Exclusion Restriction And Horizontal Pleiotropy

Why the Exclusion Restriction Matters

A genetic variant can be associated with an exposure and an outcome for more than one reason. The Mendelian randomization exclusion restriction asks whether the variant affects the outcome only through the exposure under study. It is not enough that a variant predicts the exposure, or that its association with the outcome is biologically plausible. If the variant also changes the outcome through a pathway that bypasses the exposure, the variant does not satisfy this assumption for that analysis.

The key distinction is between a genetic effect that continues downstream of the exposure and one that reaches the outcome by a separate route. The first is vertical pleiotropy and is compatible with exclusion. The second is horizontal pleiotropy and violates exclusion. A variant may have both kinds of effects.

Definition: Let \(Z\) denote a genetic instrument, \(X\) the exposure, and \(Y\) the outcome. The exclusion restriction requires that, at a fixed exposure value, changing the instrument does not change the outcome: \(Y_i(z,x)=Y_i(z',x)\) for all relevant \(z,z'\), \(x\), and individuals \(i\). In causal-diagram terms, every causal path from \(Z\) to \(Y\) must pass through \(X\).

The three instrumental-variable assumptions are relevance (the variant affects the exposure), independence (the variant is independent of causes of the exposure and outcome), and exclusion. This tutorial focuses on the third. A variant can satisfy relevance while failing exclusion; independence does not rule out a separate biological pathway from the variant to the outcome.

Vertical and Horizontal Pleiotropy

Pleiotropy means that a genetic variant influences more than one trait. The number of affected traits alone does not determine whether exclusion holds. What matters is the causal route to the outcome.

Definition: Vertical pleiotropy occurs when a genetic effect on another trait is downstream of the exposure, so the route to the outcome passes through \(X\). Horizontal pleiotropy occurs when the variant affects the outcome through a route that bypasses \(X\), such as by changing a separate biological process that affects \(Y\).

A simple vertical diagram is \(Z \rightarrow X \rightarrow M \rightarrow Y\), where \(M\) is a mediator caused by \(X\). Although the variant affects \(M\) and \(Y\) through that chain, it has no outcome pathway outside \(X\). By contrast, \(Z \rightarrow M \rightarrow Y\), alongside \(Z \rightarrow X \rightarrow Y\), is horizontal if \(M\) is affected by \(Z\) independently of \(X\). In that diagram there is a route from \(Z\) to \(Y\) that does not pass through the exposure.

These diagrams describe causal claims, not just patterns of association. If \(Z\) is associated with a second trait, that observation alone does not show whether the second trait is downstream of \(X\), a separate pathway, or a consequence of another source of association. Biological knowledge can help distinguish these explanations, but does not automatically settle them.

Worked Example: A Downstream Lipid Pathway

Hypothetical scenario. A variant changes low-density lipoprotein cholesterol (LDL-C), the exposure \(X\). Higher LDL-C increases arterial plaque burden, and plaque burden contributes to coronary disease, the outcome \(Y\). Suppose the variant affects plaque burden only because it changes LDL-C.

The proposed diagram is \(Z \rightarrow X \rightarrow M \rightarrow Y\), with \(M\) representing plaque burden. The route from the variant to coronary disease passes through LDL-C. The effect on plaque burden is therefore vertical pleiotropy, not an exclusion violation.

This conclusion depends on the stated causal structure. If the variant also changed plaque burden through a pathway independent of LDL-C, the diagram would need an additional route \(Z \rightarrow M \rightarrow Y\) that bypasses \(X\), and exclusion would fail. A second trait’s name or biological proximity does not determine the classification; its causal position does.

Worked Example: A Separate Coagulation Pathway

Hypothetical scenario. A variant changes LDL-C by \(0.20\) mmol/L per additional effect allele. Assume the causal effect of LDL-C on the log odds of an outcome is \(0.50\) log-odds units per mmol/L. The same variant also changes coagulation through an independent pathway, adding \(0.08\) log-odds units per allele to the outcome association.

