Direct And Correlated Pleiotropy

Why Direct Genetic Effects Matter

Mendelian randomization (MR) uses genetic variants as instrumental variables to estimate the causal effect of an exposure \(X\) on an outcome \(Y\). For variant \(j\), let \(\gamma_j\) denote its association with \(X\), \(\Gamma_j\) its association with \(Y\), and \(\alpha_j\) its direct effect on \(Y\) that does not operate through \(X\). Under a linear model, these associations satisfy

\Gamma_j = \beta\gamma_j+\alpha_j,

where \(\beta\) is the causal effect of \(X\) on \(Y\). If the variant affects \(Y\) only through \(X\), then \(\alpha_j=0\). A nonzero \(\alpha_j\) violates the exclusion restriction: the variant has a horizontal pathway to the outcome in addition to its pathway through the exposure. The concern is not merely that direct effects exist, but whether their pattern biases the estimator being used.

Definition: Horizontal pleiotropy occurs when a genetic variant affects the outcome through a pathway that does not pass through the exposure of interest. Vertical pleiotropy describes effects downstream of the exposure: these are transmitted along the causal pathway from the variant through \(X\) to \(Y\), and are part of the effect represented by \(\beta\), not a direct effect \(\alpha_j\).

A simple causal diagram for a valid instrument has \(G_j\to X\to Y\), with no other path from \(G_j\) to \(Y\). Horizontal pleiotropy adds \(G_j\to Y\) alongside that chain. The instrument-independence condition also requires that the genetic instrument be independent of causes of the exposure and outcome; adding a direct arrow to \(Y\) is a different problem from violating that independence condition. As in the earlier material on exclusion restriction, association alone does not establish that a variant’s outcome association is mediated by \(X\).

Three Patterns of Direct Effects

The terms balanced, directional, and correlated describe different features of the collection of direct effects across variants. They are not mutually exclusive categories: direct effects can have a nonzero average and also be correlated with instrument strength.

Definition: Balanced pleiotropy means that direct effects are distributed around zero, so positive and negative effects balance on average under the stated averaging scheme. Directional pleiotropy means that direct effects tend to point in one direction, so their average is not zero. Correlated pleiotropy means that direct effects \(\alpha_j\) are associated across variants with their exposure effects \(\gamma_j\). This is a relationship across variants, not an additional causal arrow from \(\gamma_j\) to \(\alpha_j\).

“Balanced” needs care in an estimation context. An unweighted average of direct effects equal to zero is not necessarily enough for inverse-variance weighted (IVW) estimation to be unbiased. IVW gives variants different leverage according to their exposure effects and precision. The relevant cancellation condition for the simple IVW formula is a zero weighted sum of exposure effects times direct effects, not simply a zero unweighted average.

Likewise, directional and correlated pleiotropy describe distinct properties. Direct effects that are all positive are directional, but need not be correlated with \(\gamma_j\). Direct effects can also increase with \(\gamma_j\), making them correlated, whether or not their unweighted average is zero.

How Direct Effects Enter Common Estimates

For a single variant with \(\gamma_j\ne 0\), the population Wald ratio is the outcome association divided by the exposure association. Substituting the association model shows exactly how a direct effect shifts it:

\frac{\Gamma_j}{\gamma_j} = \frac{\beta\gamma_j+\alpha_j}{\gamma_j} = \beta+\frac{\alpha_j}{\gamma_j}.

Thus, a nonzero direct effect biases the Wald ratio unless it is zero; its impact depends on its size relative to the variant’s exposure association. For independent variants and fixed positive weights \(w_j\), the simple IVW estimate is the weighted regression through the origin of \(\Gamma_j\) on \(\gamma_j\):

\hat{\beta}_{\mathrm{IVW}} = \frac{\sum_j w_j\gamma_j\Gamma_j}{\sum_j w_j\gamma_j^2}.

To see the direct-effect contribution, substitute \(\Gamma_j=\beta\gamma_j+\alpha_j\):

\hat{\beta}_{\mathrm{IVW}} = \frac{\sum_j w_j\gamma_j(\beta\gamma_j+\alpha_j)} {\sum_j w_j\gamma_j^2} = \beta+ \frac{\sum_j w_j\gamma_j\alpha_j} {\sum_j w_j\gamma_j^2}.

This identity is a population-association calculation with the weights and \(\gamma_j\) treated as fixed. It isolates the bias term; it does not account for uncertainty in estimated associations, weak-instrument bias, or other violations of the instrumental-variable assumptions. Under this simplified setup, IVW has no direct-effect bias precisely when \(\sum_j w_j\gamma_j\alpha_j=0\). Balanced direct effects can satisfy this condition, but zero unweighted mean alone does not guarantee it. Directional effects often make the sum nonzero, while correlation between \(\gamma_j\) and \(\alpha_j\) can make it nonzero even when direct effects average to zero.

