Worked Example of Refining a Locus Across Ancestries

Why a locus must be read across ancestries

At a locus with several correlated variants, the strongest marginal association is not necessarily the causal variant. A variant can have a large association statistic because it is correlated with the causal variant, and the correlations can differ across ancestry groups. Comparing those patterns can help separate variants that look similar in one group. The comparison is useful only when the summary statistics, variant alleles, and linkage disequilibrium (LD) information are aligned and interpreted under a clearly stated model.

This tutorial follows a hypothetical locus in two ancestry groups. It moves from harmonized marginal statistics to conditional analysis and then to posterior probabilities and credible sets. The goal is to show how each stage changes the resolution of the evidence—not to treat a small credible set as proof of causality.

Set up a comparable two-ancestry analysis

Suppose the locus contains three variants, V1, V2, and V3. Before comparing results, the effect allele must refer to the same allele for each variant in both ancestry groups. For example, if the reported effect allele at V1 is A in one group but T in the other, first establish whether those are opposite alleles at the same site and, if so, reverse the allele direction consistently. A sign difference caused by allele coding is not evidence of biological effect heterogeneity.

The same variants must also be represented in both sets of summary statistics, with matching variant identities and allele frequencies checked where available. Each ancestry group needs an LD matrix estimated from an appropriate reference or study sample. The entries of an LD matrix are correlations between the variants' standardized genotypes; they are not association effects. Using the wrong ancestry's LD can distort both conditional statistics and posterior rankings.

For the worked analyses, assume that variant identities and alleles have been harmonized, the listed variants are available in both groups, and each LD matrix is a valid ancestry-matched estimate. We also assume that the groups' samples are independent. When comparing credible sets below, the model allows ancestry-specific effect sizes but assumes one shared causal variant among the three candidates. This is a model assumption, not a conclusion established by the data.

Conditional analysis uses ancestry-specific LD

A simple conditional calculation illustrates why the same marginal results can imply different resolution in different ancestry groups. For two standardized variant scores with correlation \(r_{12}\), the conditional score for variant 2 given variant 1 is

The numerator removes the component of variant 2's score predicted by its LD with variant 1. The denominator standardizes the residual score. This expression follows from the bivariate score model in which the two marginal scores have unit variances and covariance \(r_{12}\). It is a two-variant conditional score, not a substitute for conditioning on all relevant variants when more are included in a locus model.

The conditional result for V1 given V2 shows how the groups provide different information about which variant better explains the association. The corresponding calculation is \(Z_{1\mid 2}=(Z_1-r_{12}Z_2)/\sqrt{1-r_{12}^2}\).

From conditional signals to posterior probabilities

Conditional analysis asks whether an association remains after accounting for selected variants. Bayesian fine-mapping instead compares candidate causal configurations under an explicit likelihood and prior. For a single-causal-variant model, let \(BF_j\) denote the Bayes factor for variant \(j\) relative to the null model, and let \(\pi_j\) be its prior probability of being causal. The posterior probability for variant \(j\), conditional on the model and candidate set, is

If the two ancestry samples are independent and the model specifies one shared causal variant, the joint evidence for a candidate can be formed from the ancestry-specific likelihoods. In a simple shared-variant comparison, the joint Bayes factor is the product of the ancestry-specific Bayes factors. The model must still specify how effects are handled; here, effect sizes may differ across ancestries even though the causal-variant identity is shared. These posterior probabilities are conditional on the candidate variants, the likelihood, the prior, and those model assumptions.

A credible set is constructed by ranking variants from largest to smallest posterior probability and adding them until their cumulative posterior probability reaches the target. The set is not a frequentist confidence interval, and its nominal coverage does not guarantee that the true causal variant is included if the candidate list or model is inadequate.

How LD differences change resolution

The reason ancestry comparison can improve fine-mapping is that association statistics reflect both causal effects and the LD structure linking tested variants to causal variants. If V1 and V2 are highly correlated in one ancestry, their marginal statistics may be similar, and that group's data may not sharply distinguish them. If they are less correlated in another ancestry, the marginal and conditional patterns can differ enough to favor one candidate. Joint evidence can then concentrate posterior probability more strongly than either ancestry's evidence alone.

This benefit depends on usable information, not simply on the number of ancestry groups. A smaller sample, lower allele frequency, poorer imputation, or an unsuitable LD reference can make one group's contribution weak or misleading. A small joint credible set is most persuasive when it is stable to reasonable prior choices and LD uncertainty and when the assumed causal-sharing structure is scientifically defensible. Posterior concentration under one model should not be confused with proof that its assumptions are true.

Common mistake: treating the strongest statistic as the answer

A common but incorrect argument is: “V1 has the largest Z statistic in both ancestries, so V1 is the causal variant.” A marginal Z statistic measures evidence for association at that variant, not the probability that it is causal. LD can produce a strong marginal association at a noncausal proxy. The correction is to examine harmonization, ancestry-specific LD, conditional results, and posterior evidence under an explicit model. Even then, the conclusion should describe how well the data distinguish candidates, rather than claim certainty.