Deriving the Stratified LD Score Regression Equation

Why the Regression Equation Has This Form

A GWAS tests each variant marginally, but the genetic signal tagged by a tested variant can come from effects at other, correlated variants. The expected chi-square statistic therefore depends not only on the effect variance assigned to the tested variant, but also on how it is correlated with variants across the genome. Stratified LD score regression expresses that relationship using annotation-specific LD scores.

The equation follows from two ingredients: a model for how per-variant effect variance depends on annotations, and the fact that a marginal association aggregates effects tagged through linkage disequilibrium. The result is an expectation across repeated samples under the model, not a formula that predicts the exact observed statistic at every SNP.

Variance Model and Annotation-Specific LD Scores

Suppose there are $M$ variants and $C$ annotations. Let $c_{kc}$ be the value of annotation $c$ at variant $k$. For a binary annotation this value is zero or one; annotations may overlap, so a variant can have nonzero values for more than one annotation. With standardized genotypes and a standardized quantitative phenotype, write the additive annotation variance model as follows.

Definition: In the additive annotation variance model, the variance of the true per-variant effect $\beta_k$ is $$ \operatorname{Var}(\beta_k)=v_k=\sum_{c=1}^{C}\tau_c c_{kc}. $$ Here $\tau_c$ is the variance coefficient for annotation $c$. The model requires the resulting $v_k$ to be a valid, nonnegative variance for each variant. A coefficient is a conditional contribution in this joint model; it is not, by itself, the fraction of total heritability in that annotation.

The annotations are used together in the sum. If a variant belongs to two binary annotations, both corresponding terms contribute to its modeled effect variance. This is why the coefficients and their interpretation depend on the joint annotation model.

Let $r_{jk}$ be the correlation between standardized genotypes at tested variant $j$ and variant $k$, with $r_{jj}=1$. The annotation-specific LD score for tested variant $j$ and annotation $c$ is the sum of squared correlations weighted by that annotation’s values:

Definition: The annotation-specific LD score is $$ \ell(j,c)=\sum_{k=1}^{M}r_{jk}^{2}c_{kc}. $$ It measures how much squared LD the tested variant has with variants carrying annotation $c$, with the annotation values providing the weights.

The ordinary LD score is the special case in which every variant has annotation value one. Annotation-specific scores separate the total LD-tagged signal into contributions associated with the modeled annotations.

Deriving the Expected Chi-Square Statistic

Let $\chi_j^2$ be the one-degree-of-freedom association chi-square statistic for variant $j$, and let $N$ be the GWAS sample size. Under the usual large-sample, small-effect approximation, the marginal genetic effect tagged by $j$ is approximately $\sum_k r_{jk}\beta_k$. Its squared size contributes approximately $N$ times that squared effect to the expected chi-square statistic.

To turn this statement into an expectation, assume the true effects have mean zero, distinct variants’ effects are uncorrelated, and their variances follow the annotation model. The zero-mean and uncorrelated assumptions remove cross-products between different variants when taking the expectation:

\begin{aligned} E\left[\left(\sum_{k=1}^{M}r_{jk}\beta_k\right)^2\right] &=\sum_{k=1}^{M}r_{jk}^{2}\operatorname{Var}(\beta_k)\\ &=\sum_{k=1}^{M}r_{jk}^{2}\sum_{c=1}^{C}\tau_c c_{kc}\\ &=\sum_{c=1}^{C}\tau_c\sum_{k=1}^{M}r_{jk}^{2}c_{kc}\\ &=\sum_{c=1}^{C}\tau_c\ell(j,c). \end{aligned}

In the absence of confounding or other departures from the ideal association model, the null sampling variation contributes approximately one to the expected chi-square statistic. An intercept allows the model to represent this baseline together with inflation that does not vary with the tested SNP’s LD scores. Denote that intercept by $a$, to distinguish it from the significance level $\alpha$.

