Measuring Fine-Mapping Information From LD Differences

Why LD Differences Can Help Fine-Mapping

When two variants are strongly correlated, either one may produce a similar pattern of marginal association across a locus. Association evidence can then be difficult to attribute to one candidate rather than the other. Across ancestries, correlations between variants can differ. If a candidate pair is less correlated in one ancestry, the association patterns expected under the two causal hypotheses may be more distinct there.

The important question is not simply whether LD differs between ancestries. It is whether the differences separate the particular candidate variants being compared, and whether the relevant ancestry contributes enough association information for that separation to matter. A numerical measure can make this reasoning explicit, provided its assumptions are kept in view.

Association Patterns Under Competing Candidates

Consider one ancestry with \(m\) variants and an ancestry-specific LD correlation matrix \(R\). Its entry \(r_{jk}\) is the correlation between standardized genotypes at variants \(j\) and \(k\). Let \(e_c\) denote the \(m\)-vector with a 1 in position \(c\) and 0 elsewhere. Under a single-causal-variant approximation, suppose the vector of marginal association \(z\)-statistics has distribution

where \(\kappa\) is the signal-strength scale for that ancestry. For example, in a simple standardized phenotype model, \(\kappa\) is approximately \(\sqrt{n}\beta\), where \(n\) is sample size and \(\beta\) is the standardized causal effect. The approximation uses the LD matrix as the covariance of the marginal \(z\)-statistics and assumes that the candidate causal variants are tested and harmonized consistently.

The mean vector under candidate \(c\) is proportional to column \(c\) of \(R\). Thus, under \(c\), the association at another variant \(j\) is expected to track its correlation with \(c\). Under candidate \(d\), the expected vector instead follows column \(d\). If those columns are similar, the candidates predict similar association patterns.

For the comparison below, assume the same \(\kappa\) under the two candidate hypotheses within an ancestry. This isolates the role of LD in distinguishing them. It does not assert that real effect sizes are known or identical across ancestries. In a trans-ancestry analysis, each ancestry can have its own signal-strength scale and LD matrix.

A Pairwise Information Measure

For two multivariate normal distributions with a common positive-definite covariance matrix \(R\) and means \(\mu_c\) and \(\mu_d\), their Kullback–Leibler divergence from the distribution under \(c\) to the distribution under \(d\) is

Here the divergence is the expected log density ratio under the first distribution. With the model above, the difference in means is \(\kappa R(e_c-e_d)\). Substitution gives

The last equality follows because \(e_c-e_d\) has a 1 at \(c\), a −1 at \(d\), and zeros elsewhere. Its quadratic form is \(r_{cc}+r_{dd}-2r_{cd}=2(1-r_{cd})\), since the diagonal entries of a correlation matrix are 1. This result says that, under the stated model, the information separating two candidates is larger when their correlation is lower and when the signal-strength scale is larger.

Justification of the Normal-Model Formula

The divergence formula can be obtained directly from the normal log densities. Write \(Z=\mu_c+\varepsilon\) under candidate \(c\), where \(E(\varepsilon)=0\) and \(\operatorname{Cov}(\varepsilon)=R\). Let \(A=R^{-1}\). The log density ratio is

Under \(c\), the second quadratic term has expectation \(\operatorname{tr}(AR)\). The first has expectation \(\operatorname{tr}(AR)+(\mu_c-\mu_d)^TA(\mu_c-\mu_d)\): its random centered component contributes the same trace, and its mean offset contributes the displayed quadratic form. Taking expectations therefore cancels the trace terms and leaves one half of the quadratic form. Positive definiteness is needed here to use the ordinary inverse and the usual nonsingular normal density.

Combining Ancestries and Reading the Weights

If ancestry-specific summary-statistic vectors are conditionally independent and each has a common covariance \(R_a\) under the competing candidates, the joint log-likelihood ratio is the sum of the ancestry-specific ratios. Consequently, the pairwise information is

where \(r_{a,cd}\) is the correlation between candidates \(c\) and \(d\) in ancestry \(a\). In the simple standardized setup, \(\kappa_a^2\) is approximately \(n_a\beta_a^2\). The weights therefore express how strongly that ancestry’s data can contribute to distinguishing the candidates; LD contrast alone is not the whole measure.

This sum is symmetric in \(c\) and \(d\) under the equal-covariance model, even though KL divergence is generally directional. The symmetry here follows because the two distributions share a covariance matrix, making the quadratic separation the same in either direction. A value of zero occurs when every weighted ancestry has \(r_{a,cd}=1\), so the model predicts no separation for the pair. Larger values mean greater expected separation in log-likelihood terms.

Using the Full LD Matrices for Several Candidates

For a pair of candidates, the relevant matrix entry is their correlation. With several candidates, it is useful to inspect the pattern of correlations rather than focus only on whether matrices differ overall. The same pairwise formula can be applied to each candidate pair, using its correlation in each ancestry. This produces a map of which pairs are predicted to be hard or relatively easy to distinguish under the model.

Singular LD Matrices and Scope of the Measure

An LD matrix need not be positive definite. Perfectly correlated variants, for instance, can make it singular. In that case the ordinary inverse \(R^{-1}\) and a full-dimensional normal density do not exist, so the inverse-based formula must not be applied as written. For the model here, however, the difference in means is \(\kappa R(e_c-e_d)\), which lies in the range of \(R\). The two normal distributions therefore have the same affine support. Their divergence on that support can be written using the Moore–Penrose pseudoinverse \(R^+\):

The second equality follows from \(RR^+R=R\). In this particular mean model, the resulting pairwise expression is still \(\kappa^2(1-r_{cd})\). Perfectly correlated candidates have \(r_{cd}=1\) and zero separation. In practice, singularity or near-singularity also calls for care in estimating LD and interpreting fine-mapping results; the formula is a model-based comparison, not a remedy for indistinguishable variants.

The derivation is most direct for a single causal variant and a common covariance across the candidate hypotheses. With multiple causal variants, ancestry-specific effects, model misspecification, or estimated rather than known LD, a complete likelihood may have additional structure. The pairwise calculation is then a useful way to understand one source of resolution, not a substitute for the specified fine-mapping model.

Common Mistakes and Practical Interpretation

• Equating different LD matrices with high information. The wrong reasoning is that any ancestry-to-ancestry change in LD must improve resolution. The correction is to examine the correlation between each candidate pair and weight its contrast by the strength of the relevant association data.
• Ignoring signal strength. A low candidate-pair correlation in an ancestry may produce little practical separation if its \(\kappa_a^2\) is small. Conversely, an ancestry with a more modest LD contrast can still contribute meaningfully when its signal is stronger.
• Calling the divergence a probability. A value in nats is an expected log-likelihood separation under a specified model. It is not the probability that a candidate is causal, nor a posterior inclusion probability.
• Using \(R^{-1}\) for a singular matrix. The formula with an ordinary inverse requires positive definiteness. For a singular matrix, use the support-aware pseudoinverse result when its conditions hold, or use an appropriate model that handles the singularity.
• Interpreting fine-mapping as causal proof. Even substantial separation between two modeled candidates does not establish that either candidate is causal. The result is conditional on the candidate set, model, harmonization, LD inputs, and signal assumptions.