Reference LD Scores and Ancestry Matching

Why Reference LD Scores Matter

Summary-statistic heritability methods use the relationship between association statistics and linkage disequilibrium (LD). In practice, the GWAS may not provide individual genotypes for estimating LD. Instead, LD scores are often calculated from a separate reference panel of genotyped or imputed individuals. Those scores are useful only insofar as the reference genotypes represent the LD patterns relevant to the GWAS population and the variants included in the analysis.

A reference panel is not merely a source of extra genotypes. Its ancestry, sample size, variant set, and genotype quality all affect the estimated LD scores. If those scores poorly represent the GWAS data, the regression may use the wrong predictor values even when the GWAS association statistics themselves are correctly computed. This tutorial focuses on constructing and evaluating the reference LD scores used in that regression.

Definition: For a focal variant $j$, its LD score is the sum of squared genotype correlations between that variant and the variants $k$ included in a specified analysis set or genomic window: $$ \ell_j = \sum_{k \in \mathcal{W}_j} r_{jk}^2. $$ Here $r_{jk}$ is the correlation between genotypes at variants $j$ and $k$, estimated in a reference sample, and $\mathcal{W}_j$ is the set of variants included for focal variant $j$. The focal variant is usually included, contributing $r_{jj}^2=1$.

Calculating an LD Score from Reference Genotypes

For a biallelic SNP, let $G_{ij}$ be the allele dosage for reference individual $i$ at variant $j$, usually between 0 and 2. A common standardization is to subtract the reference-sample mean dosage and divide by the reference-sample standard deviation. With Hardy–Weinberg genotype variance approximated by $2p_j(1-p_j)$, one may use

Z_{ij} = \frac{G_{ij}-2p_j}{\sqrt{2p_j(1-p_j)}}.

where $p_j$ is the reference-panel allele frequency for the coded allele. The genotype correlation $r_{jk}$ is the sample correlation between the dosage vectors for variants $j$ and $k$. Squaring it makes the LD score insensitive to the sign of the correlation: two variants that tend to carry the same allele and two that tend to carry opposite alleles can have the same $r^2$.

A calculation requires decisions about which variants count. A typical procedure applies genotype and variant quality filters, defines a set of analysis SNPs, and sums $r^2$ over variants in a chosen physical or genetic-distance window. Different software settings may define the window, missingness handling, and variant inclusion differently. Therefore an LD score is not an intrinsic number independent of the analysis: it is defined relative to a reference sample and a specified set of variants and genomic neighborhoods.

Worked Example: Summing Squared Correlations

Suppose a hypothetical reference panel is used to calculate the score for focal SNP $j$. The score includes $j$ and four neighboring variants. The estimated correlations with those four variants are $0.60$, $-0.30$, $0.10$, and $-0.20$. Assume these are the complete set of variants in the chosen window.

The focal SNP contributes $r_{jj}^2=1$. The other contributions are $0.60^2=0.36$, $(-0.30)^2=0.09$, $0.10^2=0.01$, and $(-0.20)^2=0.04$. Thus

\ell_j = 1+0.36+0.09+0.01+0.04=1.50.

The score is dimensionless. It summarizes the squared correlations with the variants included in this particular calculation; it is not a count of independently associated signals or a measure of causal effect size.

Reference Ancestry Must Represent the GWAS

LD depends on population history, allele frequencies, and the set of variants being considered. Consequently, a reference panel from a different ancestry may estimate different genotype correlations from those in the GWAS population. Matching labels alone is not enough: ancestry composition, admixture, and the analysis variant set can all matter. A larger reference panel can estimate its own population’s LD more precisely, but it does not make that population’s LD representative of a different GWAS population.

The relevance to summary-statistic heritability can be seen in the usual LD Score Regression relationship. In a simplified quantitative-trait setting, its expected association statistic at SNP $j$ is modeled as

E[\chi_j^2] \approx 1 + \frac{N h^2}{M}\ell_j + c.

Here $N$ is the GWAS sample size, $h^2$ is the SNP heritability on the scale specified by the analysis, $M$ is the number of SNPs used in the model, and $c$ represents a contribution to the intercept beyond the baseline 1. The slope with respect to the appropriate LD scores is $N h^2/M$. This relationship relies on assumptions about the genetic model, the GWAS statistics, and how well the LD scores represent the GWAS population. If reference scores differ systematically from the relevant LD scores, the fitted slope need not recover the intended quantity. The direction of the resulting error is not fixed: it depends on how the reference scores differ across SNPs.

Worked Example: How Mismatched Scores Can Change a Slope

Consider a deliberately small hypothetical illustration with three SNPs. Suppose their GWAS statistics follow the exact relation $\chi_j^2=1+0.02\ell_j$ for the relevant population, whose LD scores are $5,10,15$. The resulting statistics are $1.10,1.20,1.30$. Now suppose a mismatched reference panel supplies scores $5,8,17$ instead. Use ordinary least squares with a fitted intercept to compare the two slopes. This simplified calculation is for illustration; real analyses use many SNPs and regression procedures that may include weights.

With the relevant scores, each increase of 5 in $\ell_j$ corresponds to an increase of $0.10$ in $\chi_j^2$, so the slope is $0.10/5=0.0200$. With the reference scores, the mean reference score is 10 and the mean statistic is 1.20. The least-squares slope is

\hat b = \frac{(-5)(-0.10)+(-2)(0)+(7)(0.10)} {(-5)^2+(-2)^2+7^2} = \frac{1.20}{78} \approx 0.01538.

