Constructing Annotation-Specific LD Scores

Why Construct Annotation-Specific LD Scores?

A test SNP can be correlated with many other variants. If those variants belong to different functional annotations, the test SNP may tag different amounts of variation in each annotation. An annotation-specific LD score summarizes that information: it adds the squared correlation between a test SNP and each contributing variant, weighting each contribution by that variant’s annotation value.

The calculation is a property of the reference genotypes and the annotation data. It does not require a phenotype, a GWAS association statistic, or a fitted regression. In stratified heritability analysis, these scores serve as predictors: they describe how much LD-weighted annotation mass is tagged at each test SNP. They are not themselves estimates of heritability or evidence that an annotation causes a trait.

Definition: Let $j$ index a test SNP, $k$ index variants in the reference set, and $c$ index an annotation. Let $r_{jk}$ be the correlation between the standardized genotypes at $j$ and $k$, and let $a_{kc}$ be the annotation value for variant $k$ in annotation $c$. The annotation-specific LD score for test SNP $j$ and annotation $c$ is $$ \ell(j,c)=\sum_{k\in\mathcal{K}} r_{jk}^{\,2}a_{kc}, $$ where $\mathcal{K}$ is the chosen set of reference variants. The ordinary LD score over that same set is $\sum_{k\in\mathcal{K}}r_{jk}^{\,2}$.

The annotation is attached to the variant $k$ whose genetic variation may be tagged, not to the test SNP $j$ merely because $j$ is being analyzed. The test SNP is the row at which the score is calculated; each variant in the sum contributes according to its squared correlation with that test SNP and its own annotation value. Usually the test SNP itself is included in $\mathcal{K}$, in which case $r_{jj}^{\,2}=1$. If the reference set excludes it, that contribution is absent.

For a binary annotation, $a_{kc}$ is 1 when variant $k$ belongs to annotation $c$ and 0 otherwise. The score is then the sum of squared correlations with variants in that annotation. For a continuous annotation, the value $a_{kc}$ weights the contribution: a variant with a larger annotation value contributes more, all else equal. The score is dimensionless when correlations and annotation values are dimensionless.

Constructing the Scores

A practical calculation requires a defined test-SNP set, a reference variant set, genotype correlations in the reference sample, and a consistently indexed annotation table. The reference panel should be appropriate for the ancestry and variant coverage of the association analysis, because LD patterns differ across populations. Variants and alleles must be harmonized so each correlation and annotation refers to the intended variant. For each test SNP, square each correlation, multiply by the annotation value for the corresponding reference variant, and sum across the specified reference set.

The squared correlation is essential. It measures the strength of linear tagging without retaining the sign of the correlation: $r_{jk}$ and $-r_{jk}$ yield the same contribution. Thus a negative correlation does not subtract from the score. Whether an annotation is binary, overlapping, or continuous affects the weights, not the definition of LD used in the sum.

Worked Example: Two Binary Annotations

Suppose a hypothetical reference set contains four variants, including the test SNP $j$. Their squared correlations with $j$ are $1.00$, $0.64$, $0.09$, and $0.01$, respectively. The first annotation marks variants 2 and 3, while the second marks variants 1 and 4. Calculate both annotation-specific scores.

For the first annotation, the weights in variant order are $0,1,1,0$. Therefore,

\ell(j,1)=1.00(0)+0.64(1)+0.09(1)+0.01(0)=0.73.

For the second annotation, the weights are $1,0,0,1$, so

\ell(j,2)=1.00(1)+0.64(0)+0.09(0)+0.01(1)=1.01.

The second score is larger because the test SNP itself belongs to that annotation and contributes 1.00, while the other marked variant contributes 0.01. The first score is 0.73, reflecting the combined LD with its two marked variants. These numbers describe annotation-weighted LD around $j$; they do not indicate that either annotation has a larger heritability contribution.

Overlapping and Continuous Annotations

Annotations need not form mutually exclusive groups. A variant can have value 1 in several binary annotations, or have nonzero values in several continuous annotations. Its squared correlation with a test SNP then contributes to each relevant score, weighted separately by each annotation value. Consequently, annotation-specific scores can overlap in what they summarize.

Worked Example: A Variant in Two Annotations

Consider a hypothetical test SNP with squared correlations $1.00$, $0.49$, $0.25$, and $0.04$ to four reference variants. A coding annotation marks variants 1 and 2. A conservation annotation marks variants 2 and 3. Variant 2 belongs to both annotations.

The coding weights are $1,1,0,0$, giving

\ell(j,\text{coding})=1.00+0.49=1.49.

The conservation weights are $0,1,1,0$, giving

\ell(j,\text{conservation})=0.49+0.25=0.74.

The contribution $0.49$ from variant 2 appears in both scores because that variant belongs to both annotations. Adding the two scores gives 2.23, but this is not the ordinary LD score: the shared contribution has been counted twice, and variant 4 has not been counted at all. A structural condition guaranteeing that the sum of annotation-specific scores equals the ordinary LD score is that the annotation weights sum to 1 for every reference variant, as with a partition into binary annotations. For a particular test SNP, equality can also occur through cancellation, precisely when $\sum_{k\in\mathcal{K}} r_{jk}^{\,2}(\sum_c a_{kc}-1)=0$.

Continuous annotations use the same calculation without reducing values to zero or one. This preserves the specified weighting scale: rescaling annotation values rescales the corresponding LD scores. The meaning of a continuous score therefore depends on how the annotation was defined and scaled.

