Why the Pathway Matters
A genetic variant can influence several biological traits and still be a useful instrument for a particular exposure. The key question is not simply how many traits the variant affects. It is whether every causal route from the variant to the outcome passes through the exposure being studied. If the variant changes the exposure and that change then alters downstream traits that affect the outcome, those downstream effects are part of the exposure’s causal effect. If the variant can affect the outcome through a route that bypasses the exposure, the exclusion restriction is violated.
This distinction is important because the Mendelian randomization (MR) estimate targets the effect of a specified exposure on an outcome, not the effect of a genetic variant in general. The same biological effect may be downstream of one exposure definition but outside another. A pathway diagram therefore needs to be read relative to the particular exposure and outcome in the causal question.
Pathways and the Exclusion Restriction
Let \(G\) denote a genetic instrument, \(X\) the exposure, and \(Y\) the outcome. A pathway such as \(G \to X \to Y\) is compatible with the exclusion restriction. So is \(G \to X \to M \to Y\), where \(M\) is a downstream mediator caused by \(X\). In both cases, the effect of \(G\) on \(Y\) passes through \(X\). By contrast, \(G \to M \to Y\) alongside \(G \to X\) is a path from the instrument to the outcome that does not pass through \(X\). It is an exclusion-restriction violation, even if \(M\) is also affected by \(X\).
The standard instrument assumptions are relevance (the instrument is associated with \(X\)), independence (the instrument is independent of causes of the outcome, subject to the design and any appropriate adjustment), and the exclusion restriction (the instrument affects \(Y\) only through \(X\)). These assumptions are about causal structure, not merely whether associations can be detected in a data set. The earlier material on the Wald ratio gives the instrumental-variable estimate for one instrument; here the focus is how causal pathways determine whether its exclusion restriction is plausible.
For a simple linear model, suppose \(G\) is coded as allele count and the structural relationships are
Here \(U_X\), \(U_M\), and \(U_Y\) collect other causes of the respective variables. The pathway through \(M\) is downstream of \(X\): the model contains no direct effect of \(G\) on \(M\) once \(X\) is specified. For the interpretation below, assume a linear additive model, no effect modification by \(G\), the instrument assumptions just stated, and that the structural coefficients describe the causal effects of the corresponding variables. The total causal effect of \(X\) on \(Y\) includes both its direct effect \(c\) and its mediated effect through \(M\).
Worked Example: A Downstream Mediator Is Compatible With the Restriction
Consider a hypothetical study of a genetic variant, circulating exposure \(X\) measured in mmol/L, inflammation marker \(M\) measured in standardized units, and outcome \(Y\) measured in mmHg. Suppose one additional allele increases \(X\) by \(0.20\) mmol/L. A \(1\) mmol/L increase in \(X\) increases \(M\) by \(0.50\) standardized units; the effects of \(X\) and \(M\) on \(Y\) are \(0.30\) mmHg per mmol/L and \(0.40\) mmHg per standardized unit, respectively. Assume the structural model above and assume no other causal route from \(G\) to \(Y\).
The causal effect of \(X\) on \(Y\) has two parts. Its direct part is \(c=0.30\) mmHg per mmol/L. Its mediated part is \(bd=(0.50)(0.40)=0.20\) mmHg per mmol/L. Thus the total effect is \(c+bd=0.50\) mmHg per mmol/L.
The variant’s association with the outcome is its exposure association multiplied by this total effect: \(\beta_{GY}=(0.20)(0.50)=0.10\) mmHg per allele. The Wald ratio is \(\beta_{GY}/\beta_{GX}=0.10/0.20=0.50\) mmHg per mmol/L.
The model’s pathway is \(G \to X \to Y\) together with \(G \to X \to M \to Y\). Both routes from the variant to the outcome pass through \(X\). The downstream inflammation pathway contributes to the total effect of \(X\); it does not violate the exclusion restriction. This interpretation depends on the stated assumption that the variant has no separate effect on inflammation outside its effect through \(X\).
A Short Derivation: Why the Wald Ratio Includes Mediation
The distinction can be seen by substituting the mediator equation into the outcome equation. Under the structural model, a change in \(X\) changes \(M\) by \(b\) units for each unit of \(X\). The resulting change in \(Y\) is therefore the direct change \(c\) plus the mediated change \(d b\):
With \(X=aG+U_X\), the coefficient of \(G\) in the outcome equation is \(a(c+bd)\), provided there is no other causal route from \(G\) to \(Y\). The coefficient of \(G\) in the exposure equation is \(a\). If \(a \ne 0\), the Wald ratio is therefore
This is the total causal effect of \(X\) on \(Y\) in this model. The mediator does not need to be absent for the ratio to have this interpretation: it needs to be downstream of \(X\), with no additional effect of \(G\) on the outcome that bypasses \(X\). The algebra does not establish that these assumptions hold in a real study; it shows what follows if they do.
Worked Example: A Variant Also Affects a Mediator Outside the Exposure Path
Now consider a hypothetical variant that increases \(X\) by \(0.20\) mmol/L per allele and affects the same inflammation marker \(M\). Suppose \(X\) increases \(M\) by \(0.50\) standardized units per mmol/L, and the effects on \(Y\) remain \(0.30\) mmHg per mmol/L for \(X\) and \(0.40\) mmHg per standardized unit for \(M\). In addition, suppose the variant directly increases \(M\) by \(0.25\) standardized units per allele, independent of its effect on \(X\). Write this additional effect as \(qG\): \(M=bX+qG+U_M\), with \(q=0.25\).