The LDL-C-mediated contribution per allele is \(\,0.50 \times 0.20 = 0.10\,\) log-odds units. Including the independent coagulation pathway, the total variant–outcome association is \(\,0.10 + 0.08 = 0.18\,\) log-odds units per allele.

The \(0.08\) contribution is horizontal pleiotropy: the variant affects the outcome through coagulation without that effect being mediated by LDL-C. The total association of \(0.18\) therefore cannot be attributed solely to the effect of the variant on LDL-C. The arithmetic illustrates the problem; it does not establish that these values describe a real genetic association.

Why a Bypass Path Can Distort the Causal Interpretation

A simple linear model makes the consequence explicit. Suppose the exposure and outcome follow

X=\alpha Z+u,\qquad Y=\beta X+\gamma Z+\varepsilon.

Here \(\alpha\) is the effect of the instrument on the exposure, \(\beta\) is the effect of the exposure on the outcome, and \(\gamma\) represents an outcome effect of the instrument that does not pass through the exposure. For this derivation, assume linear additive relationships, \(\alpha \ne 0\), and that \(Z\) is uncorrelated with \(u\) and \(\varepsilon\). The term \(\gamma Z\) represents a horizontal pathway on this model scale; exclusion requires \(\gamma=0\).

Derivation: The outcome’s covariance with the instrument is \(\operatorname{Cov}(Z,Y)=\beta\operatorname{Cov}(Z,X)+\gamma\operatorname{Var}(Z)\), because \(\operatorname{Cov}(Z,\varepsilon)=0\). From the exposure model, \(\operatorname{Cov}(Z,X)=\alpha\operatorname{Var}(Z)\). Thus \(\operatorname{Cov}(Z,Y)/\operatorname{Cov}(Z,X)=\beta+\gamma/\alpha\). When \(\gamma=0\), this association ratio equals \(\beta\) under the model assumptions. When \(\gamma\ne0\), it includes the additional term \(\gamma/\alpha\), so it does not isolate the exposure effect.

The direction and size of the distortion depend on the signs and magnitudes of \(\gamma\) and \(\alpha\). A horizontal effect can reinforce the exposure-mediated association or oppose it. It need not make the observed variant–outcome association larger in absolute value. If \(\gamma/\alpha\) has the opposite sign to \(\beta\), the two contributions can partly or even completely cancel.

Worked Example: Horizontal Effects Can Mask an Exposure Effect

Hypothetical linear model. Suppose \(\alpha=0.10\) exposure units per allele, \(\beta=2.0\) outcome units per exposure unit, and \(\gamma=-0.10\) outcome units per allele through a separate pathway. The exposure-mediated contribution per allele is \(2.0 \times 0.10=0.20\) outcome units. The total instrument–outcome association is \(0.20-0.10=0.10\) outcome units per allele.

The association ratio in the derivation is \(0.10/0.10=1.0\) outcome unit per exposure unit, whereas the exposure effect in the model is \(2.0\) outcome units per exposure unit. The discrepancy is \(\gamma/\alpha=-0.10/0.10=-1.0\). Thus, in this hypothetical case, the horizontal pathway attenuates the association-based quantity rather than increasing it. The units are consistent: \(\gamma/\alpha\) has units of outcome per exposure.

Classifying a More Complex Genetic Example

When a variant influences several biological processes, draw the proposed causal routes before deciding whether exclusion is plausible. Ask whether each route to the outcome passes through the specified exposure. If a process is downstream of that exposure, its effect can be vertical. If the variant affects that process independently and the process affects the outcome, the route is horizontal. A mixture of routes means the variant has both vertical and horizontal effects for the analysis.

Worked Example: Insulin Resistance and Diabetes Risk

Hypothetical scenario. The exposure is insulin resistance and the outcome is type 2 diabetes. Suppose a variant increases insulin resistance, which raises diabetes risk. The variant also impairs pancreatic beta-cell function independently of insulin resistance, and that impairment raises diabetes risk.

