Interpreting The Cross-Trait LDSC Intercept

Why the Cross-Trait Intercept Needs Careful Interpretation

Cross-trait LD score regression (LDSC) relates the product of two traits’ association z-statistics at a variant to that variant’s LD score. The slope is the part that increases with LD score and, under the model’s assumptions, reflects genetic covariance. The intercept describes the fitted baseline: covariance in the association statistics that does not increase with LD score.

That baseline matters because the association statistics can covary for reasons other than shared genetic effects. For example, the two studies may include some of the same people, whose residual trait values are correlated. Population structure or other sources of confounding may also induce covariance in the statistics. The intercept can reflect these contributions, but its value alone generally cannot say which one produced them.

The Regression and Its Two Components

Let \(z_{1j}\) and \(z_{2j}\) be the association z-statistics for traits 1 and 2 at variant \(j\), with alleles aligned so that the signs refer to the same allele. Let \(l_j\) be the LD score for that variant. In the standard cross-trait LDSC model, the expected product is represented as

Here \(a_{12}\) is the intercept and \(b_{12}\) is the slope. In the usual scaling, with \(M\) variants in the reference set and suitably scaled traits and genotypes, the slope is proportional to genetic covariance:

In this expression, \(N_1\) and \(N_2\) are the sample sizes for the two traits, and \(\sigma_{g,12}\) is their additive genetic covariance on the model’s phenotype scale. Equivalently, if the traits’ genetic variances are \(h_1^2\) and \(h_2^2\), then \(\sigma_{g,12}=r_g\sqrt{h_1^2h_2^2}\) under the corresponding standardization. The intercept is not part of that genetic covariance slope: it is the component that the fitted line assigns to the baseline rather than to changing LD score.

The intercept is a model-based extrapolation to \(l_j=0\). Real variants in the analysis may all have positive LD scores, so the intercept need not equal the mean observed product \(z_{1j}z_{2j}\). Nor is it an individual variant’s measured covariance. It is a parameter of the fitted relationship across variants.

How Overlap Can Contribute

When the two studies share participants, the association statistics can be correlated even if there is no cross-trait genetic signal. If the residual values of the two traits tend to vary together among those shared participants, their contributions to the two association scores tend to move together as well. Under the standard approximation, the overlap contribution to the cross-trait intercept is

where \(N_s\) is the number of shared participants and \(\rho_e\) is their residual correlation after the relevant covariate adjustment and on the scale used by the association analyses. The result is dimensionless, like a product of z-statistics. It can be positive or negative depending on the direction of the residual correlation. This expression describes one possible contribution; it does not imply that the full fitted intercept equals this quantity.

What the Intercept Does—and Does Not—Separate

The intercept is useful because it allows the fitted regression to account for an approximately LD-independent component of cross-trait statistic covariance. It is tempting, however, to treat the intercept as a direct diagnostic of one particular mechanism. That interpretation is too strong. A fitted intercept can combine overlap, correlated confounding, and other sources of baseline covariance; estimation error also affects its reported value.

The distinction between intercept and slope is conditional on the model. If a source of confounding varies with LD score, its contribution may not be confined to the intercept. It can affect the fitted slope, which is why a nonzero intercept does not guarantee that the slope is free of bias. Conversely, an intercept near zero does not establish that every assumption needed for an unbiased slope holds.

The intercept is also not the genetic correlation. Genetic correlation is a standardized measure of genetic covariance, \(r_g=\sigma_{g,12}/\sqrt{h_1^2h_2^2}\), when the relevant genetic variances are positive. The LDSC intercept is on the scale of the product of association z-statistics. It does not have the same units or interpretation, and it should not be reported as a genetic correlation.

A Short Justification for the Overlap Term

The overlap expression follows from the shared participants’ contribution to the two association scores. To see the main scaling, consider a variant whose genotype has been standardized, and suppose the association-score noise for each trait is approximately a sum of genotype-residual products. For a participant in both studies, the expected product of the two residuals is \(\rho_e\). With standardized genotype variance one, the expected contribution of that shared participant to the cross-product of the two scores is therefore approximately \(\rho_e\).

Summing over \(N_s\) shared participants gives an expected score cross-product of approximately \(N_s\rho_e\). Each z-statistic divides its score by a scale proportional to the square root of its study sample size. Dividing the score cross-product by the product of those scales gives

This argument explains the approximate formula, rather than establishing it for every analysis design. It relies on the stated score scaling and assumptions about residuals, genotypes, and covariate adjustment. More complicated sampling or analysis choices can change the approximation.

Assumptions Behind an Intercept Interpretation

Interpreting the intercept as LD-independent baseline covariance, and the slope as genetic covariance, requires the regression model to be a reasonable description of the data. Important conditions include:

• Comparable variant definitions and allele orientation. The two z-statistics must refer to the same variant and consistently aligned alleles. Otherwise, products can be assigned the wrong sign or paired across variants incorrectly.
• Suitable LD scores. The LD reference should represent the analyzed population well enough for the scores to describe the relevant linkage disequilibrium. Poorly matched LD can distort the regression relationship.
• A suitable genetic architecture model. The relationship between expected cross-trait z-product and LD score must be adequately represented by the fitted line; in particular, the genetic effect covariance should not be systematically structured in a way the model omits.
• Confounding that is not strongly LD-score dependent. The intercept is intended to absorb baseline covariance. If confounding changes with LD score, it may contribute to the slope instead, so the intercept alone cannot protect the genetic covariance estimate.
• Appropriate association-statistic scaling. Sample sizes, phenotype scales, and any effective-sample-size conventions must be compatible with the interpretation of the slope and the overlap approximation.

These conditions are not a claim that all sources of bias have been removed. They clarify what the fitted intercept can mean within the model: a baseline component of the expected cross-trait statistic product, conditional on the regression being adequate.

Common Pitfall

Wrong reasoning: “The cross-trait intercept is positive, so the studies must overlap,” or “the intercept equals the amount of confounding.”

Correction: A positive intercept indicates positive fitted baseline covariance on the z-statistic product scale. Overlap with positively correlated residual traits is one possible contributor, and confounding can also contribute. The intercept alone does not identify or quantify those causes separately. It also does not show that any LD-dependent slope is unbiased; that depends on the assumptions governing how genetic signal and confounding relate to LD score.