Sample Overlap and LD Score Regression Intercepts

Why Sample Overlap Matters

Two genome-wide association studies can share participants even when they analyze different traits. For a given genetic variant, the association statistics from the two studies may then be correlated: the same person contributes information to both. This dependence matters when interpreting their joint association results, including the intercept from bivariate LD score regression (LDSC).

The key distinction is between an association analysis considered on its own and a comparison of association statistics across studies. Sample overlap does not, by itself, imply that the univariate LDSC intercept for either study must be elevated. In a bivariate analysis, however, overlap can contribute directly to the intercept through correlated sampling errors. The size and direction of this contribution depend on how many participants overlap and how their two phenotypes covary after relevant adjustment.

Definition: Sample overlap is the number of individuals who contribute to both association studies. In bivariate LDSC, the overlap contribution to the intercept is the component of the expected product of the two studies’ association statistics caused by shared individuals and their residual phenotypic covariance. Under the usual approximation, it is \(N_s \rho/\sqrt{N_1N_2}\), where \(N_s\) is the number of overlapping individuals, \(N_1\) and \(N_2\) are the study sample sizes, and \(\rho\) is the correlation between their appropriately adjusted, standardized phenotypes among overlapping individuals.

Where the Overlap Term Enters

For variant \(j\), let \(Z_{1j}\) and \(Z_{2j}\) be the signed association statistics for traits 1 and 2. In a standard bivariate LDSC mean model, their expected product is represented approximately as

E[Z_{1j}Z_{2j}] \approx \frac{\sqrt{N_1N_2}\rho_g}{M}l_j + a_{12}.

Here \(l_j\) is the LD score for variant \(j\), \(M\) is the number of variants represented in the model, \(\rho_g\) is the genetic covariance on the model’s scale, and \(a_{12}\) is the intercept. The slope describes the part of the expected product that changes with LD score. Under the overlap approximation, the component of the intercept due to shared sampling is

a_{12,\mathrm{overlap}} \approx \frac{N_s\rho}{\sqrt{N_1N_2}}.

This expression assumes that the association statistics are appropriately calibrated, the phenotypes are standardized on compatible scales, and the residual correlation \(\rho\) describes the phenotypic covariance relevant to the association tests after adjustment for modeled covariates. It is an approximation, not a universal identity: other sources of correlated errors can also contribute to the observed bivariate intercept.

The formula’s scale is informative. If the studies have equal sample size \(N\), the overlap term becomes \((N_s/N)\rho\). It is therefore larger when a greater fraction of the participants are shared or when the adjusted phenotypes are more strongly correlated. It can be positive or negative: the sign follows \(\rho\). With no overlap, this particular contribution is zero, although other sources of cross-study covariance may remain.

Short Derivation

A simple score-statistic argument explains the expression. Imagine that, under the null for a variant, each study’s standardized association statistic is approximately a sum of individual contributions divided by the square root of its sample size. Write those contributions as \(U_{1i}\) and \(U_{2i}\), with mean zero. Contributions from individuals observed in only one study do not create cross-study covariance. For each shared individual, suppose their contribution covariance is approximately \(\rho\), after genotype and phenotype standardization.

\operatorname{Cov}(Z_{1j},Z_{2j}) \approx \operatorname{Cov}\left( \frac{\sum_{i\in S_1}U_{1i}}{\sqrt{N_1}}, \frac{\sum_{k\in S_2}U_{2k}}{\sqrt{N_2}} \right) \approx \frac{\sum_{i\in S_1\cap S_2}\operatorname{Cov}(U_{1i},U_{2i})} {\sqrt{N_1N_2}} \approx \frac{N_s\rho}{\sqrt{N_1N_2}}.

The middle step uses independence across distinct individuals, so only matching shared individuals contribute to the covariance. This is the overlap-related covariance of the null association statistics, and it appears as an approximately LD-score-independent component in the bivariate mean model. Polygenic covariance contributes through the LD-score-dependent slope instead. The derivation relies on the stated standardization and covariance approximation; departures from them can alter the relationship.

