Designing A Two-Sample Mendelian Randomization Study

Why the Two Samples Must Be Designed Together

In two-sample Mendelian randomization (MR), genetic variants are related to an exposure in one sample and to an outcome in another. The aim is to combine those genetic association estimates to learn about the causal effect of the exposure on the outcome. Having two samples can make large genetic association data available for both traits, but it does not make the design automatically valid. The samples must measure sufficiently compatible traits, provide association estimates for the same instruments, and support the assumptions needed for a causal interpretation.

A design should therefore be specified before the estimates are combined. Investigators need to state who contributes to each sample, how the exposure and outcome are measured, which variants are considered instruments, and how each reported association is defined. These details determine what effect is being estimated and whether the exposure and outcome estimates can sensibly be paired.

Definition: A two-sample MR design uses genetic association estimates for the exposure from one sample and genetic association estimates for the outcome from another sample. For variant \(j\), write \(\hat{\beta}_{Xj}\) for its estimated association with the exposure and \(\hat{\beta}_{Yj}\) for its estimated association with the outcome. The corresponding standard errors are \(SE(\hat{\beta}_{Xj})\) and \(SE(\hat{\beta}_{Yj})\). The two samples may differ in participants, but the estimates must refer to compatible variants and clearly specified traits.

Specify the Association Estimates

For a common additive coding, let \(G_j\) be the number of copies of a specified effect allele at variant \(j\), taking values 0, 1, or 2. In the exposure sample, an association model for a continuous exposure \(X\) might be written as

E[X\mid G_j,C]=a_{Xj}+\beta_{Xj}G_j+\gamma_X^{\mathsf T}C,

where \(C\) denotes the covariates included in the association model. Here, \(\beta_{Xj}\) is the expected exposure difference per additional copy of the specified allele, conditional on those covariates. The outcome sample supplies the analogous association estimate \(\hat{\beta}_{Yj}\), with its own outcome units and model. An estimate is not fully described by its numerical value alone: its allele coding, phenotype definition, units, covariate adjustment, sample size, and standard error all matter.

For example, if the exposure is measured in standard deviation units and the outcome in mm Hg, then \(\hat{\beta}_{Xj}=0.20\) means a 0.20-standard-deviation exposure difference per additional allele copy, while \(\hat{\beta}_{Yj}=1.5\) means a 1.5 mm Hg outcome difference per copy. The ratio of the two estimates has units of mm Hg per standard deviation of exposure. If one sample reports an estimate per allele copy and the other reports an estimate for a differently defined allele, the estimates cannot simply be paired without resolving the coding. Detailed allele alignment is a separate step; the design must at least preserve enough information to perform it correctly.

For multiple variants, investigators should also document whether the variants are treated as approximately independent. Correlated variants can make association estimates statistically dependent. An analysis that treats correlated estimates as independent needs justification; alternatively, the design and analysis can account for their covariance. The set of variants and their identifiers should be fixed and recorded consistently across the two association sources.

Worked Example: Writing Down a Compatible Pair of Estimates

Suppose a hypothetical exposure GWAS reports an estimate for one variant of \(\hat{\beta}_{Xj}=0.40\) exposure standard deviations per effect-allele copy, with \(SE(\hat{\beta}_{Xj})=0.08\). A separate hypothetical outcome GWAS reports \(\hat{\beta}_{Yj}=0.12\) outcome standard deviations per copy of the same specified allele, with \(SE(\hat{\beta}_{Yj})=0.04\). Assume the two phenotype definitions and allele coding have been confirmed as compatible.

The single-variant ratio is \(\hat{\beta}_{Yj}/\hat{\beta}_{Xj}=0.12/0.40=0.30\). Its units are outcome standard deviations per exposure standard deviation. This calculation illustrates why the units and allele definition must be specified in both samples: without them, even the meaning of the ratio would be unclear. It is not, by itself, evidence that the causal assumptions hold.

A design record would retain both estimates and standard errors, the sample sizes, the trait measurement details, the covariates used, the variant identifier, and the allele to which each estimate refers. A full MR analysis would then combine the available variant-specific information using an appropriate estimator.

