One-Sample And Two-Sample Mendelian Randomization

Why the Study Design Matters

Mendelian randomization (MR) uses genetic variants as instrumental variables to investigate the causal effect of an exposure on an outcome. The instrumental-variable assumptions—relevance, independence, and exclusion restriction—remain essential regardless of study design. But the way exposure and outcome data are collected affects which analyses are possible, how precisely the genetic associations are estimated, and how weak-instrument bias behaves.

The key design question is whether the genetic association with the exposure and the genetic association with the outcome are estimated in the same participants, separate participants, or partly overlapping participants. The designs are not interchangeable labels: each implies different data requirements and different relationships between estimation errors.

What Data Each Design Requires

In one-sample MR, the analysis dataset contains a genotype or genotypes, an exposure measurement, an outcome measurement, and any covariates used in the analysis. Individual-level data allow the analyst to model the exposure and outcome within the same participants and to inspect sample-specific features such as missingness, covariate distributions, and the first-stage relationship between genotype and exposure. The same participants must have the relevant measurements for the intended analysis; otherwise, the effective sample may differ between stages.

In two-sample MR, one study estimates the genetic association with the exposure, and another estimates the genetic association with the outcome. The analyst may work with association estimates and their standard errors rather than linked individual records. The two studies need to use compatible definitions and measurement scales for the exposure and outcome, and the genetic associations must refer to comparable variants and alleles. Allele alignment is a separate essential step; it is not solved merely by using two studies with the same variant labels.

Separate studies can make it possible to use a large exposure genome-wide association study (GWAS) and a different large outcome GWAS. They also mean that differences in ancestry, phenotype measurement, covariate adjustment, or sample selection can affect comparability. “Two samples” does not itself guarantee that the samples are comparable or that the instrumental-variable assumptions hold.

Instrument Strength Depends on the Relevant Sample

Instrument strength concerns how strongly the genetic variant predicts the exposure. The relevant question is not simply whether a variant was strongly associated with exposure somewhere; it is how well the exposure association is estimated in the data used for the analysis. A discovery study and an analysis study can differ in sample size, phenotype measurement, ancestry, and covariate adjustment, so the strength in one need not carry over exactly to the other.

For a single genetic predictor in a linear regression with an intercept, the usual first-stage \(F\) statistic is related to the regression \(R^2\) by

Here \(n\) is the first-stage sample size and \(R^2\) is the proportion of exposure variation explained by the genetic predictor in that sample. The relation follows because, in a regression with one predictor, the overall \(F\) statistic equals the squared \(t\) statistic for the predictor; the regression identity \(t^2 = R^2(n-2)/(1-R^2)\) gives the displayed expression. With multiple instruments, the first-stage model and degrees of freedom must reflect the instruments and other included predictors.

For one-sample MR, assess strength in the participants used to estimate the first stage. For two-sample MR, the exposure association is estimated in the exposure sample, so strength information should be based on that sample and the association estimates actually used. A large outcome sample can improve the precision of outcome associations, but it cannot by itself make a weak genetic predictor of exposure strong.

Why Overlap Changes Weak-Instrument Bias

Weak-instrument bias is the systematic distortion that can arise when genetic predictors explain little exposure variation. In conventional one-sample instrumental-variable estimation, weak-instrument bias tends toward the confounded observational association, under the usual setting in which the instrument is valid but weak and the observational association is confounded. The direction and size of the bias are not universal: they depend on the setting, the estimator, and the relationship between exposure and outcome.

In non-overlapping two-sample MR, exposure- and outcome-association estimation errors are approximately independent when the samples are independent. Under the usual two-sample conditions, weak-instrument bias for summary-association approaches tends toward the null. This is a tendency in expectation under those conditions, not a promise that any one estimate will be closer to zero.

With participant overlap, the two association estimates can have correlated sampling errors. The amount and direction of this correlation depend on the overlap and on how the exposure and outcome residuals covary after accounting for the genetic predictor and modeled covariates. As overlap increases, weak-instrument bias can move away from the non-overlapping two-sample pattern and toward the one-sample pattern. It is not correct to assume that every partially overlapping analysis has a predictable, fixed amount of bias: overlap alone does not determine the error covariance.

A Sampling-Error Justification

Let \(\hat{\gamma}\) denote an estimated genetic association with exposure and \(\hat{\delta}\) an estimated genetic association with outcome. Write their estimation errors as \(u\) and \(v\), so that \(\hat{\gamma}=\gamma+u\) and \(\hat{\delta}=\delta+v\), where \(\gamma\) and \(\delta\) are the corresponding underlying associations. This notation isolates the design issue: whether the errors \(u\) and \(v\) are related.

If the exposure and outcome associations are estimated in independent, non-overlapping samples, the sampling errors are approximately uncorrelated under the usual independent-sampling conditions. If participants overlap, some people contribute to both estimates, so their individual-level variation can induce nonzero covariance. In general, this can be expressed as

The important implication is not that overlap always produces a particular numerical bias, but that the independence of errors cannot be presumed. That dependence changes the sampling behavior of estimators built from both associations. The standard one-sample and non-overlapping two-sample weak-instrument tendencies provide useful contrasts; partial overlap can produce behavior between them, while other sources of bias or heterogeneity can alter the result.

Design Choice, Diagnostics, and Limitations

When choosing a design, consider both the causal assumptions and the data structure. One-sample MR offers direct control over a common analysis dataset and can make participant-level diagnostics possible. Its limitations include the need for all measurements in the same participants and the potential for weak-instrument bias toward the confounded observational association. Two-sample MR can combine large studies and needs no linked individual-level records, but depends on the comparability of the source studies and, for the non-overlapping weak-instrument interpretation, independent samples. Partial overlap must be treated as a feature of the design, not ignored.

Useful checks include confirming which participants contributed to each association estimate; documenting exposure and outcome sample sizes; checking whether the exposure association comes from the same or a different sample than the outcome association; and assessing first-stage strength in the exposure-association sample. Investigate differences in ancestry, phenotype definitions, measurement units, and covariate adjustment. These checks do not prove the instrumental-variable assumptions, but they can reveal design features that make an analysis difficult to interpret.

A practical comparison is:

None of these designs repairs a failure of instrument relevance, independence, or exclusion restriction. Nor does a strong first stage make studies comparable or rule out bias from sample selection. Design affects the interpretation and performance of an estimator; it does not replace substantive justification for the causal assumptions.

Common Mistakes

• Calling any two-GWAS analysis non-overlapping. Two studies can share participants. Check the sample composition rather than inferring independence from separate study names.
• Using outcome-sample size as evidence of exposure-instrument strength. Strength concerns prediction of the exposure and must be assessed where the exposure association is estimated.
• Assuming overlap has one fixed bias direction. Overlap can correlate estimation errors, but the direction and magnitude depend on the data and model.
• Treating a strong first stage as proof of validity. Strength addresses relevance; it does not establish independence or exclusion restriction.
• Assuming that non-overlap ensures comparability. Independent studies may still differ in ancestry, phenotype measurement, or other features relevant to interpretation.