The first route, \(Z \rightarrow X \rightarrow Y\), is through the exposure. The second route, \(Z \rightarrow\) beta-cell function \(\rightarrow Y\), bypasses insulin resistance. It is horizontal pleiotropy, so exclusion does not hold for this variant under the stated causal model. If beta-cell impairment instead occurred only as a consequence of insulin resistance, its route to diabetes would pass through \(X\) and would be vertical.

This classification follows the hypothetical causal assumptions, not merely the observation that the variant is associated with both traits. If the direction or dependence of the beta-cell effect were unknown, the available description would not be enough to establish whether the pathway was horizontal.

Evidence, Diagnostics, and Limits

Exclusion is a causal assumption and is generally not verified by a single statistical test. Useful scrutiny includes reviewing the variant’s known biology, asking whether reported effects could be downstream of the exposure, and checking whether plausible independent mechanisms connect the variant to the outcome. These checks can expose a credible violation, but a lack of known alternative mechanisms is not proof that none exists.

Several failure modes deserve particular attention:

Treating every second-trait association as a violation. A second trait may lie downstream of the exposure. Trace the causal pathway rather than counting associated traits.
Treating a plausible exposure pathway as proof of exclusion. A variant can affect the outcome through the exposure and also through another route. Evidence for one valid route does not rule out a bypass route.
Interpreting a null association as proof of no direct pathway. Limited precision or opposing effects can hide a horizontal contribution. Failure to detect an association is not evidence that the causal effect is exactly zero.
Confusing exclusion with independence. Independence concerns whether the variant is associated with causes of the exposure or outcome; exclusion concerns whether the variant affects the outcome outside the exposure pathway. One assumption does not imply the other.

An observed association between a variant and an outcome cannot, by itself, distinguish the exposure-mediated contribution from a horizontal one. Likewise, adjusting statistically for the exposure does not generally prove exclusion: the result depends on the causal structure and the assumptions of that adjustment. Biological evidence and statistical patterns can inform a judgment, but may leave uncertainty about pathway direction, unmeasured mechanisms, and the extent of any bypass effect.

Feature	Vertical pleiotropy	Horizontal pleiotropy
Route from variant to outcome	Passes through the exposure	Includes a route that bypasses the exposure
Example structure	\(Z \rightarrow X \rightarrow M \rightarrow Y\)	\(Z \rightarrow X \rightarrow Y\) and \(Z \rightarrow M \rightarrow Y\)
Implication for exclusion	Compatible, if there is no other route	Violates the assumption
What an association with another trait establishes	Not enough on its own to classify the pathway	Not enough on its own to prove a bypass pathway

Key takeaway: Exclusion is about causal routes, not the number of traits a variant influences. Downstream effects are vertical and can be compatible with the assumption; an effect that reaches the outcome without passing through the exposure is horizontal and violates it.

Check Your Understanding

For each question, first classify the proposed causal route, then compare your reasoning with the solutions.

A variant changes an exposure, which changes a hormone, which then changes the outcome. Is this route vertical or horizontal?
A variant changes the exposure and independently changes kidney function; kidney function affects the outcome. Which assumption is threatened by the kidney pathway?
In the linear model, suppose \(\alpha=0.25\), \(\beta=1.2\), and \(\gamma=0.05\). What are the exposure-mediated and horizontal contributions to the instrument–outcome association per allele? What is their sum?
A variant has no statistically detectable association with a proposed second trait. Does this establish the exclusion restriction? Why or why not?

Solutions. (1) Vertical: the route from the variant to the outcome passes through the exposure. (2) Exclusion is threatened because the kidney pathway reaches the outcome without passing through the exposure. (3) The exposure-mediated contribution is \(\beta\alpha=1.2\times0.25=0.30\) outcome units per allele; the horizontal contribution is \(\gamma=0.05\) outcome units per allele; the total is \(0.30+0.05=0.35\) outcome units per allele. (4) No. A non-detectable association may reflect limited precision, and it does not rule out an unmeasured pathway or establish that all causal routes pass through the exposure.