Horizontal Pleiotropy as a False Mediation Pathway

Why a Genetic Association Can Mimic Mediation

A genetic variant associated with an exposure and an outcome can appear to support a causal pathway through a proposed mediator. But that pattern is not unique to mediation. The variant might also influence the outcome through another biological route, such as a pathway that bypasses the mediator. Its outcome association would then combine the proposed mediated effect with an effect that the mediation analysis does not intend to estimate.

This is horizontal pleiotropy: a genetic variant influences more than one trait through distinct causal pathways. In a mediation analysis, the central concern is whether the variant's effect on the outcome is transmitted through the proposed exposure and mediator pathways, or whether some part bypasses them. The distinction matters because a statistical association pattern alone does not establish which biological route produced it.

Definition: Horizontal pleiotropy, with respect to a proposed mediation model, occurs when a genetic variant affects the outcome through a causal pathway that is not part of the specified exposure-to-mediator-to-outcome pathway. A direct variant-to-outcome effect is one example; a variant effect through another trait that influences the outcome is another.

A Linear Model and Its Expected Variant Pattern

Let \(G_j\) denote a genetic variant, \(X\) the exposure, \(M\) the proposed mediator, and \(Y\) the outcome. Consider a linear, additive model with no exposure–mediator interaction. Let \(a\) be the causal effect of \(X\) on \(M\), \(b\) the effect of \(M\) on \(Y\) conditional on \(X\), and \(c'\) the direct effect of \(X\) on \(Y\) conditional on \(M\). Under this model, the indirect effect through \(M\) is \(ab\), and the total effect of \(X\) on \(Y\) is \(c'+ab\).

Write \(\beta_{GX,j}\), \(\beta_{GM,j}\), and \(\beta_{GY,j}\) for the associations of variant \(G_j\) with \(X\), \(M\), and \(Y\), respectively. These are population association effects on the stated measurement scales. For the basic expected pattern, assume the variant is relevant to \(X\), is independent of confounders of the relevant causal relationships, and has no effect on \(M\) except through \(X\), nor on \(Y\) except through \(X\) and \(M\). Also assume the causal effects are constant across variants and that the association estimates refer to aligned effect alleles and compatible populations and scales.

\beta_{GM,j}=a\beta_{GX,j}, \qquad \beta_{GY,j}=(c'+ab)\beta_{GX,j}.

The first equality says that a variant's association with the mediator is the exposure association carried through the \(X\to M\) effect. The second says that its outcome association includes both the direct exposure route and the route through the mediator. Thus, if \(\beta_{GX,j}\ne 0\), then \(\beta_{GY,j}/\beta_{GX,j}=c'+ab\). This ratio is the total effect represented by the model, not the indirect effect \(ab\) alone.

Key result: Under the stated no-pleiotropy model, the ratio of a variant's outcome association to its exposure association is \(c'+ab\). If \(a=1\), \(b=2\), and \(c'=0.4\), that ratio is \(0.4+(1)(2)=2.4\), not \(1.4\). A departure from the expected pattern can signal a violation, but does not by itself identify its cause.

How a Bypass Pathway Changes the Pattern

Suppose variant \(G_j\) has an additional effect \(\delta_j\) on \(Y\) that bypasses the proposed exposure and mediator pathways. The effect might be direct, or it might operate through another trait \(Z\) that influences \(Y\). Its outcome association then includes this extra component:

\beta_{GY,j}=(c'+ab)\beta_{GX,j}+\delta_j.

Dividing by the exposure association gives \(\beta_{GY,j}/\beta_{GX,j}=c'+ab+\delta_j/\beta_{GX,j}\). The extra term can inflate or reduce the ratio depending on its sign and size. It can also vary across variants, so variants with similar exposure associations may show different outcome associations. If the extra effects happen to align with exposure effects, the variants may still look consistent with a common slope even though that slope is biased.

A related complication is a variant that affects \(M\) directly, independently of \(X\). Call that component \(\eta_j\). It is not a bypass of the mediator if its effect on \(Y\) truly passes through \(M\), but it does bypass the proposed \(X\to M\) step. In the simple linear model, this gives \(\beta_{GM,j}=a\beta_{GX,j}+\eta_j\) and \(\beta_{GY,j}=(c'+ab)\beta_{GX,j}+b\eta_j+\delta_j\). Such a variant can therefore distort an inference about mediation of the effect of \(X\), even when one part of its outcome effect travels through \(M\). The route must be described precisely: “through the mediator” is not always the same claim as “through the exposure-induced mediator pathway.”