Suppose, only for this illustration, that $N=100{,}000$ and $M=1{,}000{,}000$. The slope-based estimate is $\hat h^2=(M/N)\hat b=10(0.01538)=0.1538$, whereas the slope from the relevant scores gives $10(0.0200)=0.2000$. The example demonstrates that mismatched scores can alter the estimate. It does not establish that ancestry mismatch always lowers an estimate; a different pattern of score differences could shift the fitted slope in another direction.

Panel Size and Sampling Noise

Reference LD scores are estimated, not known exactly. With a small reference sample, estimated correlations can vary substantially from the population correlations. Squaring a noisy correlation also creates a positive contribution even when the true correlation is zero. For independent, bivariate-normal observations with true correlation zero, the sample squared correlation $r^2$ has expectation $1/(m-1)$, where $m$ is the number of reference individuals. This is a useful approximation for understanding finite-sample noise, not a universal exact correction for genotype data.

Because an LD score adds squared correlations across variants, small sampling contributions can accumulate. The size and consequences depend on the number and dependence of the included variants, the true LD pattern, and how the scores are used in regression. Increasing reference-panel size generally reduces sampling variability, but cannot resolve ancestry mismatch, poor genotype quality, or an inconsistent variant set. A very large but poorly matched panel can therefore be less useful than a smaller, well-matched one.

Worked Example: A Finite-Panel Noise Contribution

Suppose a hypothetical reference panel contains $m=101$ individuals, and a focal SNP is compared with 100 other SNPs that are uncorrelated with it in the population. Under the independent bivariate-normal approximation, each estimated squared correlation has expectation

E[r^2] \approx \frac{1}{m-1}=\frac{1}{100}=0.0100.

By additivity of expectation, the expected sum of those 100 squared correlations is approximately $100(0.0100)=1.00$. This is a sampling-noise contribution, not evidence that the focal SNP truly has one unit of LD score from those population-independent comparisons. The calculation is illustrative: correlations in real genotype data are not generally independent normal pairs, so the exact behavior depends on the data and variant set.

Variant Coverage and Consistent Score Construction

The SNPs represented in the reference panel must be compatible with the SNPs used in the GWAS regression. If a GWAS SNP is absent from the reference data, its score may be unavailable unless an appropriate alternative procedure is used. If variants that should contribute to a score are missing, or if the reference and GWAS use substantially different filters, the calculated score may not represent the intended neighborhood. Missing variants do not have a universal effect on the final heritability estimate: removing terms changes scores, but the impact on the regression depends on which terms are missing and how the changes relate to the association statistics.

Variant identity and allele handling also require care. Since $r^2$ is unchanged by reversing allele coding at a correctly matched SNP, allele flips alone do not change a squared correlation. But incorrect variant matching, strand ambiguity, inconsistent genome builds, or different variants being assigned the same identifier can pair the wrong genotype columns and produce invalid scores. Quality control and harmonized variant identification are therefore prerequisites to interpreting the result.

Worked Example: The Effect of Omitting a Variant From a Score

Suppose a hypothetical focal SNP has an LD score of $2.10$, including itself and four other variants. One included neighbor has $r^2=0.25$. If that neighbor is absent from the reference data and the score is recalculated using only the remaining variants, the new score is

\ell_j^{\mathrm{reduced}}=2.10-0.25=1.85.

The decrease of $0.25$ is correct for this reduced variant set. It does not by itself tell us whether the estimated heritability will increase or decrease: the regression slope depends on how score changes across all analyzed SNPs, not on the change for one focal SNP alone. If the aim is to compare analyses, the SNP set and score construction should be documented and, where possible, made consistent.

A Practical Assessment of Reference LD Scores

Before using reference LD scores, check the inputs and their relationship to the GWAS. The following sequence separates distinct concerns that are sometimes mistakenly treated as interchangeable.

Specify the score definition.
Record the reference variants, focal SNPs, genomic window, quality filters, and whether the focal SNP is included.

Check ancestry relevance.
Assess whether the reference sample’s ancestry composition is appropriate for the GWAS population; size cannot substitute for a suitable match.

Check panel precision.
Consider whether the reference sample is large enough for stable LD estimates, especially when scores sum many noisy pairwise correlations.

Check variant coverage and identity.
Confirm that GWAS variants and reference variants are consistently identified, filtered, and represented in the score calculation.

Interpret the slope conditionally.
A heritability estimate relies on the LD scores representing the relevant population and analysis set; a plausible numerical estimate does not verify those assumptions.

Common Mistake

A common mistake is to reason, “The reference panel is large, so its LD scores must be accurate for this GWAS.” A large panel can reduce sampling noise in the LD estimates for its own population, but it cannot correct a population mismatch or restore variants that are not represented. The correction is to assess panel size, ancestry match, and variant coverage separately. Another mistake is to assume that omitting variants must bias heritability in one particular direction. The score changes are calculable, but the direction of the regression-slope change depends on how those changes align with the GWAS statistics across SNPs.

Key takeaway: An LD score is a sum of squared genotype correlations defined by a reference sample and a specified variant set. Reliable summary-statistic heritability estimation requires scores that are sufficiently precise, ancestry-relevant, and compatible with the GWAS variants; none of those properties is guaranteed by panel size alone.

Check Your Understanding

Answer these questions using the definitions and examples above.

A focal SNP has correlations of $0.5$, $-0.4$, and $0.2$ with three other included SNPs. What is its LD score if the focal SNP is included?
Why can a large reference panel from a different ancestry still produce unsuitable LD scores?
Under the stated null approximation, what is the expected squared sample correlation for $m=201$ individuals?
A variant is omitted from one LD score. Why does that fact alone not determine the direction of change in a heritability estimate?
Which three properties should be evaluated separately when judging reference LD scores for a GWAS?