Horizontal Pleiotropy And The Exclusion Restriction

Why the Exclusion Restriction Matters

Mendelian randomization (MR) uses genetic variation as an instrumental variable to estimate the causal effect of an exposure \(X\) on an outcome \(Y\). The central appeal is that a genetic variant \(G\) can provide variation in \(X\) that is less vulnerable to some sources of confounding than an ordinary comparison of people with different exposure levels. But this interpretation depends on what the variant does to the outcome. If \(G\) can affect \(Y\) by a route that does not pass through \(X\), its outcome association is not solely evidence about the causal effect of \(X\).

That failure is called horizontal pleiotropy when the variant affects the outcome through a pathway outside the exposure pathway. It violates the exclusion restriction. The consequence is concrete: the variant’s outcome association includes both the effect transmitted through \(X\) and an extra contribution, so the Wald ratio need not equal the causal effect.

A Model for the Variant Associations

Use a linear, additive model with a single genetic variant. Let \(G\) denote the variant, often coded as the number of copies of a specified allele. Let \(\gamma\) be its effect on the exposure, \(\beta\) the causal effect of the exposure on the outcome, and \(\alpha\) the variant’s direct effect on the outcome that is not mediated by \(X\). The model is

Here \(U_X\) and \(U_Y\) collect other determinants of the exposure and outcome. They may be associated with one another; for example, causes of \(X\) and \(Y\) can confound their observational association. For the marginal-association derivation below, however, assume explicitly that

These conditions ensure that the residual terms do not themselves contribute to the marginal associations of \(G\) with \(X\) or \(Y\). They are stronger and more precise than merely saying that \(G\) is independent of exposure–outcome confounders: independence from those confounders alone does not guarantee that \(G\) is unrelated to every component of \(U_X\) and \(U_Y\). In the usual instrumental-variable interpretation, the relevant independence condition must make the instrument exogenous with respect to the causes represented in the residuals. The model also assumes the stated linear effects and no interaction terms.

Define \(\gamma_G\) as the population slope from regressing \(X\) on \(G\), and \(\Gamma_G\) as the population slope from regressing \(Y\) on \(G\). For a nonconstant \(G\), each slope is its covariance with the outcome divided by \(\operatorname{Var}(G)\). Thus, under the residual-orthogonality conditions,

The relevance condition requires \(\gamma\ne 0\): the variant must be associated with the exposure. The exclusion restriction corresponds to \(\alpha=0\) in this model. If \(\alpha\ne 0\), the variant has an outcome effect beyond the route through \(X\). The independence and residual-orthogonality conditions are distinct from exclusion: a variant can have no residual association and still violate exclusion through a nonzero \(\alpha\).

How Horizontal Pleiotropy Changes the Wald Ratio

For one relevant variant, the Wald ratio divides its outcome association by its exposure association. In the population model above, the ratio is

The term \(\alpha/\gamma\) is the departure from the causal effect \(\beta\). Its sign depends on both the direct outcome effect and the exposure association. For a fixed \(\alpha\), a smaller absolute value of \(\gamma\) makes the departure larger. A stronger association with the exposure does not prove that \(\alpha=0\); it only reduces this particular ratio’s sensitivity to a fixed direct effect.

Why Residual Orthogonality Must Be Stated

The expression \(\Gamma_G=\beta\gamma+\alpha\) does not follow from the structural equations alone. It also requires that neither residual contribute to the marginal association with \(G\). Substituting the equation for \(X\) into the equation for \(Y\) makes the dependence clear:

Taking covariance with \(G\) and dividing by its variance gives the general association expressions

Consequently, if either residual is associated with the variant, the observed associations include extra terms. In that case, one cannot simply attribute the whole outcome association to \(\beta\gamma+\alpha\), or interpret the ratio’s departure from \(\beta\) as \(\alpha/\gamma\). This is why the assumptions supporting a formula matter as much as the formula itself.

Interpretation and a Common Pitfall

The exclusion restriction is a claim about the causal routes from the variant to the outcome, not a property that can be established by the Wald ratio’s arithmetic. In the model, a pathway from \(G\) to \(Y\) that bypasses \(X\) is represented by \(\alpha G\). If that effect is present, the outcome association contains information about both the exposure-mediated effect and the direct effect. A single variant’s two associations provide a ratio, but do not by themselves separate \(\beta\) from \(\alpha\).

A common mistake is to say, “The variant is strongly associated with the exposure, so the Wald ratio is a causal estimate.” Relevance is necessary, but it does not imply exclusion. Another mistake is to attribute every difference between an estimated ratio and a proposed causal effect to horizontal pleiotropy. Sampling error, residual association, or a mismatch between the model and the data could also matter. The algebra identifies the consequence of \(\alpha\ne0\) only under the stated model and residual conditions.