Effect Heterogeneity And Genetic Instrument Estimands

Why One Valid Instrument Need Not Mean One Universal Effect

Mendelian randomization uses genetic variation to learn about the effect of an exposure on an outcome. A valid instrument can help separate exposure variation from confounding, but validity alone does not imply that the exposure has the same effect for everyone. If effects differ across people, exposure changes, or the groups whose exposure is shifted by different variants, an instrumental-variable estimate can be causal while still not representing a population-wide average.

The key question is therefore not only whether a genetic instrument meets the assumptions for causal interpretation. It is also which causal effect the instrument identifies. That estimand may be local to people whose exposure is changed by the instrument, or it may be a weighted average of different effects. The weights depend on how the instrument changes exposure, not necessarily on how common each person or effect type is in the population.

Potential Outcomes and Instrument-Induced Exposure Changes

Let \(Z\) denote a genetic instrument, simplified here to two levels, \(Z=0\) and \(Z=1\). Let \(X_i(z)\) be the exposure person \(i\) would have under instrument level \(z\), and let \(Y_i(x)\) be that person’s outcome under exposure level \(x\). The instrument’s effect on exposure for person \(i\) is \(\Delta X_i=X_i(1)-X_i(0)\). This individual-level change can vary across people: some may have a larger exposure shift than others, and some may have no shift.

For the ratio of the instrument-outcome contrast to the instrument-exposure contrast to have a causal interpretation, the usual instrumental-variable requirements include relevance (the instrument changes exposure on average), independence (the instrument is independent of relevant potential outcomes and exposure responses, possibly conditional on appropriate covariates), and exclusion (the instrument affects the outcome only through the exposure). In genetic analyses, independence can be threatened, for example, if genetic variation is associated with confounders through population structure. The assumptions are substantive; observing an association between genotype and exposure is not enough to establish them.

A genetic variant may influence exposure in a particular group more strongly than in another. If those groups also have different exposure effects on the outcome, the ratio can give greater influence to the group with the larger instrument-induced exposure changes. The derivation below makes this weighting explicit in a simple model.

What a Wald Ratio Averages When Effects Vary

For a binary instrument, define the first-stage contrast as the difference in mean exposure between instrument levels, and the reduced-form contrast as the difference in mean outcome. The Wald ratio is the reduced-form contrast divided by the first-stage contrast. To see what it represents under heterogeneous effects, suppose each person’s outcome response is linear over the exposure change induced by the instrument, with individual slope \(\beta_i\). Exclusion then gives \(Y_i(Z)=Y_i(X_i(Z))\), and linearity implies that the instrument-induced outcome change for person \(i\) is \(\beta_i\Delta X_i\).

If instrument level is independent of these potential exposure changes and slopes, the mean contrasts equal the population averages of the individual changes. Dividing the two contrasts yields the following result.

To justify the expression, independence makes the exposure contrast \(E[\Delta X_i]\). Exclusion and the individual linear response make the outcome contrast \(E[\beta_i\Delta X_i]\). The ratio follows by dividing the latter by the former. If every \(\Delta X_i\) is nonnegative and the average change is positive, this is a weighted average of individual slopes with weights proportional to \(\Delta X_i\). Larger exposure responders receive more weight. If some changes are negative, the weights can be signed, so the ratio need not lie between the smallest and largest individual slopes.

With nonlinear responses \(Y_i(x)\), the same broad issue remains, but a single individual slope may not describe the effect over the instrument-induced interval. For a person with \(\Delta X_i\ne0\), the relevant individual contrast per exposure unit is the secant slope \([Y_i(X_i(1))-Y_i(X_i(0))]/\Delta X_i\). The ratio weights these contrasts by exposure changes when changes are all in the same direction. It therefore depends on both how much exposure changes and where on the exposure scale those changes occur. It is not automatically the average effect of increasing exposure by one unit for every member of a target population.

The derivation relies on a linear response for the simple slope formula; it does not assert that all MR estimands are linear slopes. In practice, the scale and model used for the exposure-outcome association matter. For example, an effect expressed on a risk-difference scale is not the same estimand as an effect expressed on a log-odds scale. A causal interpretation must name the scale as well as the population or instrument-defined group.

Different Variants Can Identify Different Effects

Two genetic instruments can both satisfy the relevant validity assumptions and still shift exposure in different parts of the population. One variant might mainly change exposure among people with a particular biological background, while another shifts exposure more broadly. If exposure effects differ across those groups, the variants can identify different instrument-specific averages. A discrepancy between valid estimates is therefore not, by itself, proof that one instrument is invalid; heterogeneous causal effects can also explain it.

When variant-specific ratio estimates are combined using inverse-variance weights, the resulting estimate gives more weight to estimates with greater statistical precision. Those precision weights are not generally the same as population shares or as weights proportional to each variant’s exposure changes. Consequently, an inverse-variance weighted estimate need not equal a simple population average of causal effects. Its interpretation depends on the effects each instrument identifies and on the weighting used to combine them.

Interpreting the Estimand in Practice

A useful interpretation states what contrast is estimated, for whom, and on what scale. Before describing an MR estimate as an average causal effect, ask whether the assumptions support that target or only an instrument-dependent response. In particular, distinguish the population in which the analysis is conducted from the people whose exposure is most changed by the instrument. These need not be the same weighting scheme.

Effect heterogeneity also changes how a result should be generalized. A local or instrument-specific effect can be informative: it describes the causal response associated with exposure shifts generated by that instrument. But applying it to a different population, a different exposure range, or a policy that changes exposure for everyone requires additional assumptions about how effects vary. A single estimate does not reveal that effect pattern by itself.

The assumptions used for a local-effect result should be explicit. For example, monotonicity is central to the standard LATE result for a binary instrument and binary treatment; it is not a universal requirement for every continuous-exposure MR interpretation. Conversely, simply calling a ratio “local” does not establish a complier interpretation unless the binary-treatment assumptions are satisfied.

Common Mistake

Incorrect reasoning: “The genetic instrument is valid, so the Wald ratio must be the average causal effect in the whole population.” Validity supports a causal interpretation under the instrument assumptions, but it does not guarantee homogeneous effects or equal influence across individuals. With heterogeneous effects, the ratio can weight people according to their instrument-induced exposure changes, and a binary-treatment ratio can identify the effect among compliers.

Correction: Name the estimand supported by the design and assumptions. State whether it is a population average, an instrument-weighted average response, or a complier effect, and specify the outcome scale and exposure contrast. Do not infer a population-wide average merely from instrument validity.