Weak-Instrument Bias In Mendelian Randomization

Why Sample Structure Changes Weak-Instrument Bias

A genetic instrument can be valid in the population and still yield a biased estimate in a finite sample when its association with the exposure is weak. The direction of this bias depends in part on how the exposure and outcome associations are estimated. If the same participants contribute to both associations, random errors in those estimates can be correlated. If entirely separate samples are used, those errors are usually independent, and the dominant weak-instrument effect is different.

The comparison is about expected behavior across repeated samples, not a rule that predicts the direction of every observed estimate. A single weak-instrument estimate can be far from either its target or the usual direction of bias. The key quantities are the strength of the genetic associations with the exposure and the covariance between errors in the estimated exposure and outcome associations.

For this discussion, suppose a linear causal model has exposure \(X\), outcome \(Y\), and valid genetic instruments \(G_1,\ldots,G_m\). Let \(\theta\) be the causal effect of \(X\) on \(Y\), measured in outcome units per exposure unit. Write \(\gamma_j\) for the population association of variant \(j\) with \(X\), and \(\theta\gamma_j\) for its population association with \(Y\). Validity here means the variants are relevant, independent of causes of the exposure and outcome, and affect the outcome only through the exposure. We also assume, for the basic comparison, that the variants are independent or that their dependence has been handled correctly, and that the association estimates are on compatible scales.

One-Sample Bias: The Pull Toward the Observational Association

In one-sample Mendelian randomization, the same participants are used to estimate variant–exposure and variant–outcome associations. In individual-level two-stage least squares, the first stage predicts exposure from the instruments, and the second stage relates that predicted exposure to the outcome. Because the first-stage predictions are estimated in the same data used in the second stage, they can retain accidental sample relationships with the outcome-model residuals. When the instruments explain little exposure variation, this finite-sample effect is more influential.

Under the usual linear instrumental-variable model, with valid instruments and appropriate model specification, the finite-sample bias of two-stage least squares tends toward the ordinary least-squares association between \(X\) and \(Y\), rather than toward the true causal effect. That ordinary association can be confounded. The tendency is not a claim that every one-sample estimate lies between the causal effect and the observational association; sampling variation can place it elsewhere. Nor does it mean that using one sample automatically invalidates an analysis: strong instruments reduce this problem, while weak instruments make it more consequential.

Two-Sample Bias: Attenuation Without Overlap

In two-sample Mendelian randomization, variant–exposure associations are estimated in one sample and variant–outcome associations in another. When the samples do not overlap, and the samples are otherwise independent, errors in the two sets of association estimates are approximately uncorrelated. The exposure associations in the denominator of a Wald ratio or inverse-variance weighted estimate are nevertheless noisy. Treating these noisy estimates as if they were exact generally dilutes the estimated relationship between variant–exposure and variant–outcome associations.

The following derivation makes the direction of this effect explicit for an inverse-variance weighted (IVW) slope through the origin. It is an approximation to the repeated-sampling behavior of the estimator: it uses the ratio of expected weighted sums, which is informative when many variants contribute and those sums are reasonably concentrated. It is not an exact formula for the expectation of every finite-sample ratio.

To see why, expand the numerator for each variant:

where the mean-zero errors have zero expected cross-products with the fixed \(\gamma_j\). Expanding the denominator gives

With nonoverlapping samples, the covariance term is approximately zero. The denominator still includes exposure-association error variance, so the causal signal in the numerator is divided by a larger quantity. For a positive causal effect this pulls the approximation toward zero; for a negative effect it pulls it upward toward zero. This is the errors-in-variables, or regression-dilution, explanation for the usual two-sample attenuation.

Sample Overlap: Correlated Errors Change the Direction

With overlap, some participants contribute to both association estimates. If the exposure and outcome residuals are correlated in those participants, the errors \(e_{Xj}\) and \(e_{Yj}\) can also be correlated. The covariance term in the derivation is then no longer zero. When its sign and size align with the confounded observational association, it can pull the IVW estimate away from zero and toward that association. Partial overlap often produces behavior between the nonoverlapping two-sample pattern and the one-sample pattern, but the amount and direction are not determined by the overlap fraction alone: they also depend on residual covariance, sample sizes, instrument strengths, and the analysis design.

The term “toward the observational association” describes a common pattern under standard conditions, not a universal law. If the residual covariance has the opposite sign, overlap can instead reduce attenuation or move the estimate in another direction. Differences in ancestry, measurement, or model specification between samples can also complicate the simple comparison.

What to Check in Practice

Weak-instrument bias is a design and interpretation issue, not something that can be resolved by comparing a single result with a preferred value. Check how the samples were assembled and whether participants overlap, including overlap through related cohorts or shared controls. Assess how precisely the variants predict the exposure, and whether estimates come from comparable populations and measurement scales. If an observational exposure–outcome association is available, it can help contextualize the expected direction of one-sample bias, but closeness to that association is not proof of bias.

A useful design comparison is:

Where feasible, independent exposure and outcome samples reduce the correlated-error component, although weak exposure associations can still cause attenuation. A separate sample is not automatically a cure: it must provide sufficiently informative exposure associations and a population relevant to the outcome sample. Conversely, the one-sample design is not inherently unusable, but weak instruments warrant attention to finite-sample behavior. The balance between sample size, instrument strength, and overlap matters.

Common Mistakes and Limitations

• “Two-sample MR has no weak-instrument bias.” Independence removes the overlap covariance under the idealized conditions, not the exposure-association measurement error that produces attenuation.
• “Overlap always makes the estimate more biased in the same direction.” Overlap can correlate errors, but the covariance sign and size depend on the data-generating process. The pull toward the observational association is a common result under standard conditions, not an unconditional guarantee.
• “An estimate close to the observational association proves one-sample bias.” Sampling variation, confounding, model differences, or other problems can also produce that result. Directional expectations do not diagnose a particular estimate on their own.
• “The overlap percentage determines the bias.” The numerical illustration assumed covariance proportional to overlap for simplicity. Real covariance also depends on residual relationships, sample sizes, and how associations were estimated.

The approximation derived here is deliberately limited: it treats weights as fixed, assumes mean-zero association errors, and uses a ratio of expected sums as an approximation. More complicated designs may require explicit modeling of the joint estimation errors. Weak instruments can also make ratios unstable, especially when an estimated exposure association is near zero. These limitations are reasons to interpret estimates with their design and uncertainty in view, rather than to infer that a particular bias correction is automatically appropriate.