Deriving the LD Score Regression Mean Model

Why LD Score Enters the Mean Association Signal

A variant can show an association signal even when it is not itself causal: it may be correlated with variants that contribute to the trait. The more such correlations a tested variant has, the more polygenic signal its marginal association can capture. LD score regression’s mean model expresses this idea quantitatively by relating a variant’s expected association chi-square statistic to its LD score.

The derivation is a chain of approximations, not an identity that holds for every data set. It starts with a model for genetic effects, translates those effects into marginal associations through linkage disequilibrium (LD), and then approximates the expected test statistic. Keeping those steps separate helps show both where the relationship comes from and what could make it inaccurate.

For the basic derivation, let \(X_k\) be a genotype standardized to mean zero and variance one, and let \(\beta_k\) be its effect on a standardized quantitative phenotype. Write the additive genetic component as \(G=\sum_{k=1}^{M}X_k\beta_k\), and let \(h_g^2\) denote the variance of that component. The simple random-effects model assumes effects are independent, have mean zero, and have the same variance, with \(\operatorname{Var}(\beta_k)=h_g^2/M\). This makes the expected total genetic variance \(M(h_g^2/M)=h_g^2\), under the standardized-genotype convention and the usual approximation that realized LD does not materially alter that total.

From Genetic Effects to a Marginal Association

A single-variant association test estimates the marginal association of \(X_j\) with the phenotype, rather than fitting every variant’s effect simultaneously. Under the model above, the genetic part of that marginal association is \[ \gamma_j=\operatorname{Cov}(X_j,G)=\sum_{k=1}^{M}r_{jk}\beta_k. \] This follows because \(\operatorname{Cov}(X_j,X_k)=r_{jk}\) for standardized genotypes. Thus effects at variants correlated with \(j\) contribute to the marginal effect at \(j\).

The mean-zero, independent effect assumptions let us calculate the expected squared marginal effect. When \(\gamma_j\) is squared, terms involving two different effects have expectation zero: for \(k\ne q\), independence and zero means give \(E(\beta_k\beta_q)=0\). The remaining terms are weighted by squared LD correlations.

This is the key link: under equal effect variances, the expected squared marginal effect is proportional to LD score. It is an expectation over the random effects in the model. It does not say that an individual variant’s effect, or its observed statistic, is determined by its LD score.

From Marginal Effect to Expected Chi-Square

For a one-degree-of-freedom association test, suppose the signed \(z\)-statistic is approximately normal with variance one and mean \(\sqrt{N}\gamma_j\), where \(N\) is the sample size. This approximation treats the genotype and phenotype scales as standardized and the standard error as approximately \(1/\sqrt{N}\). It is most useful when individual marginal effects are small and the sample is large. If \(Z\) is normal with mean \(\delta\) and variance one, then \(E(Z^2)=1+\delta^2\), because \(E(Z^2)=\operatorname{Var}(Z)+[E(Z)]^2\). The association chi-square statistic is \(Z^2\).

Applying this result at variant \(j\), and then averaging over the random genetic effects, gives the basic mean model:

The 1 is the expected chi-square under a correctly calibrated null test. The second term is the expected polygenic signal in the statistic under this model. The equation describes a mean across modeled effects at a given LD score; it is not a formula for the observed chi-square at a particular variant. In applications, departures from the ideal test calibration or other sources of signal may also affect observed means.

The approximation \(Z\approx N(\sqrt{N}\gamma_j,1)\) is not exact in every finite sample. In a simple regression, the residual variance and the variance of the genotype are estimated from data; with non-negligible marginal effects, the standard error need not be exactly \(1/\sqrt{N}\). The derivation uses the standardized scales and a small-effect, large-sample approximation to make the connection transparent.

What Changes When Effects Do Not Have Equal Variance?

The derivation also shows exactly what equal effect variance buys. If effects remain independent and mean zero but have variant-specific variances \(v_k=\operatorname{Var}(\beta_k)\), then \[ E(\gamma_j^2)=\sum_{k=1}^{M}r_{jk}^{2}v_k. \] This is a weighted sum of squared LD correlations. It reduces to \((h_g^2/M)l_j\) only when the variances are equal (or when a suitable approximation makes them effectively equal for the variants contributing to the score). In general, ordinary LD score alone does not retain information about which correlated variants have larger effect variances.

Assumptions, Diagnostics, and Alternatives

The mean relationship is clearest when genotypes are consistently standardized, the LD correlations represent the analyzed population, effects are independent and similarly distributed across the modeled variants, and the association statistics are well calibrated. The quantitative-trait, one-degree-of-freedom approximation also matters: different phenotype scales or test definitions can require different sample-size or variance adjustments. In particular, using LD estimates from a population that differs from the analyzed sample can make the \(r_{jk}^2\) values poor stand-ins for the correlations driving the observed marginal associations.

Several practical checks follow directly from the derivation. Confirm that the LD score uses the intended variant set and includes the tested variant if that is the chosen convention. Check that genotype scaling and the definition of \(M\) agree with the assumed effect-variance model. Inspect whether the empirical mean chi-square values broadly increase with LD score as the model predicts, while remembering that noise, imperfect LD estimates, and model departures can obscure the pattern. A few extreme observations do not by themselves refute a relationship stated for conditional means.

The principal alternatives answer different questions. A direct calculation from a fully specified genetic-effect and LD model uses the weighted sum \(\sum_k r_{jk}^2v_k\), but requires the effect variances. The simple LD-score mean model replaces those unknown variant-specific variances with a common variance and summarizes LD by one number. A genome-wide inflation factor instead summarizes the distribution of association statistics across variants; it does not derive a variant-level expected mean from LD. These summaries are not interchangeable.

A common mistake is to read \(1+(N h_g^2/M)l_j\) as an exact prediction for every SNP, or to conclude that a high-LD-score SNP must be causal. The derivation supports neither claim. It specifies an approximate conditional expectation under a polygenic random-effects model; observed statistics vary around their expectation, and LD can transmit signal from other variants. Unequal effect variances, population differences in LD, or poor test calibration can also disrupt the simple relationship.