Shared Causal Variants With Ancestry-Specific LD

Why Shared Causality Need Not Mean Shared Association Patterns

A variant can contribute to a trait in more than one ancestry group even when its nearby variants have different linkage disequilibrium (LD) relationships across groups. Because association at a noncausal variant can arise through LD with a causal variant, the marginal association pattern may differ across groups. A strong association at a proxy in one group can be much weaker in another, without requiring a different causal variant.

The key modeling distinction is between which variants are causal and what marginal associations are observed. A shared causal-variant model constrains the causal support across ancestries. It does not, by itself, require identical marginal associations. The LD matrix used to translate causal effects into marginal associations must be specific to each ancestry group.

A Shared-Support Model With Group-Specific LD

Suppose summary statistics are available for \(K\) ancestry groups and the same \(p\) variants have been harmonized so that alleles and effect directions agree across groups. For group \(a\), let \(X_a\) be the matrix of standardized genotypes, with one column per variant. Let \(R_a\) be its LD matrix, so that \(R_a=n_a^{-1}X_a^\top X_a\) in a sample of size \(n_a\). Each \(R_a\) is a correlation matrix, and different groups can have different matrices.

For a simple common-effect specification, let the standardized phenotype in each group follow a linear model with effect vector \(\gamma\). The vector is sparse: entries outside the shared set \(C\) are zero. The marginal association vector, denoted \(\theta_a\), is the vector of regression coefficients from regressing the phenotype on each variant separately within group \(a\). Under this model, the expected marginal associations are determined by that group’s LD matrix.

Thus, even when \(\gamma\) is common, \(\theta_a\) need not be common. The \(j\)th marginal association is the sum of the causal effects weighted by the correlations between variant \(j\) and the causal variants in that group. A variant outside \(C\) can therefore have a nonzero marginal association. It is an LD proxy, not necessarily a causal variant.

For large samples and a standardized linear-trait model with residual variance treated as one, the joint summary-statistic likelihood is commonly approximated by \(z_a\sim N(\sqrt{n_a}R_a\gamma,R_a)\), where \(z_a\) is the vector of association \(Z\)-statistics. This approximation uses the ancestry-specific LD matrix both in the mean and in the covariance. It is not a claim that observed \(Z\)-statistics equal their expectations. Differences in sample size also change the expected \(Z\)-statistic scale, so \(Z\)-vectors should not be compared as though their entries were marginal effect estimates.

Here \(\phi_p(\,\cdot\,;\mu,\Sigma)\) is the \(p\)-dimensional normal density with mean \(\mu\) and covariance matrix \(\Sigma\), and \(\dot{\sim}\) indicates an approximation. The product likelihood assumes that the ancestry-specific samples are independent. If samples overlap, their summary statistics can be correlated, and this product is not the appropriate joint likelihood without accounting for that dependence.

In a fine-mapping analysis, the shared-support restriction can be represented by requiring \(\gamma_j=0\) or \(\gamma_j\ne 0\) consistently across groups. The group-specific matrices \(R_a\) are not replaced by a pooled matrix: doing so would generally give the wrong expected marginal associations for at least some groups. This formulation extends the joint summary-statistic likelihood by tying causal support across groups while allowing LD to vary.

Why the Marginal Associations Equal \(R_a\gamma\)

The relationship follows directly from the regression model. Within group \(a\), write the standardized phenotype as \(y_a=X_a\gamma+\varepsilon_a\), where the error has mean zero and is uncorrelated with the genotypes. The separate regression coefficient for variant \(j\) is the sample covariance of that variant with the phenotype, since the genotype column has sample variance one. Substituting the model shows that this covariance includes contributions from every causal variant, weighted by the corresponding genotype correlation.

The error term has expectation zero under the stated assumptions. Therefore \(E(\widehat{\theta}_{aj})=\sum_k(R_a)_{jk}\gamma_k\), the \(j\)th entry of \(R_a\gamma\). If only variants in \(C\) have nonzero effects, this reduces to \(\theta_{aj}=\sum_{k\in C}(R_a)_{jk}\gamma_k\). The formula explains why a noncausal variant can show association and why the strength of that association can differ by ancestry.

Shared Causality Is Not an Identical-Pattern Constraint

An identical-pattern model places a different restriction: it requires the marginal association vectors to be equal across groups, perhaps after choosing a common effect scale. That is not implied by shared causal support. In fact, when \(R_A\ne R_B\), the common-effect shared-support model generally predicts \(\theta_A\ne\theta_B\). Conversely, forcing the marginal vectors to agree can require different underlying causal effect vectors or different causal support.

The distinction is easiest to see using marginal effect estimates rather than \(Z\)-statistics, because the latter also depend on sample size. In the example above, an identical marginal-effect pattern would require both groups to have \((0.20,0.18)^\top\). Under group B’s LD matrix, the causal-effect vector needed to produce that pattern is \(R_B^{-1}(0.20,0.18)^\top\), not the common vector \((0.20,0)^\top\).

More Than One Shared Causal Variant

The same logic applies when several variants are causal. Each marginal association combines the effects of all causal variants, with weights given by the corresponding row of the ancestry-specific LD matrix. Contributions can reinforce or oppose one another. Consequently, a proxy can have a small marginal effect even when it is correlated with causal variants, if the weighted contributions partly cancel.

Assumptions, Interpretation, and a Common Pitfall

The common-effect version is useful for isolating the role of LD, but its assumptions should be explicit. It presumes aligned alleles and variants across groups, appropriate ancestry-specific LD estimates, a compatible phenotype and effect scale, and the common causal-effect vector \(\gamma\). Its shared-support restriction says that the nonzero entries of \(\gamma\) correspond to the same causal variants in every group. This tutorial’s algebra does not establish that these assumptions hold in a particular data set, nor does an observed association at a variant alone establish that it is causal.

Causal sharing and effect equality are distinct claims. The shared-support idea can be retained while allowing effects at shared causal variants to differ by group, but that is a separate model extension. The examples here hold effects common so that the consequences of ancestry-specific LD can be seen directly. Therefore, a difference in marginal patterns should not automatically be interpreted as evidence against shared causality, and a match in marginal patterns should not be taken as proof of it.

Common pitfall: “The causal variant is shared, so the association statistics should be the same in every ancestry.” This reasoning confuses causal support with marginal association. Marginal effects depend on \(R_a\gamma\), and \(R_a\) can differ across groups; \(Z\)-statistics also depend on sample size. The correction is to state which quantity is assumed shared—causal support, causal effects, marginal effects, or some combination—and use each group’s LD matrix when deriving its expected marginal pattern.