Deriving the Variant-Level Deviation

The pattern can be derived by substituting the structural effects into the outcome equation. Under the assumed causal model, a one-unit change in \(X\) changes \(M\) by \(a\) units, and changes \(Y\) directly by \(c'\) units. The induced change in \(M\) changes \(Y\) by a further \(ab\) units. A variant that shifts \(X\) by \(\beta_{GX,j}\) therefore shifts \(Y\), through those specified routes, by \((c'+ab)\beta_{GX,j}\). If there is also a bypass effect \(\delta_j\), add it to obtain the observed genetic association.

Equivalently, define the residual from the no-pleiotropy pattern as \(r_j=\beta_{GY,j}-(c'+ab)\beta_{GX,j}\). Substitution gives \(r_j=\delta_j\) when the model is otherwise correct. This equality is a conceptual decomposition: in practice, \(c'+ab\) is estimated, association estimates have sampling error, and residuals can reflect other misspecification as well as pleiotropy.

Worked Example: A Variant With an Outcome Bypass Effect

Suppose all variables are measured on standardized scales, and the specified model has \(a=1\), \(b=2\), and \(c'=0.4\). A hypothetical variant has \(\beta_{GX}=0.10\) standard deviations of exposure per effect allele and \(\beta_{GM}=0.10\) standard deviations of mediator per allele. Under the no-pleiotropy model, calculate its expected outcome association. Then suppose it also has a bypass effect \(\delta=0.10\) standard deviations of outcome per allele.

First, the expected total effect ratio is \(c'+ab=0.4+(1)(2)=2.4\). The predicted outcome association without the bypass path is therefore \(\beta_{GY}=2.4(0.10)=0.24\) standard deviations of outcome per allele.

Including the bypass effect gives \(\beta_{GY}=0.24+0.10=0.34\) standard deviations per allele. The observed ratio is \(0.34/0.10=3.4\), rather than the no-pleiotropy value \(2.4\). The excess of \(0.10\) standard deviations per allele matches the stipulated \(\delta\). In real data, this excess would be evidence of a departure from the model, not proof that a specific bypass pathway exists.

Using Variant-Level Patterns as Diagnostics

A useful diagnostic begins with the model's predicted relationships, not with a claim that any one plot proves a causal pathway. If the variants are valid for the specified model and share the same causal effects, their exposure and mediator associations should be compatible with \(\beta_{GM,j}=a\beta_{GX,j}\), and their outcome associations should be compatible with \(\beta_{GY,j}=(c'+ab)\beta_{GX,j}\). Examine whether particular variants depart from these patterns, whether departures are systematic, and whether they are larger than expected from uncertainty in the association estimates.

Outcome excess conditional on the exposure association: A variant with a larger-than-predicted outcome association may have a positive bypass effect; a smaller-than-predicted one may have a negative bypass effect. Sampling error and other model violations can produce the same appearance.
Inconsistent mediator-to-exposure pattern: If variants with similar exposure associations have substantially different mediator associations, this may indicate direct variant effects on the mediator, heterogeneous causal effects, or estimation error. It is not specific evidence of horizontal pleiotropy to the outcome.
One influential variant: A single variant far from the overall association pattern may dominate an estimate. Earlier material on radial MR and outlier diagnostics provides tools for flagging such observations; a flag indicates sensitivity, not the biological route responsible.
Shared deviation across many variants: If many variants depart in a similar direction, a common unmodelled pathway, an incorrect causal model, or a systematic incompatibility in the data may be more plausible than one isolated outlier. A common deviation can also be hard to detect using heterogeneity alone.

Whenever possible, interpret these patterns alongside biological knowledge about the locus and the phenotype definitions. Colocalization can help assess whether genetic associations at a locus are consistent with a shared causal variant, but it does not show that the shared variant acts only through the proposed mediator. One variant can be responsible for associations with several traits through separate pathways. Conversely, evidence that a locus may be pleiotropic does not automatically show that the proposed mediation pathway is absent.

Worked Example: Comparing Two Variants With the Model Prediction

Assume again that \(c'+ab=2.4\), with associations reported per effect allele on standardized scales. Two hypothetical variants have \((\beta_{GX},\beta_{GY})=(0.10,0.34)\) and \((0.20,0.38)\). Calculate the outcome association expected under the no-pleiotropy pattern and each variant's residual from that pattern.