Worked Example: A Continuous Annotation

For a hypothetical test SNP, suppose the squared correlations with four reference variants are $0.81$, $0.36$, $0.04$, and $0.01$. Their continuous annotation values are $0.20$, $0.50$, $1.00$, and $0.25$, respectively. The annotation-specific score is

\begin{aligned} \ell(j,c) &=0.81(0.20)+0.36(0.50)+0.04(1.00)+0.01(0.25)\\ &=0.162+0.180+0.040+0.0025\\ &=0.3845. \end{aligned}

The ordinary LD score over these four variants is $0.81+0.36+0.04+0.01=1.22$. The annotation-specific score is smaller because the annotation values downweight three of the four variants. It is not a proportion of the ordinary LD score unless the weights and normalization make it one; here it is simply the weighted sum.

Why These Scores Have This Form

The sum of squared correlations has a direct justification under the additive model. Suppose genotypes $G_k$ are standardized to mean zero and variance one, and the genetic component of a trait is $G_{\mathrm{gen}}=\sum_k G_k\beta_k$. The effect $\beta_k$ is the effect on the trait scale per unit increase in the standardized genotype $G_k$. The marginal genetic association at test SNP $j$, expressed as the covariance with its standardized genotype, is

\operatorname{Cov}(G_j,G_{\mathrm{gen}})=\sum_k r_{jk}\beta_k.

If effects are centered and independent across variants, the variance of this marginal association across effect realizations is

\operatorname{Var}\!\left(\sum_k r_{jk}\beta_k\right) =\sum_k r_{jk}^{\,2}\operatorname{Var}(\beta_k).

To see why, expand the variance. The diagonal terms are $r_{jk}^{\,2}\operatorname{Var}(\beta_k)$; each cross-term contains $\operatorname{Cov}(\beta_k,\beta_m)$ for $k\ne m$, which is zero under independence. If the per-variant effect variance follows an additive annotation model, $\operatorname{Var}(\beta_k)=\sum_c a_{kc}\tau_c$, substitution and rearrangement give

\sum_k r_{jk}^{\,2}\sum_c a_{kc}\tau_c =\sum_c \tau_c\sum_k r_{jk}^{\,2}a_{kc} =\sum_c\tau_c\ell(j,c).

This identity explains why annotation-specific LD scores arise as predictors in a stratified model: each score collects the LD-weighted annotation values relevant to test SNP $j$. The argument assumes standardized genotypes, independent centered effects, and the stated additive variance model. It does not establish that a fitted coefficient is causal, nor does it supply a GWAS test-statistic expectation. The effect-variance parameters are on the phenotype scale; constructing $\ell(j,c)$ itself does not require the phenotype to have variance 1.

Interpretation and Quality Checks

A larger $\ell(j,c)$ means that test SNP $j$ is more strongly correlated, in aggregate, with variants having larger values in annotation $c$. For binary annotations it is a total of squared correlations to marked variants. For continuous annotations it is a weighted total. Scores can differ across test SNPs because their LD neighborhoods differ, even when the same annotation and reference panel are used.

Construction choices affect the resulting scores. The set $\mathcal{K}$ defines which variants can contribute; changing it changes the sum. Ancestry mismatch can make the reference correlations unrepresentative of the association sample. In a finite reference panel, estimated correlations are noisy, and squared sample correlations can be upward-biased, especially when the panel is small. Use an adequately sized, ancestry-matched panel and apply consistent variant inclusion and quality-control rules. These checks concern the scores’ construction, not the later regression procedure.

Fix the variant sets.
Specify the test SNPs and the reference variants contributing to each sum, including whether each test SNP contributes its self-correlation.

Align genotypes and annotations.
Ensure each correlation and annotation value refers to the same harmonized variant, and choose an ancestry-appropriate LD reference.

Weight and sum.
For each test SNP and annotation, calculate $r_{jk}^{\,2}a_{kc}$ for every reference variant $k$, then add those contributions.

Check interpretation.
Confirm that the values are weighted LD summaries, not heritability estimates, and account for overlapping annotation membership when comparing or summing scores.

Common Mistake

A frequent error is to annotate the test SNP and then use that one value to weight its entire LD neighborhood. For example, if test SNP $j$ has annotation value 1, a mistaken calculation might multiply every $r_{jk}^{\,2}$ by 1, regardless of whether variant $k$ belongs to the annotation. That produces an ordinary LD score for $j$, not the score for the annotation.

The correction is to use the annotation value of each contributing variant: $\ell(j,c)=\sum_k r_{jk}^{\,2}a_{kc}$. A second error is to sum signed correlations; positive and negative LD should not cancel in this score, because each term uses $r_{jk}^{\,2}$. Keep the indexing explicit and verify the weighted contributions variant by variant when checking a calculation.

Key takeaway: An annotation-specific LD score is the sum of squared correlations from a test SNP to reference variants, each weighted by that variant’s annotation value. It summarizes tagged annotation mass and becomes a predictor in stratified analyses; by itself, it is neither a heritability estimate nor a causal interpretation.

Check Your Understanding

Use the definition and the worked calculations to answer these questions.

A test SNP has squared correlations $1.00$, $0.25$, and $0.04$ with three variants. A binary annotation marks only the second and third variants. What is the annotation-specific LD score?
Why does a variant that belongs to two annotations contribute to both annotation-specific LD scores?
What changes in the calculation when annotation values are continuous rather than binary?
Under what condition does the sum of annotation-specific LD scores equal the ordinary LD score over the same variant set?
Why is it incorrect to interpret a larger annotation-specific LD score as evidence that the annotation has greater heritability?