LD Reference Mismatch in Bayesian Score Construction

Why the LD Reference Matters

Bayesian polygenic-score methods use GWAS summary statistics together with a model of linkage disequilibrium (LD) to estimate joint variant effects. The LD matrix describes correlations among genotypes: when two variants are correlated, a marginal association at one can partly reflect an effect at the other. This is why marginal effects and joint effects are not interchangeable.

The LD used in the calculation is often estimated from a reference panel rather than from the GWAS participants themselves. If that panel differs from the GWAS population in ancestry, has too few individuals, or does not include the relevant variants, its LD matrix may poorly represent the correlations that generated the summary statistics. Bayesian shrinkage can limit extreme estimates, but it cannot make an incorrect LD model correct.

The discussion below isolates LD mismatch by treating the true joint effects as the same in the target and reference populations. That is a simplifying condition, not a guarantee in real applications: effects themselves can differ across populations for biological, environmental, or measurement reasons. The effects and marginal associations here are on standardized scales, so the matrix calculations have a clear interpretation without mixing variant units.

Marginal Effects and the Assumed LD Model

Let \(\boldsymbol{\beta}\) be the vector of true joint effects in a region, and let \(\hat{\boldsymbol{b}}\) be the vector of estimated marginal GWAS effects. Under a linear model with standardized genotypes and phenotype, the expected marginal effects are approximately \(R_{\text{target}}\boldsymbol{\beta}\). In words, each marginal effect combines the joint effects of variants correlated with that variant. Sampling error adds noise to these expectations.

A useful simplified Bayesian summary-statistic model assumes \(\hat{\boldsymbol{b}}\mid\boldsymbol{\beta}\sim N(R_{\text{ref}}\boldsymbol{\beta},\,\sigma^2 R_{\text{ref}}/n)\) and an independent prior \(\boldsymbol{\beta}\sim N(\boldsymbol{0},\tau^2 I)\). Here \(n\) is the GWAS sample size, \(\sigma^2\) is residual variance on the chosen standardized scale, and \(\tau^2\) is the prior variance of a joint effect. This model assumes that the reference LD is the LD that generated the summary statistics; mismatch is precisely a violation of that assumption. Real Bayesian methods can use different likelihoods and priors, but the same issue remains: the LD information supplied to the model matters.

The formula makes two distinct sources of uncertainty visible. The GWAS sample size affects \(\lambda\): with the other quantities fixed, a smaller \(n\) means more shrinkage toward the prior mean of zero. The quality of \(R_{\text{ref}}\) affects the transformation of the marginal effects into joint effects. Strong shrinkage can reduce the size of unstable estimates, but it does not eliminate bias from using the wrong correlation pattern.

Worked Example: A Reference Correlation That Is Too High

A Short Derivation of Mismatch Bias

The posterior-mean formula also shows how a fixed mismatch shifts the expected estimate. Conditional on the true effects, suppose the target summary statistics satisfy \(E(\hat{\boldsymbol{b}}\mid\boldsymbol{\beta})=R_{\text{target}}\boldsymbol{\beta}\). The analysis nevertheless applies the reference-panel transformation. Taking the conditional expectation of its posterior mean gives

This expression is the complete justification for the mismatch comparison: if the reference matrix is the target matrix, the transformation uses the intended LD model; if the matrices differ, the expected posterior mean is generally not the true effect vector. In particular, it is not enough to say that the reference and target variants are “in the same region.” Their correlations must also be sufficiently well represented for the model being used.

Worked Example: A Reference Correlation That Is Too Low

Sample Size: GWAS Noise and Reference-Panel Noise

GWAS sample size and reference-panel sample size have different roles. In the simplified posterior formula, the GWAS sample size enters through \(\lambda=\sigma^2/(n\tau^2)\). A larger GWAS usually gives more precise marginal effects and, under this model, less shrinkage. But more precise \(\hat{\boldsymbol{b}}\) does not fix a wrong \(R_{\text{ref}}\). If the reference matrix is mismatched, reducing noise can make the distorted transformation more clearly visible rather than remove its bias.

Reference-panel size affects how accurately the correlations themselves are estimated. A small panel can produce a noisy or poorly conditioned estimate of \(R_{\text{ref}}\), even when its ancestry is appropriate. For independent observations from a bivariate normal distribution, a rough large-sample standard error for an estimated correlation \(\hat r\) is \((1-r^2)/\sqrt{m-1}\), where \(m\) is the number of reference individuals. This is an approximation, not a guarantee for genotype data with related individuals or other departures from the model.

Variant Coverage and the Meaning of the Estimated Weights

A reference panel needs to cover the variants and relationships relevant to the summary statistics. If variants are missing, the supplied matrix can describe only the retained set, or the method may need to use a proxy. Either way, a reduced set changes the model: the omitted variants’ contributions to marginal effects do not disappear simply because they are absent from the reference. Their signals can be left unexplained or attributed to correlated variants that remain.

Practical Interpretation and Common Pitfalls

An LD reference mismatch is most consequential when the score method relies on LD to distinguish correlated signals, when relevant variants are missing, or when the reference LD estimates are noisy. The resulting weights may be redistributed among correlated variants: an estimated weight can be too large, too small, or even have the opposite sign to a true joint effect. A change in weights does not by itself show that underlying biological effects differ; the LD model, sampling error, coverage, and effect differences are separate possible explanations.

When investigating a suspected mismatch, check that the reference panel is relevant to the GWAS population, that the same variants and allele coding are represented, and that the panel is large enough to estimate local correlations reliably. Compare LD patterns in regions driving the score when suitable data are available. These checks diagnose inputs to the construction; they do not establish that effects are identical across populations or guarantee a particular prediction performance.

• Pitfall: assuming a larger GWAS repairs the reference. A wrong argument is, “The association estimates are precise, so the joint weights must be accurate.” The correction is that precision of \(\hat{\boldsymbol{b}}\) and correctness of \(R_{\text{ref}}\) are different requirements. The expected estimate still depends on \((R_{\text{ref}}+\lambda I)^{-1}R_{\text{target}}\).
• Pitfall: treating a missing variant as having no effect on the remaining weights. Its marginal signal can remain in the summary statistics and be attributed to a correlated variant that is covered.
• Pitfall: interpreting a weight as a causal effect. Bayesian score weights are model-based prediction weights. Correlation, shrinkage, missing variants, and other model choices affect them; they do not establish causation.