Why Pleiotropy Needs a Multivariable Interpretation
A genetic variant may be associated with several traits. In a single-exposure Mendelian randomization (MR) analysis, an association with a second trait can look like a route around the exposure being studied. In multivariable MR (MVMR), that second trait can instead be included as another exposure. The resulting analysis can estimate the effect of each included exposure conditional on the others—but only if the instruments and the causal model support that interpretation.
The key distinction is between genetic effects that operate through the exposures in the model and genetic effects that reach the outcome by some other route. The first are represented in the MVMR model; the second are unmeasured horizontal pleiotropy and violate its exclusion assumption. A nonzero association with multiple included exposures is not, by itself, evidence that an instrument is invalid for MVMR.
The introductory material on multivariable MR establishes the setting of several exposures and genetic instruments. Here the emphasis is on what identifies the exposure-specific effects and what those effects mean, especially when the exposures are correlated.
Identification Assumptions and the Association Model
Suppose the analysis includes \(K\) exposures, \(X_1,\ldots,X_K\), and \(J\) genetic variants, \(G_1,\ldots,G_J\). Let \(\beta_{Xjk}\) denote the association of variant \(G_j\) with exposure \(X_k\), and let \(\beta_{Yj}\) denote its association with outcome \(Y\). For a linear additive causal model, write the outcome as a sum of exposure effects plus other causes. The coefficient \(\theta_k\) is the effect of \(X_k\) when the other included exposures are held fixed.
The equation is an identification model, not a guarantee supplied by the data. Its causal interpretation relies on assumptions. First, conditional relevance requires the variants to provide enough independent information about the included exposures jointly. A variant can be associated with several exposures and still contribute relevant information; the issue is whether the set of associations distinguishes their effects. Second, independence requires the variants to be independent of confounders of the included exposure–outcome relationships. Third, exclusion requires that each variant affect the outcome only through the exposures included in the model: \(\alpha_j=0\) for every instrument used.
The causal interpretation also depends on a suitable outcome model. The equation above describes linear, additive effects on the outcome scale being analyzed; it does not allow effect modification unless that is explicitly modeled. Exposure definitions and the genetic association estimates must refer to compatible populations and scales. If variants are correlated through linkage disequilibrium, their dependence must be accounted for in estimation rather than treating their associations as independent.
A useful justification of the association equation starts with the structural model \(Y=\sum_{k=1}^{K}\theta_kX_k+U\), where \(U\) collects other outcome causes. A variant's outcome association is the sum of its contributions through each exposure and any direct contribution not represented by those exposures. Thus, \(\beta_{Yj}=\sum_{k=1}^{K}\theta_k\beta_{Xjk}+\alpha_j\). Under exclusion, \(\alpha_j=0\), so the outcome association is explained by the vector of exposure associations and the conditional causal effects. This is the multivariable analogue of the instrumental-variable association restriction.
Included Exposure Effects Versus Unmeasured Pleiotropy
Suppose a variant changes both LDL cholesterol and triglycerides, and both exposures are included in the model. Its association with triglycerides is not an unmodeled route to the outcome: the model includes the triglyceride association as well as the LDL association. This is often called measured or modeled pleiotropy. It is compatible with MVMR's exclusion assumption, provided the variant has no additional effect on the outcome outside the included exposures.
By contrast, if the variant also affects the outcome through inflammation, and inflammation is not included as an exposure, that effect contributes to \(\alpha_j\). The same applies to an omitted exposure or another biological pathway that directly influences the outcome. Calling a trait “measured” is not enough: the pathway must be represented appropriately in the model for it to be accounted for. Adding many traits is not automatically a remedy, because each added exposure changes the conditional estimand and requires adequate genetic information.
