Weak-Instrument Bias Under Sample Overlap

Why Sample Overlap Changes Weak-Instrument Bias

Two-sample Mendelian randomization (MR) often estimates genetic associations with an exposure in one sample and associations with an outcome in another. When the samples are independent, sampling errors in those two sets of associations are approximately uncorrelated. If the samples share participants, however, the same individual-level exposure and outcome variation can contribute to both association estimates. Their estimation errors may then be correlated.

This matters most when instruments are weak. A weak genetic association with the exposure means that sampling error is large relative to the signal. With no overlap, that error can pull an aggregate MR estimate toward the null. With partial overlap, correlated errors can instead pull it toward the exposure–outcome association seen in the overlapping participants, which may reflect confounding as well as any causal effect. Neither direction is guaranteed in every dataset: it depends on the direction of the residual exposure–outcome association and on how much overlap and instrument information there is.

The approximations below are diagnostic, not exact finite-sample laws. They describe a linear, common-effect setting with valid instruments and a sufficiently stable aggregate first stage. They are useful for understanding bias direction and comparing plausible scenarios; they do not recover an unknown bias exactly from overlap percentage alone.

Association Errors and Their Covariance

To see the source of that covariance, consider a simple setting in which the same genotype is used in both association analyses, covariates have been accounted for, and residuals are homoskedastic. Let \(N_X\) and \(N_Y\) be the exposure and outcome sample sizes, \(n_O\) the number of shared participants, and \(v_G\) the genotype variance. Let \(r_X\) and \(r_Y\) be the exposure and outcome residuals after accounting for genotype and covariates, with residual variances \(\sigma_X^2\) and covariance \(\sigma_{XY}\). Approximate sampling variances and covariance are then

The covariance comes from the shared observations: each contributes to both association estimates. If there is no overlap, this component is zero. If residual exposure and outcome values are positively associated among the overlapping participants, the association errors tend to move together; if they are negatively associated, they tend to move in opposite directions. In a confounded observational setting, residual association can reflect confounding, but it is not automatically equal to a causal effect or to the overall observational slope.

For equal sample sizes and a common overlap fraction \(q=n_O/N_Y\), the covariance relative to the exposure-association error variance is approximately

The ratio \(\sigma_{XY}/\sigma_X^2\) is the residual outcome-on-exposure slope in this simplified setting. Thus, overlap scales the residual association that can leak into the genetic estimate. With unequal sample sizes, the relevant fraction in this expression is \(n_O/N_Y\), not necessarily the fraction of the exposure sample that overlaps.

A First-Order Aggregate Bias Approximation

For independent variants with fixed inverse-variance weights \(w_j\), an IVW-style estimate is a ratio with numerator \(\sum_j w_j\hat{\gamma}_j\hat{\Gamma}_j\) and denominator \(\sum_jw_j\hat{\gamma}_j^2\). Define the true weighted first-stage signal, weighted exposure-error variance, and weighted cross-error covariance as

These quantities give a useful approximation when the aggregate denominator is concentrated around its mean. Taking expectations of the numerator and denominator separately gives \(\beta S+C\) and \(S+V\), respectively. The ratio of these expectations is not generally the exact expectation of a random ratio. It is a first-order diagnostic, most useful when the aggregate first stage is sufficiently stable for this approximation to be informative.

When \(V>0\), define the effective first-stage strength \(F_{\mathrm{eff}}=S/V\) and the overlap-associated slope \(b_{\mathrm{ov}}=C/V\). Substitution gives a compact expression for the approximate bias:

This expression captures two central features. First, as \(F_{\mathrm{eff}}\) increases, the approximate bias shrinks. Second, the direction depends on whether \(b_{\mathrm{ov}}\) lies above or below \(\beta\). Under the equal-sample, common-overlap simplification, \(b_{\mathrm{ov}}\approx q\,\sigma_{XY}/\sigma_X^2\). In the absence of overlap, \(C=0\) and \(b_{\mathrm{ov}}=0\), so the approximation gives attenuation toward zero. With overlap, the bias can move toward a nonzero residual observational slope. This is a tendency, not a guarantee that the estimate lands between the causal and observational slopes in every setting.

Worked Examples

Assumptions and Sensitivity Analysis

The approximation is informative only when its ingredients have a sensible interpretation. In particular, it assumes a linear exposure–outcome relation with a common causal effect, valid instruments (no direct genetic effect on the outcome outside the exposure pathway), and association estimates that are comparable across the two samples. It also relies on a suitable aggregate measure of first-stage information and on a reasonable approximation to the covariance of association errors. Population structure, relatedness, heterogeneous effects, correlated variants, and different covariate adjustments can complicate these quantities. If variants are correlated, the weights and sums must account for that dependence rather than treating variants as independent.

A useful sensitivity analysis varies overlap and instrument strength separately rather than treating a single calculation as definitive. For example, an analyst can calculate approximate bias across a plausible range of \(q\) values and \(F_{\mathrm{eff}}\) values, while showing the residual slope assumed in each scenario. A simple comparison might look like this:

This comparison highlights an important qualification: overlap is not a bias magnitude by itself. If the residual exposure–outcome covariance is near zero, the overlap-related covariance term can be small even when participants are shared. Conversely, an appreciable residual association combined with weak instruments can make the bias consequential. Sample sizes and overlap counts help describe the setting, but they do not reveal the residual covariance without additional information or assumptions.

When possible, study design can reduce uncertainty by using nonoverlapping exposure and outcome samples or by reporting enough information to estimate the covariance of association errors. If overlap cannot be avoided, document the number of shared participants, sample sizes, phenotype definitions, covariate adjustments, and the basis for any residual-association assumptions. Report sensitivity results as scenario-dependent diagnostics, not as proof that a particular bias has been removed.

Common Pitfall

• Wrong reasoning: “The exposure and outcome samples overlap, so the MR estimate must be biased toward the observational association by a known amount.”
• Correction: Overlap can correlate association errors, but the direction depends on the residual exposure–outcome covariance and the size of the causal effect; the magnitude also depends on aggregate instrument strength and the overlap pattern. Overlap alone does not identify the bias.
• Another common error: Treating the ratio-of-expectations approximation as the exact expectation of a weak-instrument ratio estimator. It is not. If the aggregate denominator is unstable, this approximation can be poor, and a scenario calculation should not be presented as an exact finite-sample correction.