Instrument Relevance And Weak-Instrument Bias

Why Relevance Is More Than a Formal Assumption

In Mendelian randomization, a genetic variant is used as an instrument to learn about the effect of an exposure \(X\) on an outcome \(Y\). The relevance assumption says that the variant predicts the exposure. But prediction can be very weak: the variant may be associated with \(X\) in the population while explaining only a small fraction of its variation. Such an instrument is relevant in principle, yet can yield unstable estimates and systematic finite-sample bias.

Let \(Z\) denote a genetic variant, \(X\) the exposure, and \(Y\) the outcome. Let \(\beta\) be the causal effect of \(X\) on \(Y\), expressed in outcome units per exposure unit. For a variant coded as 0, 1, or 2 effect alleles, a linear first-stage model might be \(X=a+\pi Z+V\), where \(\pi\) is the mean exposure difference per additional effect allele and \(V\) contains other exposure variation. Relevance means \(\pi\ne 0\) after accounting for the model’s covariates. The direction of the allele coding changes the sign of \(\pi\), not whether the variant predicts the exposure.

Relevance alone does not establish that a variant is a valid instrument. The three instrumental-variable assumptions include independence and exclusion restriction as well as relevance. The discussion below concerns the effects of weak prediction on estimation, conditional on the other assumptions being appropriate; it does not establish those assumptions. In particular, an apparent exposure association in a sample is not proof that the population association is nonzero or strong.

Why the Study Design Changes the Bias

In a one-sample design, the exposure and outcome are measured in the same people. In a two-sample design, exposure associations and outcome associations are estimated in separate samples. The difference matters because, in a one-sample analysis, sampling errors in the exposure association and outcome information can be related. With weak instruments, conventional instrumental-variable estimates can then be pulled toward the observational association between exposure and outcome. That observational association may be confounded and need not equal the causal effect.

In a two-sample design with independent samples, the error in an estimated genetic association with \(X\) is independent of the error in the corresponding association with \(Y\), under the usual sampling assumptions. This removes the same-sample error correlation that contributes to one-sample bias. A different effect remains: noisy estimated exposure associations enter both the numerator and denominator of ratio-based estimation. In the standard large-sample approximation, this extra denominator variation attenuates the estimate toward the null.

One-Sample Bias: Shared Sampling Noise

Consider the linear structural model \(Y=\beta X+U\), where \(U\) represents outcome causes not included in the model. Suppose the variant satisfies \(\operatorname{Cov}(Z,U)=0\), and the first stage is \(X=\pi Z+V\). If \(V\) is associated with \(U\), then exposure and outcome have an observational association beyond the causal effect. For example, a factor affecting both exposure and outcome can contribute to \(\operatorname{Cov}(V,U)\).

An instrumental-variable estimate uses the sample association between \(Z\) and \(Y\) divided by the sample association between \(Z\) and \(X\). The structural equation gives the identity

In the population, instrument independence makes the covariance in the second numerator zero. In a finite sample it is generally not exactly zero. When the first-stage association is weak, its sample estimate can be small and noisy, so the ratio can be strongly affected by sampling fluctuations. Because the exposure and outcome are observed in the same people, the errors in these sample associations are not generally independent when \(V\) and \(U\) are associated. Under the usual linear-model conditions, this dependence produces the familiar finite-sample tendency toward the observational exposure–outcome association.

This is a direction-of-bias result under specified conditions, not an exact formula for every sample or instrument. If the observational association exceeds the causal effect, the tendency is upward; if it is below the causal effect, the tendency is downward. If there is no confounding so that the observational association equals \(\beta\), the direction description does not imply a shift away from \(\beta\). Weak instruments can also produce estimates far from either quantity in an individual sample.

Two-Sample Bias: Noisy Exposure Associations in the Denominator

For variants \(j=1,\ldots,J\), let \(\gamma_j\) be the true association of variant \(j\) with \(X\), and let \(\Gamma_j=\beta\gamma_j\) be its association with \(Y\) when the variants satisfy the instrumental-variable assumptions and the common-effect model applies. Let \(\hat{\gamma}_j=\gamma_j+\delta_j\) and \(\hat{\Gamma}_j=\Gamma_j+\epsilon_j\) be estimates from the exposure and outcome samples. In a non-overlapping two-sample design, assume the samples are independent, so \(\delta_j\) and \(\epsilon_j\) are independent. For a simple inverse-variance weighted estimator with fixed weights \(w_j>0\),

The key feature is that \(\hat{\gamma}_j\), including its error, appears in both parts of the ratio. If the errors have mean zero, independence gives

The denominator’s expected value includes exposure-association error variance, while the numerator’s expected value does not include a matching term under sample independence. When the ratio is sufficiently stable for a ratio-of-expectations approximation, this yields

The fraction is between zero and one when the weights and variances are nonnegative and at least one exposure-association variance is positive. Thus a positive \(\beta\) is attenuated downward toward zero, while a negative \(\beta\) is attenuated upward toward zero. This justification is an approximation: the expectation of a ratio is not generally the ratio of expectations. Its usefulness depends on having enough information for the weighted denominator to be reasonably stable.

Conditions, Diagnostics, and Limits

The bias directions above depend on design and assumptions. The one-sample result concerns conventional instrumental-variable estimation with weak first-stage prediction and the usual finite-sample relationship between instrument and outcome sampling errors. The two-sample derivation assumes independent exposure and outcome samples, valid variants, correctly aligned association estimates, and a common causal effect for the displayed IVW model. Sample overlap can reintroduce correlated estimation errors; with partial overlap, bias may lie between the non-overlapping two-sample tendency and the one-sample tendency, depending on overlap and the design.

Weakness is not diagnosed by looking only at the estimated exposure effect. A small \(\hat{\gamma}_j\) might be precise in a very large sample, while a larger estimate can be uncertain in a smaller sample. Relevant diagnostics compare first-stage signal with its uncertainty and account for the number of instruments and analysis design. The later assessment of F statistics formalizes a common diagnostic; the key point here is that instrument strength is a property of the association and its precision, not simply the numerical size of the coefficient.

Warning signs include highly imprecise or unusually large ratio estimates, substantial sensitivity to small changes in exposure-association estimates, and a weak first stage relative to its sampling error. These symptoms do not by themselves show that weak instruments caused the problem: allele harmonization errors, invalid instruments, model misspecification, or other issues can also affect estimates. Conversely, a plausible point estimate does not demonstrate that instruments are strong.

The practical comparison is therefore not that one design removes all weak-instrument concerns. A one-sample analysis can be vulnerable to finite-sample movement toward the observational association; a non-overlapping two-sample analysis avoids that particular same-sample correlation but can still suffer denominator-driven attenuation and imprecision. Neither design makes an irrelevant variant useful, and neither bias direction verifies independence or exclusion restriction.