Worked Example: Balanced Effects That Cancel in IVW

Consider two hypothetical independent variants, equal weights \(w_1=w_2=1\), exposure associations \(\gamma_1=0.10\) and \(\gamma_2=0.20\), and direct effects \(\alpha_1=0.02\) and \(\alpha_2=-0.01\). Suppose the causal effect is \(\beta=0.50\) outcome units per exposure unit. The direct effects have opposite signs. Check whether they cancel for IVW.

The outcome associations implied by the model are \( \Gamma_1=(0.50)(0.10)+0.02=0.07 \) and \( \Gamma_2=(0.50)(0.20)-0.01=0.09 \), in outcome units per allele.

The direct-effect contribution to the IVW numerator is \( \gamma_1\alpha_1+\gamma_2\alpha_2=(0.10)(0.02)+(0.20)(-0.01)=0.002-0.002=0 \). Therefore,

\hat{\beta}_{\mathrm{IVW}} = \frac{(0.10)(0.07)+(0.20)(0.09)} {(0.10)^2+(0.20)^2} = \frac{0.007+0.018}{0.01+0.04} = \frac{0.025}{0.05} = 0.50.

Here the direct effects cancel under the IVW weighting, so the estimate equals the stipulated causal effect. The result does not mean each variant is valid: the individual Wald ratios are \(0.07/0.10=0.70\) and \(0.09/0.20=0.45\), each shifted by its own direct effect.

Worked Example: Directional Effects Shift IVW

Now use two hypothetical variants with \(\gamma_1=0.10\), \(\gamma_2=0.20\), equal weights, causal effect \(\beta=0.50\), and direct effects \(\alpha_1=\alpha_2=0.02\). The direct effects point in the same direction, so they are directional. The associations are \(\Gamma_1=(0.50)(0.10)+0.02=0.07\) and \(\Gamma_2=(0.50)(0.20)+0.02=0.12\).

The IVW estimate is

\hat{\beta}_{\mathrm{IVW}} = \frac{(0.10)(0.07)+(0.20)(0.12)} {(0.10)^2+(0.20)^2} = \frac{0.007+0.024}{0.01+0.04} = \frac{0.031}{0.05} = 0.62.

The estimated effect is \(0.12\) outcome units per exposure unit above the stipulated value \(0.50\). The bias expression gives the same result: \( \sum_j\gamma_j\alpha_j/\sum_j\gamma_j^2=(0.002+0.004)/0.05=0.12 \). The individual Wald ratios are \(0.70\) and \(0.60\); they are both shifted upward, though by different amounts because \(\alpha_j/\gamma_j\) differs.

Correlated Pleiotropy and What Heterogeneity Can Miss

Correlated pleiotropy matters because IVW’s bias term depends on the relationship between \(\gamma_j\) and \(\alpha_j\). For example, suppose three hypothetical variants have \(\gamma_j=0.10, 0.20, 0.30\), equal weights, and \(\alpha_j=0.10\gamma_j\). The direct effects are \(0.01, 0.02, 0.03\), so they increase with the exposure associations. If \(\beta=0.50\), then \( \Gamma_j=(0.50+0.10)\gamma_j \) for every variant. Each Wald ratio is exactly \(0.60\), and IVW is also \(0.60\), despite there being no disagreement among the ratios.

This example shows why visible heterogeneity is not required for pleiotropic bias: a shared proportional direct effect shifts every ratio by the same amount. Conversely, different ratios may arise from direct effects that cancel in IVW, as in the first example. The relationship between effects and the causal structure matters; a single pattern in the ratios does not identify the mechanism.

Even a zero unweighted average can fail to protect IVW when effects are correlated with instrument strength. For three variants, let \(\gamma=(0.10,0.20,0.30)\) and \(\alpha=(-0.01,0,0.01)\), with equal weights. The direct effects average to zero, but \( \sum_j\gamma_j\alpha_j=-0.001+0+0.003=0.002 \), while \( \sum_j\gamma_j^2=0.14 \). The IVW shift is therefore \(0.002/0.14\approx0.0143\) outcome units per exposure unit. Here larger exposure associations tend to accompany larger direct effects, so the weighted cancellation condition fails.

A causal diagram can show a direct route \(G_j\to Y\), but it does not by itself display whether direct effects across different variants are balanced, directional, or correlated. Those are properties of the collection of variant-specific effects. In particular, “correlated pleiotropy” does not mean that the genetic variants are correlated through linkage disequilibrium. Correlated variants require accounting for their association structure in estimation; correlation between \(\gamma_j\) and \(\alpha_j\) is a separate biological and statistical issue.

