Patterns of Horizontal Pleiotropy in Mendelian Randomization

Direct Effects and the MR Association

Mendelian randomization (MR) uses genetic variants as instrumental variables to estimate the causal effect of an exposure \(X\) on an outcome \(Y\). A central concern is that a variant may affect the outcome through a route that does not pass through the exposure. Such an effect is horizontal pleiotropy: it violates the exclusion restriction for that variant. By contrast, a genetic effect that reaches the outcome only through the exposure is vertical pleiotropy and is part of the causal pathway.

For a variant \(G_j\), the intended causal path is \(G_j \rightarrow X \rightarrow Y\). Horizontal pleiotropy adds a direct path, \(G_j \rightarrow Y\), that bypasses \(X\). Let \(\beta_{Xj}\) denote the genetic association with the exposure, \(\beta_{Yj}\) the genetic association with the outcome, and \(\alpha_j\) the direct effect on the outcome not mediated by \(X\). Under a common linear causal effect \(\theta\), the association relation is

\beta_{Yj} = \theta\beta_{Xj}+\alpha_j.

This equation separates the association expected from the exposure pathway, \(\theta\beta_{Xj}\), from the direct contribution \(\alpha_j\). It is a useful summary model when variants are harmonized to the same allele, the associations refer to compatible populations and scales, and the causal effect is represented by a common \(\theta\). In the examples below, the associations are treated as known population values to isolate the effect of pleiotropy; real estimates also contain sampling uncertainty.

Definition: Horizontal pleiotropy occurs when a genetic variant affects the outcome through a pathway that does not pass through the exposure. In the association model, that route is represented by a nonzero direct effect \(\alpha_j\). The pattern across variants—rather than simply whether any one \(\alpha_j\) is nonzero—distinguishes balanced, directional, and correlated pleiotropy.

Three Patterns of Direct Effects

Pleiotropy can be described by the distribution of the direct effects across variants and by how those effects relate to the variants’ exposure associations. These descriptions are not three mutually exclusive labels: for example, a set of direct effects can have mean zero and still be correlated with \(\beta_{Xj}\).

Pattern	Direct-effect feature	Potential concern for IVW
Balanced	Direct effects vary but average to zero under a stated averaging scheme	Sampling and weighting can leave a nonzero net contribution
Directional	Direct effects tend, on average, to point in one direction	The net direct contribution can shift the estimate away from \(\theta\)
Correlated	Direct effects are associated with the exposure associations \(\beta_{Xj}\)	The pleiotropic contribution can systematically track instrument strength

“Balanced” and “directional” describe the average direction of direct effects. The averaging scheme matters: it might be an unweighted average across variants, or an average that gives some variants more influence. “Correlated” describes a relationship between direct effects and instrument-exposure associations, not simply a nonzero average. In common MR discussions, independence between direct effects and instrument-exposure associations is called the InSIDE assumption. Its role is considered in more detail in the later treatment of MR-Egger regression.

Worked Example: Balanced Direct Effects

Consider four hypothetical independent variants, all oriented so that their exposure associations are positive. Suppose the true causal effect is \(\theta=0.5\) outcome units per exposure unit, and the associations are as follows. Use equal weights for the illustrative IVW calculation.

Variant	\(\beta_{Xj}\)	\(\alpha_j\)	\(\beta_{Yj}=0.5\beta_{Xj}+\alpha_j\)
1	1	-0.2	0.3
2	2	0.2	1.2
3	3	0.2	1.7
4	4	-0.2	1.8

The unweighted mean direct effect is \((-0.2+0.2+0.2-0.2)/4=0\), so the effects are balanced by this definition. With equal weights, the through-origin IVW estimate is

\hat{\theta}_{\mathrm{IVW}} =\frac{\sum_j \beta_{Xj}\beta_{Yj}}{\sum_j \beta_{Xj}^2} =\frac{1(0.3)+2(1.2)+3(1.7)+4(1.8)}{1^2+2^2+3^2+4^2} =\frac{15}{30}=0.5.

