Meta-Analysis With Overlapping Cohorts

Why Cohort Overlap Changes Meta-Analysis

A meta-analysis often combines genetic association estimates from several cohorts. In the standard inverse-variance calculation, the estimates are treated as independent: the variance of a weighted sum is calculated without covariance terms. That assumption fails when cohorts share participants. A participant appearing in two analyses can contribute to both estimates, making their sampling errors correlated even when the association results are reported in separate files.

The consequence is not merely that the reported sample sizes appear larger than the number of distinct people. The central inferential issue is that the usual pooled standard error is based on an incorrect variance calculation. Treating positively correlated estimates as independent generally makes the pooled result look more precise than it is. The remedy is to account for the covariance, or to construct analyses on disjoint participant sets so that the relevant estimates are independent under the study’s sampling assumptions.

The discussion here concerns pooling estimates of one common genetic effect on a harmonized scale, as in a fixed-effect analysis. That requires the estimates to target the same effect and to be unbiased, or approximately unbiased, for that target. It also requires suitable estimates of their variances and covariances. Shared participants are one source of dependence; the covariance must reflect the actual analyses rather than be inferred mechanically from overlap counts alone.

Two Estimates: The Covariance Term

Write the two cohort estimates as \(\hat{\beta}_1\) and \(\hat{\beta}_2\), with variances \(v_1\) and \(v_2\), and covariance \(c\). For weights \(w_1,w_2\) summing to one, the pooled estimate is \(\hat{\beta}=w_1\hat{\beta}_1+w_2\hat{\beta}_2\). Its variance includes a cross-product term:

If \(c=0\), this reduces to the familiar independent-estimates calculation. If \(c>0\), the cross-product increases the variance for positive weights. Shared participants often produce positive covariance when the phenotype and effect allele are aligned, but the appropriate covariance depends on the analysis and should be estimated or derived for that setting. It should not be assumed to equal a simple fraction of either cohort’s variance.

Deriving the Covariance-Aware Estimator

The two-estimate formulas are a special case of a general result. Let \(\hat{\boldsymbol{\beta}}\) be the vector of \(K\) cohort estimates, let \(\mathbf{1}\) be a vector of \(K\) ones, and assume \(E(\hat{\boldsymbol{\beta}})=\theta\mathbf{1}\) and \(\operatorname{Var}(\hat{\boldsymbol{\beta}})=\Sigma\). A linear pooled estimate \(w^\mathsf{T}\hat{\boldsymbol{\beta}}\) is unbiased for the common effect \(\theta\) when \(w^\mathsf{T}\mathbf{1}=1\). Its variance is \(w^\mathsf{T}\Sigma w\).

To find the unbiased linear estimator with minimum variance, minimize \(w^\mathsf{T}\Sigma w\) subject to \(w^\mathsf{T}\mathbf{1}=1\). Using a Lagrange multiplier \(\lambda\), differentiate \(w^\mathsf{T}\Sigma w-\lambda(w^\mathsf{T}\mathbf{1}-1)\) with respect to \(w\). For symmetric positive-definite \(\Sigma\), setting the derivative to zero gives \(2\Sigma w-\lambda\mathbf{1}=0\), so \(w=(\lambda/2)\Sigma^{-1}\mathbf{1}\). Applying the constraint determines the constant and hence the weights:

For the variance expression, substitute the weight vector into \(w^\mathsf{T}\Sigma w\). The numerator becomes \(\mathbf{1}^\mathsf{T}\Sigma^{-1}\Sigma\Sigma^{-1}\mathbf{1}=\mathbf{1}^\mathsf{T}\Sigma^{-1}\mathbf{1}\), while the squared denominator is \((\mathbf{1}^\mathsf{T}\Sigma^{-1}\mathbf{1})^2\). Their ratio is the reciprocal shown above. These formulas require an invertible covariance matrix; inference also typically uses an approximate normal distribution for the estimates and a suitably estimated \(\Sigma\).

Where the Covariance Comes From

A useful simplified illustration comes from two sample means calculated on overlapping sets of independent, identically distributed observations with individual variance \(\sigma^2\). If the sets have sizes \(n_1\) and \(n_2\), and share \(m\) observations, then each shared observation contributes to both sample means. Summing the shared terms in the covariance gives

This expression illustrates why overlap induces covariance, but it is not a universal GWAS formula. Association estimates are regression coefficients, and their covariance can depend on genotypes, phenotypes, covariate adjustment, relatedness, and the estimation procedure. In practice, covariance may be derived from a joint model, estimated using participant-level information or appropriate influence-function methods, or obtained through a carefully designed nonoverlapping analysis. Merely knowing the number of shared people may not be enough to calculate it.

Using Nonoverlapping Participant Sets

When a reliable covariance estimate is unavailable, one alternative is to re-run association analyses on participant sets that do not overlap. This can make the resulting estimates independent with respect to participant sampling, provided the remaining samples are independent and there are no other relevant sources of dependence. The price is reduced information: removing shared participants can increase standard errors. The analysis must also preserve the intended target and apply consistent variant, allele, phenotype, and covariate definitions.

A nonoverlap strategy must be implemented from data or summary components that allow the estimates to be recomputed. One cannot generally recover a valid estimate for a cohort’s unique participants by simply subtracting one published regression coefficient from another. Regression estimates are not participant-level sums in that way.

Common Mistakes and Practical Checks

• Using independent weights and only changing the final standard error. The covariance can change both the pooled variance and the optimal weights. Use the covariance matrix in the pooling calculation itself.
• Counting the same participants twice as independent information. The estimates may each be valid, but their shared sampling errors mean the combined information is less than an independence calculation suggests.
• Equating the overlap fraction with the correlation. Overlap counts alone do not generally determine the correlation between GWAS regression estimates. The phenotype and analysis details matter.
• Combining estimates on incompatible scales or allele orientations. Covariance adjustment does not repair differences in effect-allele coding, phenotype units, or target effect. Those must be aligned before pooling.
• Ignoring whether the covariance matrix is valid. A covariance matrix must be symmetric and positive semidefinite. For two estimates this requires \(v_1\geq0\), \(v_2\geq0\), and \(v_1v_2-c^2\geq0\). An invalid matrix can signal an estimation or implementation problem.

Overlap-aware analysis matters because meta-analysis is used to gain precision, and its claimed precision is only credible when dependence is represented correctly. Report how overlap was handled, how the covariance was obtained or why independence was plausible, and whether estimates were reanalyzed on disjoint sets. For the analysis, a covariance-aware model can retain information that would be discarded by removing participants; a nonoverlapping approach can be easier to justify when the necessary covariance information is unavailable.