Worked Example: A Variant Associated With Two Included Exposures
Consider hypothetical per-allele associations for two variants. Let \(X_1\) be LDL cholesterol in mmol/L, \(X_2\) be triglycerides in mmol/L, and let \(Y\) be a continuous outcome in outcome units. Suppose the conditional effects are \(\theta_1=2\) outcome units per mmol/L of LDL and \(\theta_2=1\) outcome unit per mmol/L of triglycerides, with no unmeasured direct effects.
| Variant | \(\beta_{Xj1}\) (mmol/L) | \(\beta_{Xj2}\) (mmol/L) | Predicted \(\beta_{Yj}\) (outcome units) |
|---|---|---|---|
| \(G_1\) | 0.10 | 0.02 | 0.22 |
| \(G_2\) | 0.02 | 0.10 | 0.14 |
| \(G_3\) | 0.08 | 0.06 | 0.22 |
For \(G_1\), the model predicts \(2(0.10)+1(0.02)=0.22\) outcome units per allele. For \(G_2\), it predicts \(2(0.02)+1(0.10)=0.14\). For \(G_3\), it predicts \(2(0.08)+1(0.06)=0.22\). Each variant's association with both exposures is accounted for by the two terms in the model; none requires a nonzero \(\alpha_j\) in this hypothetical example.
This calculation shows what “measured pleiotropy” means in the association model. It does not establish from observed data that the effects are truly 2 and 1, or that the exclusion assumption holds. It illustrates the interpretation if the model and assumptions are correct.
Worked Example: An Unmeasured Direct Pathway
Keep the same two exposures and hypothetical conditional effects, but suppose \(G_3\) also affects \(Y\) through an omitted pathway, adding \(\alpha_3=0.05\) outcome units per allele. For \(G_3\), the exposure-mediated component is still \(2(0.08)+1(0.06)=0.22\). Its total outcome association is instead \(0.22+0.05=0.27\) outcome units per allele.
If the analyst uses the valid-model prediction of 0.22, the discrepancy is \(0.27-0.22=0.05\) outcome units per allele. That difference is the unmeasured direct contribution in this constructed example. In real data, the causal effects are not known in advance, so a discrepancy from a fitted model does not uniquely identify a pathway or prove which variant is invalid. It indicates that the observed associations do not fit the stipulated model exactly.
For a small arithmetic illustration of how such a discrepancy can affect a joint fit, take three variants with exposure associations \((0.10,0)\), \((0,0.10)\), and \((0.10,0.10)\), and outcome associations \(0.20\), \(0.10\), and \(0.35\). The first two agree with effects \((2,1)\); the third has a 0.05 direct contribution beyond its predicted 0.30. An unweighted regression through the origin gives \(X^\mathsf{T}X=\left(\begin{smallmatrix}0.02&0.01\\0.01&0.02\end{smallmatrix}\right)\) and \(X^\mathsf{T}Y=(0.055,0.045)^\mathsf{T}\), where \(X\) is the matrix of exposure associations. Solving the two normal equations gives \(\hat\theta_1=2.167\) and \(\hat\theta_2=1.167\). These differ from 2 and 1 because the misspecified third association pulls the joint fit. This calculation is illustrative, not a recommended estimator or a test of validity.
What a Direct Effect Means When Exposures Are Correlated
The word direct is relative to the included exposures. In the two-exposure model, \(\theta_1\) is the effect of changing \(X_1\) while holding \(X_2\) fixed; \(\theta_2\) is the effect of changing \(X_2\) while holding \(X_1\) fixed. These are not automatically total effects, nor are they necessarily effects that exclude every biological mediator. A mediator omitted from the exposure set is outside the model and may contribute to a variant's \(\alpha_j\).
When exposures are correlated, the distinction between conditional and marginal effects matters. A genetic variant's association with \(Y\) reflects all its included exposure associations together. A single-exposure analysis may therefore mix the effects of correlated exposures. MVMR attempts to separate them by using differences in variants' joint exposure-association patterns. If those patterns contain little information that distinguishes \(X_1\) from \(X_2\), the direct effects can be imprecise or unstable even when the exposures are strongly associated with the outcome. Conditional instrument strength, covered in earlier material on conditional instrument strength in MVMR, concerns this joint information.