Here, the direct effects cancel in the weighted numerator as well as in their unweighted mean: \(\sum_j\beta_{Xj}\alpha_j=-0.2+0.4+0.6-0.8=0\). Thus this particular balanced pattern does not shift the illustrative IVW estimate. Balance alone, however, does not guarantee that cancellation.

How Direct Effects Enter the IVW Estimate

For independent variants, the conventional summary-data IVW estimate uses weights \(w_j=1/SE(\beta_{Yj})^2\) and fits the outcome associations against the exposure associations through the origin. In this expression, uncertainty in the exposure associations is typically not included in the weights. The exact estimate is

\hat{\theta}_{\mathrm{IVW}} =\frac{\sum_j w_j\beta_{Xj}\beta_{Yj}}{\sum_j w_j\beta_{Xj}^2}.

Substitute the direct-effect relation \(\beta_{Yj}=\theta\beta_{Xj}+\alpha_j\) into the numerator:

\begin{aligned} \hat{\theta}_{\mathrm{IVW}} &=\frac{\sum_j w_j\beta_{Xj}(\theta\beta_{Xj}+\alpha_j)} {\sum_j w_j\beta_{Xj}^2}\\ &=\theta+\frac{\sum_j w_j\beta_{Xj}\alpha_j} {\sum_j w_j\beta_{Xj}^2}. \end{aligned}

This identity is the key justification for why pleiotropy can bias IVW: the estimate’s departure from \(\theta\) is determined by the weighted cross-product of the exposure associations and direct effects. If that cross-product is zero, the direct effects do not shift this particular IVW estimate in the association model. If it is nonzero, the estimate is shifted; its direction depends on the sign of the cross-product.

The identity also clarifies why a zero average direct effect is not enough to establish no bias. The relevant quantity is not generally \(\sum_j\alpha_j\), but \(\sum_j w_j\beta_{Xj}\alpha_j\). Variants with stronger exposure associations or greater precision can contribute more to the IVW estimate. A pattern that cancels without weighting may not cancel after weighting.

Worked Example: Directional Pleiotropy

Suppose three hypothetical variants have \(\beta_{Xj}=1,2,3\), a true causal effect of \(\theta=0.5\) outcome units per exposure unit, and the same direct effect \(\alpha_j=0.2\) outcome units per allele for each variant. The direct effects point in the same direction, so their unweighted mean is \(0.2\). Their outcome associations are \(0.7,1.2,1.7\). With equal weights,

\hat{\theta}_{\mathrm{IVW}} =\frac{1(0.7)+2(1.2)+3(1.7)}{1^2+2^2+3^2} =\frac{8.2}{14} =0.5857.

The estimate is \(0.0857\) outcome units per exposure unit above the stipulated causal effect. The direct-effect contribution gives the same result:

\frac{\sum_j\beta_{Xj}\alpha_j}{\sum_j\beta_{Xj}^2} =\frac{1(0.2)+2(0.2)+3(0.2)}{14} =\frac{1.2}{14} =0.0857.

This is a simple directional pattern in which the positive direct effects shift the estimate upward. A nonzero mean does not, by itself, determine the direction or exact size of the shift in every possible design: the weighted cross-product, not the mean alone, gives that answer.

Worked Example: Correlated Direct Effects With a Zero Mean

Now use four hypothetical variants with \(\beta_{Xj}=1,2,3,4\), \(\theta=0.5\), and direct effects \(\alpha_j=-0.3,-0.1,0.1,0.3\). The mean direct effect is zero. But the direct effects increase with the exposure associations, so they are positively correlated with instrument strength. The corresponding outcome associations are \(0.2,0.9,1.6,2.3\). Under equal weights,

\hat{\theta}_{\mathrm{IVW}} =\frac{1(0.2)+2(0.9)+3(1.6)+4(2.3)}{1^2+2^2+3^2+4^2} =\frac{16}{30} =0.5333.