For the first variant, the prediction is \(2.4(0.10)=0.24\). Its residual is \(0.34-0.24=0.10\) standard deviations of outcome per allele. For the second, the prediction is \(2.4(0.20)=0.48\), and its residual is \(0.38-0.48=-0.10\) standard deviations per allele.

The two residuals point in opposite directions. A simple average of the residuals is zero, but that cancellation would not establish the absence of pleiotropy: each variant could still have a nonzero bypass effect. The exposure-weighted through-origin slope for these two points is \((0.10(0.34)+0.20(0.38))/(0.10^2+0.20^2)=0.110/0.050=2.2\). That differs from \(2.4\), illustrating that variant-specific pleiotropic effects can change a combined estimate even when their unweighted residuals cancel. This calculation is illustrative; an actual analysis must account for the uncertainty and dependence structure of its association estimates.

When the Pattern Is Not Enough

A departure from the predicted variant pattern is a warning about the assumptions, not a unique diagnosis. It may arise from horizontal pleiotropy, but also from imprecise association estimates, allele misalignment, differences in measurement or sample composition, linkage between variants, or a causal structure that is not represented by the linear model. Likewise, a close fit to the predicted pattern does not prove that every variant is valid: pleiotropic effects can be proportional to exposure associations, or opposing effects can cancel in an aggregate estimate.

The causal diagram is therefore a tool for stating the question clearly. The target pathway is \(G\to X\to M\to Y\), alongside the allowed direct exposure route \(X\to Y\). A threat is a route such as \(G\to Z\to Y\), where \(Z\) is outside the proposed mediation chain. That extra arrow changes the interpretation of the variant-outcome association even if the observed associations with \(X\), \(M\), and \(Y\) are all statistically clear.

Worked Example: Distinguishing a Mediator Route From a Bypass

Suppose a hypothetical variant is associated with an exposure and with the outcome. The proposed model says that \(X\) affects \(M\), which affects \(Y\); the model also allows a direct \(X\to Y\) effect. A biological investigation suggests that the variant may additionally affect a separate trait \(Z\), which affects \(Y\), without changing \(X\) or \(M\).

The additional route is \(G\to Z\to Y\), so it contributes to \(\delta_j\), not to \(ab\). For example, if the model predicts an outcome association of \(0.24\) standard deviations per allele and the \(Z\)-path contributes \(0.06\), the total association is \(0.24+0.06=0.30\) standard deviations per allele. Attributing the full \(0.30\) to the exposure-mediated model would overstate the association transmitted by its specified routes. The numerical decomposition is valid only if the stated component effects are on compatible scales and the additive model is appropriate.

Common Mistakes and Practical Interpretation

Calling every genetic association with \(Y\) mediation. An association with the outcome can contain direct, mediated, and pleiotropic components. State the causal routes assumed before interpreting the association.
Confusing the total effect ratio with the indirect effect. Under the stated model, \(\beta_{GY,j}/\beta_{GX,j}=c'+ab\), not \(ab\) alone. The direct exposure effect \(c'\) is part of the total association pattern.
Treating a clean-looking pattern as proof of validity. A common slope can coexist with proportional pleiotropic effects. Diagnostics can reveal inconsistencies, but consistency cannot verify every exclusion assumption.
Claiming a particular pathway from a residual alone. A residual is a departure from a model prediction. It does not identify whether the cause is \(G\to Z\to Y\), another form of pleiotropy, or a different problem.
Ignoring the role of scale and allele alignment. Comparisons require compatible effect scales and correctly aligned alleles. A sign error or incompatible units can create apparent departures unrelated to biology.

Key takeaway: Under a linear, additive mediation model with valid variants, the expected variant-level outcome association is \((c'+ab)\beta_{GX,j}\). A pathway that bypasses the specified model adds an unmodelled component and can make the genetic pattern look like evidence for mediation. Variant-level departures are diagnostic clues, not proof of a specific mechanism.

Check Your Understanding

Answer each question using the model and definitions in this tutorial.

If \(a=0.5\), \(b=2\), and \(c'=0.3\), what is the expected ratio \(\beta_{GY,j}/\beta_{GX,j}\) under the no-pleiotropy model? Which part is the indirect effect?
A variant has \(\beta_{GX}=0.08\), and \(c'+ab=2.5\). What outcome association is predicted under the model?
In one sentence, distinguish a variant effect through \(G\to Z\to Y\) from the proposed \(X\to M\to Y\) pathway.
Why does a variant-level residual from the predicted pattern not, by itself, identify the biological cause of the departure?
How can pleiotropic effects be present even when the variants appear to follow a common association slope?