Causal Estimands For Mendelian Randomization

Why the Estimand Must Come First

Mendelian randomization (MR) uses genetic variation to investigate whether an exposure has a causal effect on an outcome. Before asking how an estimate is calculated, specify which causal effect is being targeted. “The effect of the exposure on the outcome” is incomplete unless the exposure contrast, outcome scale, and target population are clear.

A genetic variant can be associated with both an exposure and an outcome. Those are associations involving the variant; neither association, on its own, defines the causal effect of changing the exposure. The causal estimand is a property of a specified exposure intervention in a specified population. Whether genetic data can identify that estimand is a separate question.

Potential Outcomes and a Target Population

Let \(X\) denote an exposure and \(Y\) an outcome. For a person who could receive exposure level \(x\), write \(Y(x)\) for the outcome that would occur under that exposure level. These are potential outcomes: for any one person, only the outcome under the exposure actually received is observed. A causal contrast compares potential outcomes under alternative exposure interventions, rather than simply comparing people who happen to have different observed exposures.

The subscript \(T\) in an expectation indicates the target population. For example, \(E_T[Y(x)]\) is the mean potential outcome under exposure level \(x\) among members of that population. The target might be a defined population of adults, people with a particular clinical condition, or another group to which the scientific question refers. Specifying it matters: an average effect in one population need not be the average effect in another.

Potential outcomes must refer to sufficiently well-defined interventions. If “set exposure to 10” could mean several materially different interventions with different consequences, the notation \(Y(10)\) may not identify a single outcome. The interpretation also presumes that the outcome for one person is not changed by another person’s exposure assignment (no interference). These considerations concern the definition of the causal question, not whether an MR analysis can answer it.

Specify the Exposure Contrast

For a continuous exposure, an estimand should state the two exposure levels being compared or the size of the change. A contrast from 8 to 10 units is not automatically the same scientific quantity as a contrast from 10 to 12 units. The average change per exposure unit over a specified interval can be written as

where \(\delta\) is the stated exposure change. This is an average contrast per unit over that interval; it is not necessarily a constant effect at every exposure level. Calling it a “per-unit effect” without saying which interval or assuming a suitable constant-effect model can conceal this distinction.

The exposure’s units also matter. An effect per 1 mmHg is numerically different from an effect per 10 mmHg, although both may describe the same underlying contrast when correctly converted. For a transformed exposure, such as log concentration, the unit is on that transformed scale; it should not be described as a one-unit change in the original concentration.

Specify the Outcome Scale

The same exposure intervention can have distinct causal estimands on different outcome scales. For a binary outcome, let \(Y(x)\) equal 1 if the outcome occurs under exposure level \(x\), and 0 otherwise. Then \(E_T[Y(x)]\) is the risk of the outcome under that intervention. Comparing two risks can produce a risk difference, a risk ratio, or an odds ratio; these quantities answer different questions.

Genetic Associations Are Not the Causal Estimand

Let \(G\) denote a genetic variant. A genetic association with the exposure describes how the distribution or mean of \(X\) differs across values of \(G\). A genetic association with the outcome describes how the distribution or mean of \(Y\) differs across values of \(G\). These are statements about genotype groups in observed data. The causal estimand, by contrast, compares outcomes under interventions on \(X\).

For example, a difference in mean outcome between two genotype groups is not, just by being a genetic association, the mean outcome difference that would result from intervening to change the exposure. Genetic data can provide a basis for learning about an exposure’s causal effect only under additional identification conditions. Defining the target estimand does not establish those conditions, and this tutorial does not treat an association coefficient as a causal effect.

The Target Population Changes the Average

An average causal effect combines the effects for people in the target population. If effects differ across subgroups, changing the population’s subgroup proportions can change the average, even when each subgroup-specific effect stays the same. Thus, an estimand should say whose average is intended, not merely name the exposure and outcome.

For two subgroups, suppose the same exposure contrast and outcome scale are used in both. If subgroup 1 has proportion \(w\) in target population \(T\), and subgroup 2 has proportion \(1-w\), the population average effect is the weighted average of their subgroup-specific effects:

This expression follows by partitioning the target-population mean under each exposure intervention by subgroup, then subtracting the two means. The weights are population proportions, not sample sizes unless the sample represents the target population in those proportions.

A Practical Specification Checklist

Before interpreting an MR result as addressing a causal question, write down the intended estimand in plain language and, where useful, as a formula. Check that the words, units, and mathematical contrast agree.

Common Mistake

Wrong reasoning: “The variant is associated with the outcome, so this association is the causal effect of the exposure on the outcome.” This confuses the variable defining the genetic groups with the exposure being intervened on. The association is about genotype and outcome; the causal estimand is about outcomes under alternative exposure interventions.

Correction: State the exposure contrast, outcome scale, and target population first. Then assess whether the genetic analysis can identify that causal quantity under its assumptions. A carefully defined estimand is necessary for interpretation, but it is not evidence that identification assumptions hold.