Genetic Instrument Independence

Why Independence Is Not Guaranteed by Inheritance

Mendelian randomization uses genetic variants as instruments for an exposure. The appeal is that alleles are assigned before many later-life exposures and outcomes occur. But inheritance does not make a genetic variant independent of every factor that confounds the exposure–outcome relationship in the population being studied. Ancestry, family environment, mate choice, and participation in a study can all create or reveal associations between genotype and those factors.

The independence assumption concerns the relation between the instrument and confounders. It is distinct from instrument relevance, which concerns whether the instrument predicts the exposure, and from the separate requirement that the instrument not affect the outcome through a pathway other than the exposure. A variant should be associated with the exposure under relevance; independence does not mean that the variant is unrelated to the exposure or outcome.

Here, \(U\) can include measured or unmeasured factors that affect both exposure and outcome. It can also include causes of those factors that induce genotype–confounder dependence. Independence is therefore a substantive assumption about how genetic variation is distributed in the analyzed data, not a property guaranteed simply by describing a variant as genetic.

How Dependence Can Distort an Estimate

A short linear-model derivation shows why this assumption matters. For illustration, suppose variables are centered and the exposure and outcome follow

Assume \(\operatorname{Cov}(G,\varepsilon_X)=0\), \(\operatorname{Cov}(G,\varepsilon_Y)=0\), and \(\operatorname{Cov}(G,U)\) is the only source of dependence between \(G\) and the outcome-confounding term in this simplified model. The coefficient \(\alpha\) represents instrument relevance, while \(\delta\) describes the effect of \(U\) on \(Y\) not mediated by \(X\). This model deliberately omits other pathways so that the effect of violating independence can be isolated.

The ratio equals \(\beta\) only if the extra covariance term is zero under these model assumptions. Thus, when \(G\) and \(U\) are associated and \(U\) affects \(Y\), the instrument–outcome association contains a component related to confounding. The direction of distortion depends on the signs of \(\delta\), \(\operatorname{Cov}(G,U)\), and \(\operatorname{Cov}(G,X)\); it is not necessarily toward or away from zero.

Worked Examples: Four Routes to Dependence

Recognizing and Investigating Violations

The four mechanisms above can overlap. Population structure means that ancestry-related differences in allele frequencies coincide with differences in confounders. Adjusting for ancestry can reduce this source of dependence, but a few ancestry summaries may not capture fine-scale structure, recent admixture, or socially patterned environmental differences. Adjustment is a design and analysis strategy, not proof that independence holds.

Assortative mating occurs when mates are more similar or dissimilar on relevant traits than would be expected under random pairing. It can alter correlations among inherited traits and between genetic variants and family characteristics. Dynastic effects, sometimes called genetic nurture in this context, arise when parental genotype influences a child’s environment or behavior. A child inherits part of the parental genome, so the child’s genotype can be associated with that family environment even if allele transmission is random conditional on parental genotypes.

Selection is a separate concern: even if the instrument and confounders are independent in the source population, restricting analysis to people who survive, enroll, meet eligibility criteria, or have complete measurements may create dependence. The selection process need not explicitly use genotype. It is enough that selection is affected by factors associated with genotype and by confounders or their causes.

Useful diagnostics and design checks include:

• Compare instrument frequencies and measured confounders across ancestry groups, recruitment sites, and other plausible population strata. Large differences motivate closer investigation, though absence of a detected difference is not proof of independence.
• Check whether the variant or instrument set is associated with measured confounders. Associations can reveal problems, but null tests may have low power, and testing many variants creates multiple-comparison concerns.
• Describe who entered the analysis and why. Compare participants with the eligible source population where possible, and assess whether inclusion or missingness could depend on both genetic and confounder-related factors.
• Consider family-based comparisons when appropriate. Comparing relatives can control some shared ancestry and family environment, but it does not automatically remove all individual-level confounding, measurement error, or selection bias; such analyses may also be less precise.
• Use subject-matter knowledge about ancestry, mate choice, parental effects, and recruitment. No single statistical test can establish independence with respect to every measured and unmeasured confounder.

An observed genotype–confounder association is a warning to investigate, not a complete diagnosis of the bias in a causal estimate. Conversely, failure to detect association does not establish independence: unmeasured confounders may remain, tests may be underpowered, and selection can induce dependencies that are difficult to measure directly.

Choosing a Response and Knowing Its Limits

The appropriate response depends on the suspected mechanism. Ancestry covariates or stratified analyses may address broad population structure; family-based designs can reduce some ancestry and dynastic concerns; and careful eligibility, recruitment, and missing-data analyses can clarify selection. Each approach changes what comparisons are being made and may require additional assumptions. For example, adjustment can leave residual structure, while a family-based analysis may target a different population or have substantially less precision.

These remedies should not be treated as interchangeable. Ancestry adjustment does not necessarily remove family-level effects; family comparisons do not undo selection into a study; and restricting to a more homogeneous sample can itself change selection and the population to which results apply. Report the target population, the covariates or design features used, and the remaining plausible sources of dependence.

Finally, keep independence conceptually separate from the other instrumental-variable assumptions. A variant can be relevant but associated with a confounder, so relevance alone is insufficient. Likewise, diagnosing independence does not establish that every other requirement for a causal interpretation holds. Each assumption addresses a different route by which the observed genetic association could fail to identify the exposure effect.

Common Mistake

Incorrect reasoning: “Alleles are assigned at conception, so genotype is independent of confounders in any adult study.” This treats inheritance as if it randomized variants across ancestry groups, families, and the selected study sample. It also overlooks that parental genotype can shape family environments and that participation can depend on multiple causes.

Correction: Ask whether the instrument is independent of relevant confounders in the actual target and analyzed population, possibly conditional on justified covariates. Then consider population structure, assortative mating, dynastic effects, and selection explicitly. Genetic inheritance provides a rationale for the design, not an automatic guarantee of the assumption.