Effect-Size Heterogeneity Across Ancestry Groups

Why Compare Genetic Effects Across Ancestry Groups?

A polygenic score may predict an outcome less accurately in one ancestry group than another, but that performance gap does not by itself show that the underlying genetic effects differ. Differences in allele frequencies, linkage disequilibrium, measurement, and prediction can all affect portability. Here the question is narrower: do comparable estimates of a genetic effect differ across groups beyond what sampling variation would plausibly produce?

The comparison is meaningful only when the estimates target the same quantity. For a variant, this generally means the same allele contrast, the same phenotype definition and scale, and a sufficiently comparable association model. A difference in marginal association can arise because a variant tags a causal variant differently across groups; that is not necessarily a difference in the causal effect. The earlier material on portability and LD mismatch is relevant to that distinction.

Set Up a Comparable Comparison

Let \(\hat{\beta}_{gj}\) denote the estimated effect for variant \(j\) in ancestry group \(g\), and let \(SE_{gj}\) be its standard error. For example, \(\beta_{gj}\) might be the change in a quantitative phenotype, in its stated units, per additional copy of the effect allele. For a binary phenotype, it might instead be an effect on the log-odds scale. Those are different estimands, so effects from different scales should not be compared as though they were interchangeable.

Before testing heterogeneity, verify that allele coding is aligned, phenotype units and transformations are consistent, and the association models target comparable effects. In particular, a per-allele effect on a raw phenotype scale need not equal a per-allele effect after group-specific standardization. Also check whether the estimates come from independent samples. If samples overlap, their estimates may be correlated and the covariance must be included in uncertainty calculations.

For two groups, the estimated difference is \(\hat{\Delta}_j=\hat{\beta}_{1j}-\hat{\beta}_{2j}\). If the estimates are independent and approximately normal, its standard error is the square root of the sum of their variances. If they are correlated, subtract twice their covariance.

Under the null hypothesis \(H_0:\beta_{1j}=\beta_{2j}\), a Wald statistic is \(z=\hat{\Delta}_j/SE(\hat{\Delta}_j)\). With a normal approximation, the two-sided p-value is \(2P(Z\geq |z|)\), where \(Z\sim N(0,1)\). This approximation is most credible when the estimates are sufficiently precise and the association model and standard errors are appropriate. A small p-value is evidence against equal effects under the model; it is not the probability that the null hypothesis is true.

Assessing Heterogeneity Across Several Groups

For \(G\) groups, a common-effect model assumes each group estimates one shared effect, with differences among estimates arising from sampling error. An inverse-variance weighted estimate under that model is \(\hat{\beta}_{\mathrm{FE}}=\sum_g w_g\hat{\beta}_g/\sum_g w_g\), where \(w_g=1/SE_g^2\). Cochran’s \(Q\) measures the weighted squared departures from that common estimate.

If estimates are independent, approximately normal, their standard errors are valid, and a common underlying effect is true, then \(Q\) is approximately distributed as a chi-squared random variable with \(G-1\) degrees of freedom. A large \(Q\) relative to that reference distribution is evidence that the estimates vary more than the common-effect sampling model predicts. The test can have low power when there are few groups or imprecise estimates, so a non-significant result is not proof of identical effects.

For two independent groups, \(Q\) equals the squared difference divided by its variance. This shows why the two-group Wald test and the two-group heterogeneity test give the same evidence when their assumptions match.

To see the identity, write \(w_g=1/SE_g^2\). With two groups, the weighted mean lies between the estimates, and its distances from them are \(\hat{\beta}_1-\hat{\beta}_{\mathrm{FE}}=w_2(\hat{\beta}_1-\hat{\beta}_2)/(w_1+w_2)\) and \(\hat{\beta}_2-\hat{\beta}_{\mathrm{FE}}=-w_1(\hat{\beta}_1-\hat{\beta}_2)/(w_1+w_2)\). Substituting these into \(Q\) gives \(Q=(\hat{\beta}_1-\hat{\beta}_2)^2w_1w_2/(w_1+w_2)\). Since \(w_1w_2/(w_1+w_2)=1/(SE_1^2+SE_2^2)\), the stated identity follows.

Quantifying Variation and Interpreting Its Source

A heterogeneity test answers whether observed differences are unusual under a common-effect model; it does not express the size of between-group variation in the original units. A random-effects model instead represents group effects as \(\beta_g=\mu+u_g\), where \(u_g\) has mean zero and variance \(\tau^2\). Here \(\tau^2\) is between-group effect variance, in squared effect units, and \(\tau\) is its standard deviation in effect units.

One simple method-of-moments estimate is the DerSimonian–Laird estimate: \(\hat{\tau}^2=\max\{0,(Q-(G-1))/C\}\), where \(C=\sum_g w_g-\sum_g w_g^2/\sum_g w_g\). It is an estimate, not a test result or a biological explanation. With few groups it can be unstable, and its value depends on the groups and effect scale under analysis. A confidence interval for \(\tau^2\) or \(\tau\) is preferable when the aim is to quantify uncertainty in the magnitude of heterogeneity.

Even strong statistical evidence for heterogeneity leaves several explanations open. Ancestry-correlated environmental exposures, imperfect control of population structure, phenotype ascertainment, and differences in measurement can bias group-specific estimates. For example, a genetic association can partly reflect a non-genetic exposure if that exposure is correlated with genotype within a group. Differences in allele frequency or LD can also change the marginal association of a tag variant even when the causal variant’s effect is shared. Thus, heterogeneity is a pattern in estimates; it does not automatically identify a difference in causal biology.

Diagnostics, Alternatives, and Limitations

A useful analysis examines estimates and uncertainty before focusing on a single p-value. Plot group-specific estimates with confidence intervals, verify allele alignment and units, check sample overlap and model specification, and investigate whether a small number of groups or extreme estimates dominate \(Q\). For genome-wide comparisons, account for the fact that many variants are tested; nominal evidence at one variant does not automatically control the overall false-positive rate. Related variants also are not independent evidence, so a collection of variant-level tests cannot be treated as independent without justification.

The common-effect approach is useful when the goal is to test consistency under one shared effect, but it can be misleading if effects truly vary. A random-effects model allows variation and estimates its spread, but does not solve confounding, and its estimate may be imprecise with few groups. Pairwise difference tests are direct and easy to interpret for two groups, but multiple pairwise comparisons increase the testing burden. These approaches answer related but distinct questions; none alone identifies the cause of a difference.

When estimates overlap in samples, use their covariance rather than the independent-sample formula. When allele coding differs, align alleles and signs before comparison. If the phenotype scale differs, transform or re-estimate effects on a common, scientifically meaningful scale rather than comparing raw coefficients mechanically. Finally, ancestry groups are broad labels for heterogeneous populations, not discrete biological mechanisms. A group-specific estimate is conditional on the sampled people, methods, and context; generalization beyond them requires additional evidence.