Worked Example: Read the Causal Diagram Before Interpreting an Effect

Suppose a hypothetical variant \(G\) affects an inflammatory marker \(X\), and \(X\) affects an outcome \(Y\). The variant also changes a separate metabolic pathway that affects \(Y\) directly. State which path is horizontal and how it enters the association model.

The path \(G\to X\to Y\) is the exposure-mediated path and contributes \(\beta\gamma\) to the variant-outcome association. The separate path \(G\to Y\), which bypasses \(X\), is horizontal pleiotropy and contributes \(\alpha\). Thus the total association is \( \Gamma=\beta\gamma+\alpha \). For hypothetical associations \(\gamma=0.10\), \(\beta=0.50\), and \(\alpha=0.02\), the outcome association is \( \Gamma=(0.50)(0.10)+0.02=0.07 \); the Wald ratio is \(0.07/0.10=0.70\), not \(0.50\). This interpretation depends on the separate pathway truly bypassing the exposure. A pathway downstream of \(X\) instead belongs to the mediated causal effect.

Assumptions, Diagnostics, and Limits

The calculations above assume correctly aligned alleles, nonzero instrument-exposure associations, and population associations that can be represented by the stated linear model. The simple IVW expressions also assume independent variants and fixed positive weights; when variants are correlated, the association covariance must be incorporated rather than treating the variants as independent. The causal interpretation additionally requires the instrumental-variable assumptions: relevance, independence from confounders of exposure and outcome, and exclusion restriction. Pleiotropy is the particular concern that the last condition fails.

Useful checks include reviewing variant biology for plausible direct pathways, comparing variant-specific ratios, and asking whether the conclusions are sensitive to instruments with unusual effects. Such checks can flag incompatibility with a simple common-effect model, but they cannot prove that instruments are valid. Heterogeneity may arise for reasons other than horizontal pleiotropy, and proportional direct effects can bias all ratios similarly without producing conspicuous disagreement. Likewise, observing no obvious biological route is not evidence that \(\alpha_j=0\).

The main alternatives to a single-variant Wald ratio or standard IVW are methods designed to tolerate particular patterns of invalid instruments. Their robustness depends on additional assumptions about the direct effects, so no method removes the need to reason about the causal structure. For example, allowing an average direct effect to differ from zero does not, by itself, resolve correlation between direct effects and instrument strength. The appropriate estimator depends on which assumptions are plausible, not just which diagnostic gives the most reassuring result.

A common mistake is to reason, “the average direct effect is zero, so IVW must be unbiased.” The correction is to check the estimator-specific condition: for the simplified IVW model, the relevant quantity is \(\sum_j w_j\gamma_j\alpha_j\). Another mistake is to treat similar variant-specific ratios as proof of no pleiotropy. The proportional example shows why this fails: a common direct-effect component can shift every ratio by the same amount. Diagnostics describe patterns in the observed associations; they do not reveal the causal pathway with certainty.

Key takeaway: Direct effects enter a Wald ratio as \(\alpha_j/\gamma_j\) and shift simple IVW by \(\sum_j w_j\gamma_j\alpha_j/\sum_j w_j\gamma_j^2\). Balanced effects can cancel, but only under the relevant weighting; directional or correlated effects commonly create a nonzero shift. Similar ratios do not rule out bias.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

A single variant has \(\gamma=0.08\), \(\beta=0.40\), and \(\alpha=-0.008\). Find its outcome association and Wald ratio.
For two equal-weight variants, \(\gamma_1=0.10\), \(\gamma_2=0.20\), and \(\alpha_1=0.01\), \(\alpha_2=-0.005\). Does the direct-effect contribution to IVW cancel?
Explain how direct effects can average to zero across variants and still bias IVW.
Do identical variant-specific ratios establish that horizontal pleiotropy is absent? Explain.

Solutions. (1) \( \Gamma=(0.40)(0.08)-0.008=0.024 \). The Wald ratio is \(0.024/0.08=0.30\), which is below \(0.40\) by \(0.10\), equal to \(\alpha/\gamma\). (2) The weighted numerator contribution is \( (0.10)(0.01)+(0.20)(-0.005)=0.001-0.001=0 \), so it cancels in this equal-weight IVW calculation. (3) Zero unweighted mean does not imply \( \sum_j w_j\gamma_j\alpha_j=0 \); if direct effects are associated with instrument strength, the estimator-weighted contributions need not balance. (4) No. A proportional direct effect can shift all ratios by the same amount, leaving them identical while violating the exclusion restriction.