Result: Under the annotation variance model and the approximations above, the stratified LD score regression expectation is $$ E[\chi_j^2]=a+N\sum_{c=1}^{C}\tau_c\ell(j,c). $$ The intercept $a$ is approximately one when the baseline model is well calibrated and there is no additional LD-independent inflation. The sample-size-scaled sum is the expected genetic contribution tagged by variant $j$.

The derivation uses squared correlations because the expected squared marginal effect depends on $r_{jk}^2$, not on the signed correlation alone. It uses sample size because, approximately, the noncentrality of a test grows in proportion to $N$ for a fixed marginal effect. The expectation is approximate: it relies on standard GWAS large-sample and weak-effect conditions, and on the assumed relationship between effects and annotations.

Worked Examples

Worked Example: Calculating the Expected Statistic With Two Annotations

Hypothetical data. Suppose a quantitative-trait GWAS has $N=12{,}000$ participants. For a tested variant, the annotation-specific LD scores are $\ell(j,1)=2.0$ and $\ell(j,2)=0.75$. The fitted variance coefficients are $\tau_1=1.5\times10^{-5}$ and $\tau_2=0.8\times10^{-5}$, and the intercept is $a=1.04$. Assume the annotation variance model and the effect-independence approximation are suitable for this calculation.

Substitute these values into the expectation:

\begin{aligned} E[\chi_j^2] &=a+N\{\tau_1\ell(j,1)+\tau_2\ell(j,2)\}\\ &=1.04+12{,}000\{(1.5\times10^{-5})(2.0) +(0.8\times10^{-5})(0.75)\}\\ &=1.04+12{,}000(3.0\times10^{-5}+0.6\times10^{-5})\\ &=1.04+12{,}000(3.6\times10^{-5})\\ &=1.04+0.432=1.472. \end{aligned}

The model predicts an expected chi-square statistic of 1.472 for a variant with these LD scores, sample size, coefficients, and intercept. This is an expectation over repeated samples under the model; an observed statistic at that SNP may be higher or lower.

Worked Example: Tracking an Overlapping Variant

Hypothetical data. Consider three variants with squared correlations to a tested SNP equal to $0.70$, $0.20$, and $0.05$. Annotation A has values $(1,1,0)$, and annotation B has values $(0,1,1)$. Thus, the second variant belongs to both annotations. Let $N=25{,}000$, $\tau_A=2\times10^{-5}$, $\tau_B=4\times10^{-5}$, and $a=1.12$. Assume these coefficients imply valid variances for the variants and that the model’s effect assumptions hold.

First calculate the two annotation-specific LD scores:

\ell(j,A)=0.70+0.20=0.90,\qquad \ell(j,B)=0.20+0.05=0.25.

The shared variant contributes its squared correlation to both scores. Substituting the scores and coefficients gives

\begin{aligned} E[\chi_j^2] &=1.12+25{,}000\{(2\times10^{-5})(0.90) +(4\times10^{-5})(0.25)\}\\ &=1.12+25{,}000(1.8\times10^{-5}+1.0\times10^{-5})\\ &=1.12+25{,}000(2.8\times10^{-5})\\ &=1.12+0.70=1.82. \end{aligned}

The overlap is not a reason to assign the shared variant to only one annotation: the stated model adds both annotation contributions to that variant’s effect variance. The expected statistic is 1.82, with 0.70 contributed by the sample-size-scaled genetic term and 1.12 by the intercept.

Worked Example: Verifying the Equation From Variant-Level Variances

Hypothetical data. A tested SNP has squared correlations $0.40$, $0.10$, and $0.30$ with three variants. Annotation 1 has values $(1,0,1)$, and annotation 2 has values $(0,1,1)$. Let $\tau_1=1\times10^{-5}$, $\tau_2=2\times10^{-5}$, $N=10{,}000$, and $a=1.06$. Assume the effects are uncorrelated across variants and the model is applicable.

