Interpreting Heterogeneity Diagnostics in Fine-Mapping

Why Examine Effect Heterogeneity?

Trans-ancestry fine-mapping uses information from multiple ancestry groups, but a combined analysis can rely on different assumptions about genetic effects. A shared-effect model assumes that a variant has the same effect on the chosen outcome scale in each group. An ancestry-stratified analysis makes it possible to inspect that assumption rather than treating it as automatic.

Ancestry-specific estimates will rarely be identical. Even if the underlying effects are equal, sampling variation produces different estimates, especially when standard errors differ. Heterogeneity diagnostics help assess whether the observed differences are larger than expected under a shared-effect model. They do not, by themselves, establish why effects differ or identify which variant is causal.

Check Comparability Before Calculating

A heterogeneity statistic is meaningful only if its inputs can be compared. Each estimate should refer to the same variant and the same effect allele, with alleles aligned so that a positive estimate has the same direction in every group. The outcome and effect scale must also match. For example, log-odds coefficients should not be compared directly with risk differences as though they were measurements in the same units.

For the usual inverse-variance calculation, assume the ancestry-specific estimates are approximately unbiased and normally distributed around their true effects, and that their standard errors are valid. The basic Cochran \(Q\) calculation also assumes the estimates are independent across groups. If participants overlap or estimates have other dependence, their covariance must be included; treating dependent estimates as independent can miscalibrate the test. These diagnostics assess effect estimates, not differences in linkage disequilibrium (LD) patterns, which can change association patterns even when the causal effect is shared.

Inverse-Variance Summary and Cochran’s Q

Under the shared-effect model, each estimate receives weight \(w_k=1/SE_k^2\). An estimate with a smaller standard error receives more weight because it provides more precise information. The inverse-variance pooled estimate and its standard error are

The subscript \(\mathrm{FE}\) denotes a fixed-effect calculation: it summarizes estimates under the assumption of one common effect. Cochran’s heterogeneity statistic measures the weighted dispersion of the ancestry-specific estimates around that pooled value:

When the shared-effect model and the stated sampling assumptions hold, \(Q\) is approximately distributed as chi-square with \(df=K-1\). A large \(Q\), relative to that reference distribution, is evidence against a common effect on the analyzed scale. The p-value is the probability, assuming the shared-effect model and its assumptions, of obtaining a \(Q\) at least as large as the observed one. It is not the probability that the shared-effect model is true.

Confidence Intervals and the Heterogeneity Test

A confidence interval for one ancestry-specific effect describes uncertainty about that group’s effect, under the model used to construct the interval. It is useful to inspect the intervals, but visual overlap is not a formal heterogeneity test. Conversely, the absence of overlap is not itself the definition of heterogeneity. The test of a difference should account for the standard errors of the estimates and, where relevant, their covariance.

For two independent estimates, the estimated difference is \(\hat{\beta}_1-\hat{\beta}_2\), and its standard error is \(\sqrt{SE_1^2+SE_2^2}\). This makes the role of precision explicit: a numerical gap can be unremarkable when its uncertainty is large, while a smaller gap can be compelling when estimated precisely.

Low Precision Can Hide Heterogeneity

A non-small heterogeneity p-value does not demonstrate that effects are equal. When one ancestry group has a large standard error, its estimate can differ substantially from the others without making \(Q\) large. This is especially important with few groups: the test may have limited power to detect moderate differences, and the chi-square reference distribution does not turn a weak diagnostic into evidence of equivalence.

What a Heterogeneity Signal Does—and Does Not—Mean

When \(Q\) is large and its p-value is small, first verify that the estimates are genuinely comparable. Check allele alignment, variant identity, phenotype definition, effect scale, imputation and measurement quality, and the standard-error calculations. Also check whether the independence assumption is reasonable. An apparent difference caused by a mismatched allele or scale is not biological effect heterogeneity.

If those checks are satisfactory, the result is evidence that the common-effect model may be inadequate for these estimates. It is not a test of whether the same causal variant is present across ancestries, nor does it identify a biological mechanism. A shared causal variant can have different estimated effects, and a heterogeneity test alone cannot distinguish genuine effect modification from residual bias or other model misspecification. Likewise, a non-significant test does not establish effect portability.

The fixed-effect pooled estimate is most defensible when a common effect is a reasonable working assumption. If that assumption is doubtful, do not present the pooled value as though it necessarily describes every group. Report the ancestry-stratified estimates and intervals, the heterogeneity statistic and its degrees of freedom, and the assumptions behind the comparison. A p-value should be read alongside the size and precision of the differences, not as a stand-alone measure of their importance.