The part of the variant’s outcome association that passes through \(X\) is \((0.20)[0.30+(0.50)(0.40)]=(0.20)(0.50)=0.10\) mmHg per allele. The extra route through \(G \to M \to Y\) contributes \((0.25)(0.40)=0.10\) mmHg per allele. Hence the total variant–outcome association is \(0.10+0.10=0.20\) mmHg per allele.
The exposure association is still \(0.20\) mmol/L per allele. The Wald ratio is \(0.20/0.20=1.00\) mmHg per mmol/L, whereas the total causal effect of \(X\) in the model is \(0.30+(0.50)(0.40)=0.50\) mmHg per mmol/L. The ratio is shifted by the bypassing pathway: its extra contribution is \(dq/a=(0.40)(0.25)/0.20=0.50\) mmHg per mmol/L. The path \(G \to M \to Y\) does not pass through \(X\), so the exclusion restriction fails.
This model also shows why a mediator’s name or role is not enough to classify a pathway. Inflammation is downstream of \(X\) along one route, but the variant’s separate effect on inflammation creates another route. The causal structure, not the biological label, decides the classification.
Multiple Downstream Routes
A biological exposure can affect several downstream processes, each of which can contribute to the outcome. Provided every such process is affected by the variant only through \(X\), they remain vertical pathways. For example, if \(X\) changes two mediators \(M_1\) and \(M_2\), and both affect \(Y\), the total effect includes the direct effect of \(X\) and both mediated effects. This is still a single exposure effect in the specified model, not a collection of exclusion-restriction violations.
Worked Example: Two Mediators Downstream of the Exposure
Suppose, hypothetically, one allele raises exposure \(X\) by \(0.10\) units. The exposure changes mediator \(M_1\) by \(0.40\) units per unit of \(X\), and \(M_2\) by \(0.20\) units per unit of \(X\). Let the direct effect of \(X\) on outcome \(Y\) be \(0.50\) outcome units per unit of \(X\); let each unit of \(M_1\) affect \(Y\) by \(0.30\) outcome units and each unit of \(M_2\) by \(0.60\) outcome units. Assume both mediators are caused by \(X\), and there are no other routes from \(G\) to \(Y\).
The mediated contribution through \(M_1\) is \((0.40)(0.30)=0.12\) outcome units per unit of \(X\). Through \(M_2\), it is \((0.20)(0.60)=0.12\). Adding the direct effect gives a total effect of \(0.50+0.12+0.12=0.74\) outcome units per unit of \(X\).
The variant–outcome association is \((0.10)(0.74)=0.074\) outcome units per allele. Dividing by the variant–exposure association gives the Wald ratio \(0.074/0.10=0.74\) outcome units per unit of \(X\). The two mediator paths are compatible with the exclusion restriction because each begins \(G \to X\) and then proceeds downstream. If either mediator also received an independent causal input from \(G\), the resulting route to \(Y\) would need to be assessed separately.
How to Assess a Proposed Biological Pathway
A useful assessment begins by stating the exposure and outcome precisely, then tracing every plausible causal route from the genetic variant to the outcome. For each route, ask whether it contains the specified exposure. The following sequence keeps the reasoning explicit.
State which measured or biologically defined trait is \(X\). A pathway can be downstream of one definition of \(X\) but bypass a narrower or different definition.
For each proposed route from \(G\) to \(Y\), record whether it passes through \(X\). A mediator affected by \(X\) is downstream on that route, but it could also be reached through a separate route from \(G\).
Routes that pass through \(X\) contribute to its total effect in the stated model. A route to \(Y\) that avoids \(X\) is an alternative pathway and conflicts with the exclusion restriction.
Biological knowledge can support or challenge a pathway diagram, but a plausible diagram does not prove the assumptions. Report uncertainty about unmeasured routes rather than treating a lack of known pathways as proof that none exist.
Common Mistake and Practical Importance
A common mistake is to reason, “The gene affects more than one phenotype, so it must violate the exclusion restriction.” That conclusion is too broad. If all effects on the outcome arise because the gene changes \(X\), and \(X\) then changes one or more downstream traits, the additional traits are part of the causal pathway from \(X\) to \(Y\). The correction is to trace the arrows: count a route as a violation when it reaches \(Y\) without passing through the specified \(X\), not merely when it passes through more than one phenotype.
The reverse mistake is also possible: calling every observed mediator “vertical” because it is biologically related to the exposure. A mediator may have both an \(X\)-dependent component and an independent genetic component. The independent component can create a bypassing path. Further, biological pathway diagrams are simplifications: an unmeasured route, an incorrectly ordered pathway, or an imprecisely defined exposure can change the classification.
This distinction matters for interpreting MR estimates. Under the stated assumptions, downstream mediation is included in the total effect of the exposure; it is not a reason by itself to discard an instrument. But a genuine path from the instrument to the outcome that bypasses the exposure means the Wald ratio need not estimate that total causal effect. The pathway assessment is therefore part of justifying the causal interpretation, not just describing biology after the estimate is calculated.
Check Your Understanding
For each question, identify the relevant route relative to the specified exposure.
- A variant changes exposure \(X\), which changes mediator \(M\), which changes outcome \(Y\). Does \(G \to X \to M \to Y\) violate the exclusion restriction? Explain.
- A variant changes both \(X\) and \(M\), and \(M\) affects \(Y\). What additional causal information is needed to decide whether the mediator pathway is vertical or bypasses \(X\)?
- In the linear model with \(M=bX+U_M\) and \(Y=cX+dM+U_Y\), write the total effect of \(X\) on \(Y\), including mediation through \(M\).
- If a variant has an independent effect \(q\) on \(M\) and \(M\) affects \(Y\) by \(d\) per unit, what is the resulting contribution of that bypassing route to the variant–outcome association?
- Why can a pathway be downstream of one definition of an exposure but bypass a different, narrower definition?