The annotation model gives per-variant effect variances $1\times10^{-5}$, $2\times10^{-5}$, and $3\times10^{-5}$, respectively. Weighting each variance by its squared correlation gives

(0.40)(1\times10^{-5})+ (0.10)(2\times10^{-5})+ (0.30)(3\times10^{-5}) =1.5\times10^{-5}.

Now calculate the annotation-specific scores: $\ell(j,1)=0.40+0.30=0.70$, and $\ell(j,2)=0.10+0.30=0.40$. The equation gives the same weighted sum:

\begin{aligned} \tau_1\ell(j,1)+\tau_2\ell(j,2) &=(1\times10^{-5})(0.70)+(2\times10^{-5})(0.40)\\ &=0.7\times10^{-5}+0.8\times10^{-5} =1.5\times10^{-5}. \end{aligned}

Therefore,

E[\chi_j^2]=1.06+10{,}000(1.5\times10^{-5}) =1.06+0.15=1.21.

This check shows explicitly that the annotation-score expression is just a rearrangement of the LD-weighted sum of per-variant effect variances.

Assumptions and Meaning of the Intercept

The equation is informative only to the extent that its assumptions are reasonable. Its main requirements and approximations are:

Effect variance follows the stated model. The annotation values and coefficients must describe the per-variant effect variances, with nonnegative implied variance at each variant.
Effects at distinct variants are uncorrelated. If their effects have covariance, cross-products can contribute to the expected squared marginal effect, and the simple sum of annotation-specific LD scores need not capture that contribution.
LD scores are relevant to the GWAS population. Correlations used in the scores should represent the ancestry and variant set underlying the association statistics well enough for the approximation.
Association statistics are in the appropriate large-sample regime. The proportionality to $N$ is an approximation for standard GWAS statistics, not an exact identity for every sample design or effect size.
The intercept is interpreted carefully. It accommodates baseline sampling variation and inflation that is approximately independent of the LD scores. It does not guarantee that every form of confounding or misspecification has been removed.

In particular, an intercept near one is not proof that the model is correct. A deviation from one can reflect LD-independent confounding or other sources of inflation, while deviations that themselves track LD scores may instead affect the slope relationship. The intercept is a component of the expectation, not a general correction for all possible problems.

Common Mistake: Confusing an Annotation Coefficient With Its Heritability Share

A common error is to interpret $\tau_c$ as the proportion of total heritability contributed by annotation $c$, or to read it as the expected chi-square increase at every SNP in that annotation. Neither interpretation follows from the equation. The contribution of annotation $c$ to the expected statistic at SNP $j$ is $N\tau_c\ell(j,c)$, which depends on that SNP’s annotation-specific LD score as well as the coefficient and sample size. Total heritability and enrichment require additional aggregation and comparisons.

Another error is to omit the LD weights and use only the annotation status of the tested SNP. A tested SNP can tag effects at other variants, including variants carrying an annotation it does not carry itself. The squared correlations in $\ell(j,c)$ represent precisely this indirect tagging.

Key takeaway: The stratified LD score regression equation follows by weighting each variant’s annotation-modeled effect variance by its squared LD with the tested SNP, then multiplying by sample size and adding an intercept. Annotation-specific LD scores summarize those weights; they do not make the intercept or the variance coefficients causal interpretations.

Check Your Understanding

Use the definitions and derivation above to answer these questions.

In the equation $E[\chi_j^2]=a+N\sum_c\tau_c\ell(j,c)$, which terms represent sample size, annotation-specific tagging, and LD-independent inflation?
A variant has values one for two overlapping annotations. How does the additive variance model include its contribution?
Why do squared correlations, rather than signed correlations, appear in the annotation-specific LD score?
Under what assumption does the expectation of the squared marginal effect reduce to a sum of squared correlations times per-variant effect variances?
Why does an intercept near one not establish that all confounding or model misspecification is absent?