Worked Example: Calculating an Overlap Contribution

Suppose two hypothetical studies analyze different standardized traits. Study 1 has \(N_1=100{,}000\) participants, study 2 has \(N_2=80{,}000\), and \(N_s=20{,}000\) participants are in both studies. Assume their adjusted phenotypes have residual correlation \(\rho=0.25\). Under the overlap approximation, calculate the intercept contribution.

First calculate the denominator:

\sqrt{N_1N_2} =\sqrt{100{,}000\times80{,}000} =\sqrt{8{,}000{,}000{,}000} \approx 89{,}443.

Then substitute into the overlap formula:

a_{12,\mathrm{overlap}} \approx \frac{20{,}000\times0.25}{89{,}443} =\frac{5{,}000}{89{,}443} \approx 0.0559.

The overlap component is approximately \(0.0559\) on the expected \(Z_{1j}Z_{2j}\) scale. This is not a claim that the fitted intercept must equal \(0.0559\): the fitted intercept can also reflect other sources of covariance and estimation uncertainty. The calculation assumes the sample sizes and overlap count are appropriate for the analyzed samples and that \(0.25\) is the relevant adjusted residual correlation.

What Overlap Does—and Does Not—Mean for an Intercept

It is essential to identify which intercept is being discussed. A univariate LDSC regression relates squared association statistics from one GWAS to LD score. The overlap formula above concerns the product of statistics from two studies. Sharing participants with an external GWAS does not automatically add this cross-study covariance term to the univariate regression for one study.

Overlap becomes directly relevant to a univariate analysis if it changes how that study’s own data were assembled—for example, if participants are counted more than once in a meta-analysis that treats overlapping cohorts as independent. In that case, the independence assumptions behind the combined standard errors and test statistics are violated. One should not expect the bivariate overlap formula to repair that analysis or to describe its univariate intercept.

Nor does a nonzero bivariate intercept uniquely identify sample overlap. Overlap with nearly uncorrelated adjusted phenotypes can contribute little, while correlated errors can occur for reasons other than shared participants. Accordingly, the overlap formula is best used to explain a predicted component under stated assumptions, not to infer overlap automatically from an observed intercept.

Worked Example: Substantial Overlap but a Small Contribution

Consider hypothetical studies with \(N_1=60{,}000\), \(N_2=50{,}000\), and \(N_s=30{,}000\). Although half or more of each study overlaps, suppose the adjusted residual phenotype correlation is only \(\rho=0.02\). Calculate the overlap contribution.

The denominator is

\sqrt{60{,}000\times50{,}000} =\sqrt{3{,}000{,}000{,}000} \approx54{,}772.

Therefore,

a_{12,\mathrm{overlap}} \approx\frac{30{,}000\times0.02}{54{,}772} =\frac{600}{54{,}772} \approx0.0110.

The predicted contribution is small despite extensive overlap because the adjusted phenotypes are only weakly correlated. This example assumes the residual correlation is correctly specified for the shared individuals. It illustrates why reporting an overlap count alone is insufficient to predict the intercept contribution.

Worked Example: Interpreting an Intercept Under an Explicit Assumption

Suppose two hypothetical studies have \(N_1=50{,}000\) and \(N_2=40{,}000\). A fitted bivariate LDSC intercept is \(0.040\), and an analyst estimates the relevant adjusted residual correlation as \(\rho=0.20\). If, purely for illustration, the entire intercept were attributed to sample overlap, what overlap count would the formula imply?

Rearrange \(a_{12,\mathrm{overlap}}\approx N_s\rho/\sqrt{N_1N_2}\) to obtain

N_s\approx \frac{a_{12,\mathrm{overlap}}\sqrt{N_1N_2}}{\rho}.

Substitution gives

N_s\approx \frac{0.040\sqrt{50{,}000\times40{,}000}}{0.20} = \frac{0.040\times44{,}721}{0.20} \approx8{,}944.