Worked Example: Interpreting a Conditional Effect and a Joint Change
Suppose a hypothetical MVMR analysis estimates \(\theta_1=1.5\) outcome units per standard-deviation increase in \(X_1\), conditional on \(X_2\), and \(\theta_2=0.5\) outcome units per standard-deviation increase in \(X_2\), conditional on \(X_1\). The estimate for \(X_1\) does not mean that a real-world increase in \(X_1\) will necessarily leave \(X_2\) unchanged.
Under the additive model, a specified joint change of \(\Delta X_1=1\) standard deviation and \(\Delta X_2=0.4\) standard deviations corresponds to a predicted outcome change of
This is the model's contrast for that joint change. It is not a claim that the natural effect of intervening on \(X_1\) includes a 0.4-standard-deviation change in \(X_2\). That additional change must be justified by the intervention or by a separate causal model. The calculation also assumes the effects are additive and constant over the specified changes.
Diagnostics, Alternatives, and Limitations
A poor fit between the observed variant-outcome associations and the values predicted by the joint exposure model can be a warning. Heterogeneity, influential variants, or markedly different estimates after excluding a variant may motivate investigation. Biological knowledge can also help identify plausible omitted pathways. Such checks are diagnostic, not proofs: a good fit cannot rule out pleiotropy shared in a way that mimics exposure-mediated effects, and a poor fit can arise for reasons other than invalid instruments.
The main alternative is a single-exposure MR analysis, which answers a different question and may attribute effects of correlated exposures to the one exposure analyzed. Pleiotropy-robust approaches from the earlier material can be useful under their own assumptions, but they do not automatically identify MVMR direct effects when exposures are correlated. MVMR's advantage is that it explicitly models multiple exposure associations; its limitation is that identification can be difficult when those associations provide little independent variation or when exclusion fails.
- Weak joint information: conditional instrument weakness can lead to imprecise estimates and, depending on the design, bias. Strong marginal associations with each exposure do not by themselves ensure strong conditional instruments.
- Omitted exposure or pathway: a genetic effect through a trait not included in the model can appear as \(\alpha_j\), violating exclusion.
- Overinterpretation: a conditional coefficient is not automatically a total effect or the effect of a naturally occurring change in one exposure alone.
- Uncertainty about validity: neither a diagnostic nor a statistically compatible fit can verify all instrumental-variable assumptions.
A common mistake is to say, “This variant affects two traits, so it is horizontally pleiotropic and must be removed.” The correction is to ask whether both traits are included exposures. If they are, their associations are modeled jointly; the exclusion concern is an effect on the outcome through a route outside the included exposure set. A second mistake is to interpret a conditional effect as the overall effect of changing one exposure in a population where the other exposure also changes. The model gives the effect holding the other included exposures fixed; any joint change must be stated and interpreted separately.
Check Your Understanding
Answer each question, then compare with the brief solutions.
- A variant affects both included exposures, and its outcome effect is fully explained by those exposure associations. Is this, by itself, a violation of MVMR exclusion?
- For \(\theta_1=3\) outcome units per unit of \(X_1\) and \(\theta_2=-1\) outcome unit per unit of \(X_2\), calculate the model-predicted outcome change for \(\Delta X_1=0.5\) and \(\Delta X_2=2\).
- What does \(\theta_1\) mean in a two-exposure MVMR model?
- Why can highly correlated exposures yield imprecise direct-effect estimates even if each exposure has strong marginal genetic associations?
Solutions. (1) No. If both genetic effects operate through included exposures and there is no additional outcome pathway, they are represented in the model. (2) \(\Delta Y=3(0.5)+(-1)(2)=1.5-2=-0.5\) outcome units. (3) It is the effect of \(X_1\) on the outcome conditional on holding the other included exposure fixed, under the model assumptions. (4) The variants may not provide enough distinct joint association patterns to separate the exposure effects, weakening conditional information even when marginal associations are strong.