncertainty in a Difference

Suppose \(\hat\theta_1\) and \(\hat\theta_2\) estimate the same type of quantity in two cohorts, where \(\theta\) may be a genetic covariance or a genetic correlation. If their standard errors are \(SE_1\) and \(SE_2\), and their sampling covariance is \(\operatorname{Cov}(\hat\theta_1,\hat\theta_2)\), then the variance of their difference follows from the variance rule for a linear combination:

For independent cohorts, the covariance term is zero. For overlapping samples or estimates that otherwise share data, it need not be zero. Treating dependent estimates as independent can therefore misstate the uncertainty of the contrast.

When the contrast has an approximately normal sampling distribution, a two-sided Wald test of \(H_0:\theta_1-\theta_2=0\) uses \(z=(\hat\theta_1-\hat\theta_2)/SE(\hat\theta_1-\hat\theta_2)\). The approximate two-sided \(p\)-value is \(2P(Z\geq |z|)\), where \(Z\sim N(0,1)\). An approximate 95% confidence interval is the estimated contrast plus or minus 1.96 standard errors. These procedures require a suitable normal approximation; for genetic correlations, that approximation can be poor near the parameter boundaries or when estimates are unstable. In such settings, use an inference method that reflects the estimator’s constraints and sampling distribution, rather than assuming every estimate lies within the parameter bounds.

Comparing Genetic Covariances

The same contrast method applies to genetic covariances, but their units and scale deserve attention. If the two traits are measured in different units or their scales have been transformed differently across cohorts, the covariance estimates may not be directly comparable. Before testing a covariance contrast, establish that the traits have compatible definitions and scales, or explicitly define the scale on which the comparison is intended.

Overlapping Samples and Joint Uncertainty

When cohort estimates share participants, their sampling errors may be correlated. A participant can contribute information to both trait analyses, so the estimates need not fluctuate independently across repeated samples. The relevant quantity is the covariance of the two estimates, not merely whether the studies have any overlap. It should be estimated using a method appropriate to the data and analysis; it cannot generally be recovered from the two standard errors alone.

More Than Two Cohorts

With \(K\) cohorts, a global test can assess whether the estimates are compatible with a common value. Let \(\hat{\boldsymbol\theta}\) be the vector of \(K\) estimates and \(V\) their sampling covariance matrix. Under an approximate multivariate normal model, compare the estimates with a common-value fit. If the estimates are independent with standard errors \(SE_k\), the inverse-variance weighted common estimate and heterogeneity statistic are:

Under the common-value null and the stated approximation, \(Q\) is compared with a chi-squared distribution with \(K-1\) degrees of freedom. For correlated estimates, the covariance matrix \(V\) must be used in the corresponding generalized least-squares comparison; the independent-estimate formula is not appropriate. A global test indicates whether the estimates are collectively inconsistent with a common value under the model. It does not reveal which cohort differs or explain the source of any discrepancy. Follow-up contrasts should account for their uncertainty and, when many are tested, the multiplicity of comparisons.

Deriving the Contrast Standard Error

The uncertainty formula follows directly from the variance of a difference. For random estimates \(A\) and \(B\), the variance of a linear combination is \(\operatorname{Var}(aA+bB)=a^2\operatorname{Var}(A)+b^2\operatorname{Var}(B)+2ab\operatorname{Cov}(A,B)\). Set \(a=1\) and \(b=-1\). Then:

Taking the square root gives the standard error of the difference. Independence is one sufficient condition for the covariance term to be zero, but it should not be assumed when participants or other sources of sampling information are shared.

Interpreting Heterogeneity Carefully

A statistically significant contrast is evidence that the cohort-specific estimands differ more than expected from sampling variation under the test assumptions. It does not, on its own, establish a biological mechanism. Differences may reflect genetic architecture, population-specific linkage disequilibrium or allele frequencies, study ascertainment, or non-comparable trait definitions and measurement. These possibilities require substantive evaluation rather than inference from a \(p\)-value alone.

Conversely, a non-significant contrast does not prove equality. Wide confidence intervals may be compatible with both small and substantial differences. When the scientific aim is to support practical similarity, define an acceptable difference in advance and use an equivalence analysis designed to assess whether the interval is sufficiently contained within that range. Do not replace that question with a failure to reject a zero-difference null.

Common Mistakes

• Comparing point estimates without their uncertainty. A visible gap is not enough to establish heterogeneity. Calculate a contrast and its standard error, confidence interval, or test statistic.
• Assuming estimates are independent because cohorts have different names. Shared participants can induce dependence. Use the covariance of the estimates when it is nonzero.
• Treating every estimated genetic correlation as bounded. The parameter is bounded, but an unconstrained estimate can fall outside \([-1,1]\). Check how the estimate was obtained and whether the inference method respects the parameter space.
• Calling a non-significant result proof of equality. It means the analysis did not reject the specified null at its chosen significance level; it does not show that the cohorts have the same genetic correlation.
• Interpreting heterogeneity as necessarily biological. A test of a difference does not distinguish genetic effects from differences in trait definition, scale, measurement, or study design.