Two-Step Mendelian Randomization for Mediation

Why Use Two-Step Mendelian Randomization?

A genetic variant associated with an exposure may also be associated with an outcome because the exposure affects the outcome through a mediator. Two-step Mendelian randomization (MR) estimates this pathway in two parts: first, the causal effect of the exposure on the mediator; second, the causal effect of the mediator on the outcome. The product of those two effects estimates the indirect effect along the specified pathway, under substantive assumptions.

The procedure is useful when the two relationships are informed by different genetic instruments. It does not, by itself, estimate the total effect of the exposure on the outcome. Nor does using separate instrument sets automatically make either estimate valid: each set must satisfy the assumptions for the causal effect it is being used to estimate.

Two Steps and the Product Estimate

For the first step, genetic variants associated with \(X\) are used as instruments to estimate the effect of \(X\) on \(M\). For the second, variants associated with \(M\) are used as instruments to estimate the effect of \(M\) on \(Y\). The variants used in the two steps need not be the same. In either step, the genetic association estimates must refer to clearly defined scales: for example, a change in the mediator per one unit of exposure, followed by a change in the outcome per one unit of mediator.

With one instrument in a step, a basic estimate is the ratio of the variant’s association with the downstream trait to its association with the instrumented trait. With several independent variants, inverse-variance weighted (IVW) estimation combines their ratios. In the first step, write \(\hat{\gamma}_{Xj}\) for the estimated association of variant \(j\) with \(X\), and \(\hat{\gamma}_{Mj}\) for its association with \(M\). A first-order IVW estimate is:

The second step has the corresponding form, using variants \(k\) for the mediator and their associations with the outcome:

These first-order weights treat uncertainty in the instrument–exposure association in a step as small relative to the downstream association’s uncertainty. They are most defensible when those instruments are sufficiently strong. The formulas also describe a weighted regression through the origin; their use relies on compatible association scales and appropriate instrument validity, not just on the arithmetic.

Conditions for a Causal Interpretation

For either MR step, the variants used as instruments must satisfy the standard instrumental-variable requirements for that step. Relevance means they predict the trait they instrument: \(X\) in the first step and \(M\) in the second. Independence means they are independent of causes of the relevant downstream trait, conditional on appropriate design and ancestry control. Exclusion means their effects on that downstream trait operate through the instrumented trait, rather than through another route. These are distinct requirements, and a variant’s association with a trait alone establishes relevance, not the other two conditions.

For example, a variant used to estimate \(M \rightarrow Y\) cannot have an independent effect on \(Y\) outside the mediator pathway. If it changes \(X\), and \(X\) affects \(Y\) through a route other than \(M\), that route can also violate the exclusion condition for this step. Merely calling a variant a mediator instrument does not rule out such pathways.

The product interpretation also requires a mediation model in which the two effects combine in the stated way. In a simple linear additive model with no exposure–mediator interaction, write:

Here \(\alpha\) is the effect of \(X\) on \(M\), \(\beta\) is the effect of \(M\) on \(Y\) holding the direct \(X\) pathway fixed, and \(\delta\) is the direct effect of \(X\) on \(Y\) in this model. Increasing \(X\) by one unit changes \(M\) by \(\alpha\) units. The resulting change in \(Y\) through \(M\) is therefore \(\beta\alpha\). The direct change is \(\delta\), so the total change is \(\delta+\beta\alpha\). This justifies the product as the indirect component within this model; it does not establish the model or instrument assumptions from data alone.

A causal interpretation additionally depends on the target population, compatible effect scales, and the absence of relevant model misspecification. In particular, the two estimated effects must be meaningful components of the same pathway in the same causal setting. The product can still be calculated when assumptions are doubtful, but then it should be reported as a product of MR estimates rather than asserted to be an identified causal indirect effect.

Common Mistakes and Practical Checks

• Calling the product the total effect. The product estimates only the specified mediated pathway. A total-effect estimate includes all causal pathways represented by its model.
• Assuming separate instruments guarantee validity. Different variant sets do not prevent either set from violating relevance, independence, or exclusion.
• Ignoring units or scales. Check that the mediator units cancel in the product and that both steps refer to compatible scales and populations.
• Interpreting a product causally without a pathway model. The arithmetic is always possible; the indirect-effect interpretation needs valid instruments and assumptions that justify composing the two effects.
• Subtracting from a total effect automatically. The difference is a direct effect only under a compatible model and identification conditions, not simply because one estimate is labelled “total.”

Before interpreting a result, verify the instrument set for each step, the direction and units of both estimates, and the assumptions connecting them. Also distinguish the estimate’s sign and magnitude from its uncertainty: the point estimate alone does not show whether the pathway is estimated precisely.