Sample Overlap In Cross-Trait Summary Analyses

Why Sample Overlap Matters

Cross-trait summary analyses compare association statistics for two traits across many genetic variants. If the studies include some of the same participants, the two statistics can be correlated even when the variant has no causal effect on either trait. The shared participants contribute to both statistics, and correlated trait values in those people create a shared sampling fluctuation.

This overlap-induced covariance is distinct from covariance due to shared genetic effects. In cross-trait LD score regression (LDSC), the central diagnostic is how the expected product of the two association statistics varies with LD score: shared polygenic effects contribute an LD-dependent component, whereas overlap often contributes to the approximately LD-independent intercept. The distinction is useful, but not absolute: the intercept can also reflect other sources of confounding, and overlap information alone does not establish genetic sharing.

The approximation assumes that the traits and genotypes have been handled on compatible scales, that association statistics are appropriately standardized, and that the relevant overlap participants have a common trait correlation. Real analyses may require adjustments for covariates, case-control sampling, or other design features. The formula is a useful baseline, not a substitute for checking how the studies and statistics were constructed.

Deriving the Overlap Contribution

Consider one variant and, for clarity, suppose each trait is measured on a standardized scale. Let \(g_i\) denote the centered, variance-one genotype for participant \(i\), and let \(u_{1i}\) and \(u_{2i}\) denote the standardized residual values for the two traits under a null model for this variant. In a simple score-test approximation, the association statistics have the form

Here \(S_1\) and \(S_2\) are the two study samples. Suppose participants are independent of one another, genotype is independent of the residual traits under the tested-variant null, and the residual correlation for an overlapping participant is \(\rho_o\). Terms from different participants have zero covariance under these assumptions. The only shared terms are from the \(N_s\) overlapping participants. For each such participant, the covariance of the two score contributions is

because \(E(g_i^2)=1\) and the residual correlation is \(\rho_o\). Summing the \(N_s\) shared contributions and applying the two standardizing factors gives

The sign follows the trait correlation among overlapping participants. Positive correlation produces positive overlap covariance; negative correlation produces negative covariance. The term is dimensionless, like the product of two \(z\)-statistics, and it is not itself a genetic correlation.

How Overlap Appears in Cross-Trait LDSC

For variant \(j\), let \(z_{1j}\) and \(z_{2j}\) be its association statistics in the two studies, and let \(\ell_j\) be its LD score, defined as the sum of squared correlations between that variant and the variants included in the analysis. Under the standard cross-trait LDSC approximation, the expected product of the association statistics can be written as

Here \(M\) is the number of variants in the genetic-architecture model, and \(C_g\) is the total additive genetic covariance of the two standardized traits on the scale used by the model. The first term is the genetic component: its expected size changes with LD score. The second is the overlap contribution: in the simple model, it does not depend on the tested variant’s LD score. In practice, the fitted intercept can also absorb other approximately LD-independent contributions, such as certain forms of confounding.

This equation explains why overlap can matter even if the scientific target is genetic covariance. If it is ignored, an upward or downward shift in the association-statistic products can be mistaken for a genetic pattern, particularly when the analysis or its assumptions do not adequately separate the intercept from the slope. LDSC uses variation in LD score to estimate the LD-dependent slope and the intercept; it does not identify the cause of every intercept contribution.

Using the Intercept as a Diagnostic

If the number of overlapping participants and their trait correlation are known, the formula provides a predicted overlap contribution to compare with the fitted cross-trait LDSC intercept. Agreement is consistent with overlap explaining much of the intercept under the assumed model. A discrepancy is a reason to investigate the assumptions and other sources of intercept inflation; it does not by itself prove that the overlap count or the LDSC analysis is wrong.

Conversely, an intercept can sometimes be used to make a rough overlap calculation if the overlap trait correlation is known and other intercept sources are negligible. Rearranging the formula gives

where \(I\) is the intercept contribution being attributed to overlap. This inversion is unstable when \(\rho_o\) is close to zero, and it is not justified when the intercept also contains meaningful confounding or other contributions. An intercept is therefore a diagnostic, not a definitive census of shared participants.

Assumptions, Interpretation, and Common Pitfalls

The simple calculation is most direct when the association statistics are comparably standardized, the sample sizes are correctly represented, and \(\rho_o\) describes the relevant trait correlation among the overlapping participants. If the shared participants differ from the full studies, a single correlation may be an imperfect summary. Covariate adjustment, relatedness, ascertainment, or different phenotype scales can also make the simple approximation less exact. In applied work, use overlap counts and trait-correlation estimates that match the analysis design, and interpret the LDSC intercept in light of other plausible sources of covariance.

A key distinction is between the correlation of observed trait values and genetic covariance. Correlated observed traits among shared participants can induce correlated sampling errors. Genetic covariance, by contrast, concerns shared additive genetic effects and is represented by the LD-dependent component in the cross-trait LDSC model. The two components can have different magnitudes or even opposite signs.

• Mistake: “A positive intercept proves shared genetic effects.” The intercept can reflect sample overlap and other approximately LD-independent effects. The model associates shared polygenic effects with the LD-dependent slope, not with an intercept alone.
• Mistake: “No overlap means no covariance between the statistics.” No overlap removes this particular shared-participant contribution under the simple model. It does not remove covariance caused by shared genetic effects or other sources that affect both studies.
• Mistake: “The overlap formula gives the genetic correlation.” It gives an approximate sampling-covariance contribution to the association statistics. Genetic correlation is a distinct parameter based on genetic variances and genetic covariance.
• Mistake: “The fitted intercept exactly equals the overlap term.” That equality requires the overlap model to be adequate and other intercept contributions to be negligible. Treat the comparison as a diagnostic rather than an identity guaranteed by the data.