Compatibility of Ancestry and Phenotypes

The exposure and outcome samples need not be identical, but differences between them should not be ignored. A basic design question is whether each variant’s association refers to a comparable genetic contrast in both samples. Ancestry composition matters because allele frequencies and patterns of linkage disequilibrium can differ across populations. If a reported variant is a proxy for a causal variant in one sample, but tags it differently in the other, the two association estimates may not represent the intended same genetic contrast. Directly using the same variant in both sources and documenting ancestry and variant definitions helps make this issue assessable; it does not guarantee that all genetic effects are identical across populations.

Phenotype compatibility is equally important. The exposure GWAS should measure the exposure that defines the causal question, and the outcome GWAS should measure the specified outcome. A diagnosis, a laboratory measurement, and a self-reported symptom are not interchangeable simply because they concern a related condition. Differences in measurement instruments, thresholds, timing, or units can change what an association estimate represents. When measures differ, investigators should explain why they are sufficiently comparable for the intended question, or state that the resulting estimand concerns the particular operational definitions used.

Covariate adjustment and sample characteristics should be reported for each association source. Adjusted estimates do not necessarily become comparable merely because both models include some covariates; the covariates, sample inclusion criteria, and phenotype construction still need to be understood. The goal is not to force the two GWAS models to be identical, but to establish that their estimates can be interpreted as associations with the intended exposure and outcome in populations relevant to the causal question.

Worked Example: Detecting an Ancestry and Measurement Mismatch

Consider a hypothetical study whose exposure estimate comes from a GWAS of adults from one ancestry group, using a measured biomarker in mmol/L. The outcome estimate comes from a multi-ancestry GWAS of a related but not identical biomarker, recorded as a clinical diagnosis. For one variant, the reported estimates are \(\hat{\beta}_{Xj}=0.25\) mmol/L per effect-allele copy and \(\hat{\beta}_{Yj}=0.08\) log-odds units per copy.

The arithmetic ratio is \(0.08/0.25=0.32\) log-odds units per mmol/L. That arithmetic is correct, but it does not resolve the design concerns. The exposure is a continuous measurement, whereas the outcome is a diagnosis; the population composition also differs. The investigator should not describe this result as the effect of the measured biomarker on the same measured biomarker, nor assume without argument that the genetic associations transport identically across the populations.

Before proceeding, the study should define the causal question in terms of these actual phenotypes, explain the rationale for treating the diagnosis as the outcome of interest, and assess whether the variant associations are sufficiently comparable for that question. If the mismatch cannot be justified, a more compatible outcome source or a narrower interpretation is needed. Reporting the ratio alone would hide the main limitation.

Independence Conditions for Causal Interpretation

A two-sample layout describes where the association estimates came from; it does not establish that the genetic instruments are valid. For a causal interpretation, the usual instrumental-variable assumptions are required. In brief, the variants must be associated with the exposure, independent of factors that confound the exposure–outcome relationship, and related to the outcome only through the exposure being studied. The last condition is the exclusion restriction. These are assumptions about the genetic instruments and the causal system, not conclusions that can be guaranteed by choosing separate datasets.

The independence assumption is particularly important in study planning. If a variant is associated with a confounder of the exposure and outcome, its association with the outcome may not reflect a pathway through the exposure alone. Ancestry composition, for example, should be described and considered when selecting and analyzing association sources; the details of population-structure adjustment require careful treatment. Likewise, a plausible biological story is not a substitute for the formal assumptions. The study should state why the selected variants are intended to serve as instruments and what limitations remain.

There are two distinct uses of “independence” to keep separate. The instrumental-variable independence assumption concerns the relationship between the instruments and confounders. Statistical independence between the exposure and outcome association estimates is a separate sampling issue. Using non-overlapping participant samples is a straightforward design target for a conventional two-sample analysis. If participants overlap, the association estimates may be correlated, so an analysis that assumes independent estimation errors may need reconsideration. The amount and source of any overlap should therefore be recorded rather than left implicit.

Design conditions: Before interpreting an MR estimate causally, specify the intended instruments and assess (1) relevance: the instruments are associated with the exposure; (2) independence: the instruments are independent of exposure–outcome confounders; and (3) exclusion restriction: the instruments affect the outcome only through the exposure. Also state whether the two association samples are separate, and do not confuse sample-level independence with the instrumental-variable independence assumption.