Although the average direct effect is zero, its weighted cross-product is not:

\sum_j\beta_{Xj}\alpha_j =1(-0.3)+2(-0.1)+3(0.1)+4(0.3)=1.0.

Thus the estimate is \(1/30=0.0333\) outcome units per exposure unit above \(\theta\). This example shows why balanced pleiotropy and correlated pleiotropy can coexist: balance by the unweighted mean does not rule out a relationship with instrument strength or a shift in IVW.

Interpreting the Patterns in Practice

The direct-effect diagram is the same for all three patterns: each variant can have a path through the exposure and, under horizontal pleiotropy, an additional path directly to the outcome. What changes is the set of \(\alpha_j\) values across variants. A balanced pattern has positive and negative direct effects that cancel under a specified average. A directional pattern has an average that is above or below zero. A correlated pattern has direct effects that tend to change with \(\beta_{Xj}\). These features can overlap rather than exclude one another.

A positive weighted cross-product shifts IVW upward relative to \(\theta\); a negative cross-product shifts it downward. For example, if all variants are oriented so \(\beta_{Xj}>0\), direct effects that are mostly positive tend to make the cross-product positive. But this is not a universal shortcut: unequal weights, mixed signs, and the relationship between effect size and precision matter. The formula is the reliable way to determine the shift in a specified example.

These patterns matter because a precise IVW estimate is not automatically a valid causal estimate. More precise outcome associations can narrow uncertainty around an estimate while direct effects still shift its target. Nor does observing varied variant-specific ratios, \(\beta_{Yj}/\beta_{Xj}\), identify the pattern by itself: ratio differences can also reflect sampling error or other departures from the simple model. Diagnostics can reveal some inconsistencies, but cannot prove that every instrument satisfies the exclusion restriction.

The examples make simplifying choices: equal weights, independent variants, positive exposure associations, and a common linear causal effect. In an actual analysis, allele harmonization, instrument strength, population structure, and the study design also matter. The preceding material on MR diagnostics emphasizes that no single diagnostic establishes that pleiotropy is absent. Here, the main aim is to understand the structure of the direct-effect contribution, not to select a correction method.

Common Mistake

Common mistake: “The direct effects average to zero, so horizontal pleiotropy cannot bias IVW.” This reasoning ignores the weights and the exposure associations. The relevant term is \(\sum_j w_j\beta_{Xj}\alpha_j\), not just \(\sum_j\alpha_j\). The correct conclusion is that a zero average can be reassuring under a specified balance assumption, but it does not by itself guarantee that the weighted direct-effect contribution is zero.

A related mistake is to treat directional and correlated pleiotropy as synonyms. Directional pleiotropy concerns the average direction of direct effects; correlated pleiotropy concerns their relationship with instrument-exposure associations. One can occur without the other, and both can occur together.

Key takeaway: Horizontal pleiotropy is represented by direct effects \(\alpha_j\) on the outcome outside the exposure pathway. Whether those effects are balanced, directional, or correlated helps describe their pattern, but the IVW shift depends on their weighted cross-product with \(\beta_{Xj}\).

Check Your Understanding

Use the direct-effect relation and the IVW bias expression to answer these questions.

In the association model, what does \(\alpha_j=0\) mean for variant \(G_j\), and which IV assumption does a nonzero \(\alpha_j\) threaten?
Can direct effects have an unweighted mean of zero while still shifting the IVW estimate? Explain using the weighted cross-product.
If the weighted cross-product \(\sum_j w_j\beta_{Xj}\alpha_j\) is negative, how does the IVW estimate compare with \(\theta\) under the stated model?
How does directional pleiotropy differ from correlated pleiotropy?
Why does a precise IVW estimate not, by itself, establish that horizontal pleiotropy is absent?