Annotation-Dependent LD and Heritability Attribution

Why Annotation-Dependent LD Matters

A GWAS association statistic belongs to a tested SNP, but the genetic effects contributing to it may arise at other variants correlated with that SNP. This distinction matters when variants differ by functional annotation or allele frequency. If one category of SNPs tends to tag another category well, its GWAS statistics can be large even when the causal effects are concentrated elsewhere. Conversely, causal variants in a category may be poorly tagged by the SNPs tested in the GWAS.

The central task is therefore to keep two questions separate: which variants carry modeled causal-effect variance? and which variants tag the resulting association signal? Stratified LD score regression links these questions using annotation-specific LD scores. The link is useful, but it does not make the two quantities interchangeable.

An LD score is defined from a specified set of variants and an LD reference panel. It is not a measure of causal status. A SNP can have a high \(l(j,c)\) because it is correlated with many annotated variants, or strongly correlated with a few of them. If annotations overlap, one SNP can contribute to multiple category-specific LD scores.

From Causal Variance to Tagged Association

Consider a quantitative phenotype standardized to variance one, and standardized genotypes \(X_k\). Under an additive model, write the genetic component as \(g=\sum_k X_k\beta_k\), where \(\beta_k\) is the effect of SNP \(k\) on the standardized phenotype. For the usual stratified LD score model, effects are mean zero and uncorrelated across SNPs, conditional on the annotations, with variance \( \operatorname{Var}(\beta_k)=\sum_c a_{kc}\tau_c \). Here \(\tau_c\) is the per-unit annotation contribution to effect variance. With binary annotations, it is the variance increment associated with membership in category \(c\), conditional on the other modeled annotations. With continuous annotations, it is the increment per unit annotation value.

The equation shows how causal effect variance at SNPs in category \(c\) contributes to association statistics at SNP \(j\): its contribution is proportional to \(N\tau_c l(j,c)\). If \(j\) is in a category but has little LD with that category’s variants, its annotation-specific LD score may be small. If \(j\) is outside a category but tags its variants, the score may be large.

To see the justification, the marginal association of tested SNP \(j\) is driven by its correlations with the genetic effects at all SNPs. Under the model, the signal component is approximately \(\sqrt{N}\sum_k r_{jk}\beta_k\). Its expected squared value is \(N\sum_k r_{jk}^2\operatorname{Var}(\beta_k)\): cross-products vanish because distinct SNP effects have zero covariance. Substituting the annotation variance model and rearranging the finite sums gives the displayed expression. The additional baseline and intercept account for sampling noise and residual inflation.

Heritability Attributed to a Category

Under the model, the expected variance contribution from a set of causal variants is obtained by summing their effect variances. For a binary, non-overlapping category \(c\), its model-based contribution is \(h_c^2=\sum_k a_{kc}\tau_c\), where the sum is over the variants included in the model. More generally, the additive annotation model assigns the term \(h_c^2=\tau_c\sum_k a_{kc}\) to annotation \(c\). These terms sum to the model’s total SNP heritability when the model and variant set define the full partition.

This attribution is a statement about the fitted variance model, not a direct identification of causal variants from GWAS summary statistics. In a joint model with overlapping annotations, a variant can contribute through more than one annotation term. A coefficient is conditional on the other annotations included; its meaning is not the same as the effect of analyzing that category alone. Estimated coefficients can also be negative because of sampling error or model limitations, even though a true variance cannot be negative. In that case, the fitted coefficient should not be described as a literal negative biological variance.

Tagged Heritability Is a Different Summary

A tagged summary groups association statistics according to the annotation of the tested SNPs, or otherwise summarizes signal captured by a set of tag SNPs. Such a summary describes where association signal is observed or captured. It need not describe where the modeled causal-effect variance lies. In the expectation for a set of tested SNPs \(J\), the contribution from category \(c\) is proportional to \(\tau_c\sum_{j\in J}l(j,c)\). This can be substantial even if none of the tested SNPs in \(J\) is causal, provided they tag category-\(c\) variants.

A simple count or sum of marginal signals across tag SNPs is also not automatically a partition of total heritability. Correlated tag SNPs can reflect the same underlying causal signal repeatedly, and annotation sets can overlap. The LD-score regression model uses the LD structure and the specified annotation terms to estimate variance contributions; merely assigning each association statistic to the tested SNP’s category does not do that.

Allele Frequency and LD Patterns

LD depends on the allele frequencies and haplotype structure of the variants being compared. Consequently, annotations that differ in their allele-frequency distributions can also have different patterns of LD scores: for example, SNPs in one functional class may be tagged by common variants more effectively than SNPs in another class. The direction and size of this difference are properties of the population and variant set; they cannot be inferred from annotation labels alone. A frequency-specific analysis should therefore use LD estimates and variant definitions appropriate to the population and frequency range being studied.

Frequency also affects the scale connecting raw genotype effects to standardized-genotype effects. If a genotype is coded as an allele count with frequency \(p\), its variance under Hardy–Weinberg equilibrium is \(2p(1-p)\). A raw per-allele effect \(\beta_{\mathrm{raw}}\) corresponds to a standardized-genotype effect \(\beta_{\mathrm{std}}=\sqrt{2p(1-p)}\beta_{\mathrm{raw}}\). Thus equal raw effects do not imply equal variance contributions on the standardized scale. This is one reason frequency definitions and effect scales must be stated when interpreting partitions.

Interpretation and Common Pitfalls

Annotation-dependent LD is important because it determines how a category’s modeled causal variance appears across the GWAS statistics used for estimation. A category can be enriched for heritability, have high LD scores at many tested SNPs, or both; these are related but distinct properties. The fitted model uses the pattern of LD scores across tested SNPs to estimate annotation coefficients, rather than treating each SNP’s marginal association as a direct measurement of its own causal effect.

• Wrong reasoning: “The strongest associated SNP is annotated as regulatory, so the heritability must be caused by regulatory variants.” Correction: That SNP may tag causal variants in another category. Inspect the annotation-specific LD structure and interpret the fitted variance model.
• Wrong reasoning: “A high LD score for an annotation means that annotation has high heritability.” Correction: The LD score measures tagging of annotated variants. Heritability attribution also depends on the estimated \(\tau_c\).
• Wrong reasoning: “The category-specific heritability term is the same as the fraction of association statistics at SNPs labeled with that category.” Correction: Tagged statistics can include signal from correlated variants outside the category, and repeated tags can represent overlapping signal.
• Wrong reasoning: “Rare or common variants must always be better tagged.” Correction: Tagging depends on the actual allele frequencies and haplotypes in the relevant population, so it must be evaluated from an appropriate LD reference.