Why the Ratio Targets a Causal Effect

A short linear-model argument shows why pairing the two association estimates requires compatible genetic contrasts and valid instruments. Suppose, for one instrument \(G_j\), the exposure follows \(X=\alpha_jG_j+U_X\), and the outcome follows \(Y=\theta X+U_Y\). Assume the instrument is independent of the residual causes \(U_X\) and \(U_Y\), and has no direct route to \(Y\) other than through \(X\). Taking covariances with \(G_j\) gives

\operatorname{Cov}(G_j,X)=\alpha_j\operatorname{Var}(G_j), \qquad \operatorname{Cov}(G_j,Y)=\theta\operatorname{Cov}(G_j,X).

If \(\operatorname{Cov}(G_j,X)\ne 0\), dividing the second relation by the first yields

\frac{\operatorname{Cov}(G_j,Y)}{\operatorname{Cov}(G_j,X)}=\theta.

The genetic association slopes are these covariances divided by the same \(\operatorname{Var}(G_j)\), so their ratio also equals \(\theta\). In a two-sample design, estimates of the numerator and denominator are obtained in different samples. For the ratio to estimate the same target, the genetic contrast and the meanings of \(X\) and \(Y\) must be sufficiently compatible, as well as the instrumental-variable assumptions being credible. This argument is a justification for the design requirements, not a test that those assumptions are true.

Worked Example: Auditing Independence in a Hypothetical Design

Suppose a hypothetical outcome sample includes two subpopulations with different outcome risks, and a candidate variant is more common in one subpopulation. Let \(U\) indicate membership in the higher-risk group. Assume \(P(U=1\mid G=1)=0.80\) among carriers and \(P(U=1\mid G=0)=0.20\) among non-carriers. Suppose the outcome is \(Y=2U+\varepsilon\), where \(E[\varepsilon\mid G,U]=0\), and the variant has no direct effect on \(Y\).

The unadjusted mean outcome difference between carriers and non-carriers is \[ E[Y\mid G=1]-E[Y\mid G=0] =2(0.80)-2(0.20)=1.20. \] Thus the variant appears associated with the outcome by 1.20 outcome units even though the model contains no causal effect of the variant on the outcome. The association arises because \(G\) and the outcome-risk factor \(U\) are not independent in this sample.

This example illustrates why an association table is not enough to establish instrument validity. A design should examine whether the instruments can be independent of relevant confounders in the populations contributing the estimates, and report the ancestry and adjustment choices needed to evaluate that question. A numerical ratio cannot repair a failed independence assumption.

Common Design Mistakes

Pairing estimates because they share a variant label. The variant, effect-allele definition, and genetic contrast must correspond across sources; a matching label alone is insufficient.
Reporting estimates without their units or standard errors. The units determine the scale of the MR effect, and standard errors describe the uncertainty in the association estimates used by the analysis.
Assuming related phenotypes are interchangeable. State the actual exposure and outcome measurements and justify their relevance to the causal question.
Treating separate GWAS sources as proof of instrument validity. Separate samples do not establish relevance, independence from confounders, or the exclusion restriction.
Ignoring correlations among variants or association estimates. Record linkage disequilibrium and any participant overlap, and ensure the chosen analysis is consistent with the resulting dependence.
Interpreting a ratio as causal simply because it is computable. The calculation can be defined even when the design assumptions are implausible; causal language requires a defensible design and explicit limitations.

Key takeaway: A well-designed two-sample MR study makes the exposure and outcome association estimates traceable and comparable: it specifies the samples, traits, units, variant coding, ancestry, covariate adjustments, and dependence structure. It also states the instrumental-variable assumptions required for causal interpretation rather than treating them as consequences of the two-sample format.

Check Your Understanding

Answer each question by referring to the design requirements, not just to the arithmetic of an MR estimator.

What information should accompany an exposure association estimate and its corresponding outcome association estimate?
Why can differences in ancestry composition make two variant-association estimates difficult to interpret as the same genetic contrast?
State the three instrumental-variable assumptions needed for a causal interpretation and distinguish instrument independence from independence of the two association estimates.
A ratio has units of outcome standard deviations per exposure standard deviation. What does this tell you about the reported association scales, and what does it not establish?
Why should a study report whether its exposure and outcome GWAS samples overlap, even when it describes the analysis as two-sample MR?