This is an algebraic implication of a deliberately strong assumption, not a reliable estimate of the actual overlap. If other sources contribute to the fitted intercept, or if the residual correlation is inaccurate, the inferred count can be wrong. Overlap should be established from cohort information where possible, rather than reverse-engineered from the intercept.

Diagnostics, Reporting, and Alternatives

A useful diagnostic begins with the study design, not the fitted intercept. Determine whether the same participants contributed to both traits, whether overlap occurred in all or only some cohorts, and whether any participants were duplicated within either meta-analysis. Compare the actual analyzed sample sizes with the reported totals; different missingness patterns can make the overlap count vary across traits or variants. For case-control studies, state how sample size was defined and how any effective sample size was calculated.

When overlap is expected, compare its predicted contribution with the fitted bivariate intercept, while recognizing that the comparison is approximate. A large discrepancy can prompt checks of overlap records, phenotype and covariate definitions, sample-size accounting, and summary-statistic harmonization. It is not, on its own, proof that the LDSC regression has failed. Inspect the intercept’s uncertainty and the stability of the fit as well: a point estimate without its standard error may suggest more certainty than the analysis supports.

When reporting, give readers enough information to assess the overlap term and its assumptions. At minimum, report:

the number of participants in each analyzed GWAS and the number shared, or a clear statement that overlap was absent or could not be quantified;
which cohorts contributed to each trait and whether the reported overlap is exact, estimated, or only partially known;
how sample sizes were defined, including relevant case-control or effective-sample-size conventions;
the phenotype and covariate context used to interpret the residual correlation, and its value if it was used to calculate an expected overlap contribution;
whether the result is a univariate or bivariate LDSC intercept, together with the estimate, standard error, and any relevant modeling choices.

The main alternative to relying on an intercept-based interpretation is to document overlap directly from cohort membership and study records. For a meta-analysis, the appropriate response to overlapping cohorts is to account for their dependence in the analysis rather than to treat them as independent studies. For a cross-trait LDSC result, reporting the expected overlap component alongside the fitted intercept can make the interpretation more transparent, provided the assumptions are stated.

Common mistake: “The bivariate intercept is positive, so the studies must have overlapping participants.” This conclusion does not follow. Overlap contributes approximately \(N_s\rho/\sqrt{N_1N_2}\) under the stated assumptions, but other sources of covariance can also affect the intercept. Conversely, overlap can make little contribution when the adjusted residual correlation is near zero. Report overlap from study records and describe the intercept as an estimate from the specified regression, not as a definitive overlap detector.

Key takeaway: In bivariate LDSC, shared participants can produce an intercept contribution proportional to both the overlap count and the adjusted residual phenotype correlation. Distinguish this cross-trait effect from a univariate intercept, and report overlap, sample-size definitions, and uncertainty explicitly.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

Two studies have \(N_1=90{,}000\), \(N_2=40{,}000\), \(N_s=10{,}000\), and adjusted residual correlation \(\rho=0.30\). What is the approximate overlap contribution to the bivariate intercept?
If two studies overlap substantially but their adjusted residual phenotype correlation is zero, what does the overlap formula predict for this contribution?
Does overlap between two GWAS automatically raise the univariate LDSC intercept of each GWAS? Explain the distinction.
Why is it not valid to infer an exact overlap count from a fitted bivariate intercept without additional assumptions?

Solutions. (1) The denominator is \(\sqrt{90{,}000\times40{,}000}=60{,}000\). Thus \(a_{12,\mathrm{overlap}}\approx(10{,}000\times0.30)/60{,}000=0.0500\). (2) The predicted contribution is zero because \(N_s\rho/\sqrt{N_1N_2}=0\) when \(\rho=0\), even though participants are shared. (3) No. The formula describes covariance between two studies’ statistics in a bivariate analysis; overlap with another study does not by itself add that term to a single study’s univariate regression. (4) The fitted intercept may include covariance from sources other than overlap, and the residual correlation, sample sizes, or overlap count may be uncertain. Inferring \(N_s\) requires treating the other quantities as known and assuming the intercept is entirely due to overlap.