The Three Instrumental-Variable Assumptions

Why Three Assumptions Are Needed

In Mendelian randomization, a genetic variant \(Z\) is used as an instrument for an exposure \(X\) to learn about its causal effect on an outcome \(Y\). The attraction is that inherited genetic variation can, under suitable conditions, provide variation in exposure that is less affected by some sources of confounding than ordinary observational comparisons. But a genetic association is not automatically a valid instrument. The causal interpretation depends on three assumptions: relevance, independence, and exclusion restriction.

These are causal assumptions, not three statistical tests that certify an instrument. Relevance concerns whether the instrument predicts the exposure. Independence and exclusion concern how the instrument relates to the potential outcomes and causal pathways. The distinction matters: a strong association with the exposure does not, by itself, establish that the instrument is independent of confounders or affects the outcome only through the exposure.

The Three Assumptions in Potential-Outcome Terms

For two instrument values \(z=0,1\), relevance requires that changing the instrument can change the exposure for at least some members of the target population. A common population-level expression is \(E[X\mid Z=1,C]\ne E[X\mid Z=0,C]\) for relevant values of \(C\). This mean contrast is a convenient expression of relevance; it is not a claim that every individual’s exposure changes.

Independence requires that, conditional on \(C\), instrument assignment be independent of the potential exposure and outcome values. One strong formulation is \(Z\perp\{X(0),X(1),Y(x,z):x,z\}\mid C\). This says that, within covariate strata, the instrument does not sort people according to the potential values that would determine their exposure or outcome. Independence is sometimes described as the absence of instrument–outcome confounding.

Exclusion restriction states that the instrument has no effect on the outcome except through the exposure. In potential-outcome notation, it requires \(Y(x,1)=Y(x,0)\) for every relevant \(x\). Thus, once exposure is held fixed, changing the instrument cannot change the outcome. A direct effect of the variant on the outcome, or an effect through a pathway that bypasses \(X\), violates this assumption.

Reading the Assumptions in a DAG

A directed acyclic graph (DAG) gives a complementary representation. A simple graph compatible with the assumptions contains the arrows \(Z\to X\to Y\), and may also contain \(U\to X\) and \(U\to Y\), where \(U\) is an unmeasured common cause of exposure and outcome. The pathway through \(U\) explains why an ordinary association between \(X\) and \(Y\) may be confounded. The instrument is useful if its variation shifts \(X\) without carrying that confounding into the comparison.

Relevance corresponds to a causal arrow or other causal dependence from \(Z\) to \(X\). Independence requires that there be no open noncausal path from \(Z\) to the potential outcomes after conditioning on \(C\). For example, if ancestry \(A\) affects both allele frequencies and the outcome, the path \(Z\leftarrow A\to Y\) can violate independence unless the design and adjustment adequately address it. Exclusion requires that every causal path from \(Z\) to \(Y\) pass through \(X\). An arrow \(Z\to Y\), or a path \(Z\to M\to Y\) through another trait \(M\), violates exclusion if that path does not operate through the defined exposure.

A DAG is a statement about causal structure, not a picture that proves the structure. No arrow from \(Z\) to \(Y\) represents the exclusion restriction; it cannot be confirmed simply because a statistical test fails to detect a direct association. Likewise, adjustment for \(C\) is justified only when the covariates and conditioning set are appropriate. Conditioning on a collider can itself create an association, so adding more covariates is not automatically protective.

How the Assumptions Support Identification

Identification means that a causal estimand can be expressed in terms of the distribution of observed data under stated assumptions. The three assumptions do not, on their own, identify every possible causal effect. The identified estimand depends on the treatment and instrument scales and on additional conditions. The following binary case makes this explicit.

Let \(Z\) and \(X\) both be binary. Define the potential exposure \(X(z)\) and the potential outcome \(Y(x)\), where exclusion allows the outcome to be indexed by \(x\) alone. In addition to consistency, suppose instrument independence holds, relevance gives \(E[X\mid Z=1]\ne E[X\mid Z=0]\), and the instrument is monotone: \(X(1)\ge X(0)\) for every individual. Monotonicity rules out individuals whose exposure would be reduced by changing \(Z\) from 0 to 1. Under these conditions, the Wald ratio identifies the average causal effect among compliers—individuals with \(X(1)=1\) and \(X(0)=0\)—not necessarily the average effect for everyone.

To see why, let \(C_{\mathrm{comp}}\) indicate complier status. Under independence and consistency, the difference in mean exposure between instrument groups equals \(P(C_{\mathrm{comp}}=1)\): always-takers and never-takers have no exposure difference across instrument values. Under exclusion and independence, the difference in mean outcome equals the complier proportion multiplied by the mean causal effect among compliers. Dividing the outcome contrast by the exposure contrast therefore removes the complier proportion.

This result is a local causal effect: it applies to the people whose exposure is changed by the instrument. For continuous exposures, a ratio of instrument–outcome and instrument–exposure associations needs further modeling or identifying conditions—for example, a linear structural model with an adequately specified homogeneous causal effect. The three assumptions alone do not make a ratio a general, nonparametric average treatment effect.

What Violations Do to the Causal Interpretation

A violation can produce a ratio that is numerically well-defined but causally misleading. Independence failure allows the instrument groups to differ in potential outcomes for reasons other than the instrument’s effect on exposure. Exclusion failure adds an outcome pathway that the exposure contrast does not capture. The next examples isolate these problems so that their effects on the arithmetic are visible.

Assessment, Alternatives, and Limitations

The assumptions differ in how directly observed data can inform them. Relevance can be assessed by estimating the instrument–exposure association, though the size and uncertainty of that association matter. Independence and exclusion are generally not fully testable from the observed instrument, exposure, and outcome alone: the relevant potential outcomes are never jointly observed, and an absent direct pathway is a causal claim. Background knowledge, study design, measured covariates, and sensitivity analyses can make these claims more or less credible, but no single diagnostic proves them.

In genetic studies, plausible threats include population structure that links allele frequencies to outcome determinants, assortative mating or family-level pathways, selection into the analyzed sample, and variant effects on the outcome through traits other than the target exposure. These possibilities are reasons to scrutinize the causal graph and target population. They are not resolved merely by labeling a variant “genetic.”

Instrumental-variable analysis is an alternative to estimating an exposure–outcome effect by ordinary covariate adjustment. Adjustment can identify a causal effect if all relevant confounders are appropriately measured and modeled; IV can identify a different estimand under its own assumptions, even when some exposure–outcome confounding remains. A randomized intervention on the exposure would avoid the need for an instrument, but may be infeasible or answer a different intervention question. The appropriate approach depends on the scientific estimand and the credibility of its assumptions.

A common mistake is to reason, “The variant is associated with the exposure, so it is a valid instrument.” That establishes, at most, evidence for relevance. A valid causal interpretation also requires independence and exclusion, and the estimand may require additional conditions such as monotonicity or a specified structural model. Another mistake is to interpret the binary Wald ratio as the population average treatment effect without qualification. Under the conditions derived here, it is the complier average causal effect.