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An instrumental variable changes an exposure or treatment but affects the outcome only through that exposure. Together, instrument relevance, instrumental-variable independence, and the exclusion restriction identify causal effects under suitable structural assumptions.

Mendelian randomization uses inherited genetic variants associated with an exposure as instrumental variables. Its causal interpretation requires instrument relevance, instrumental-variable independence, and the exclusion restriction.

The core instrumental variable assumptions are:

For a local average effect one additionally uses instrumental-variable monotonicity. Because the valid instrument supplies variation in education independent of the confounders and has no direct route to the outcome, the instrument-outcome association can be attributed to the education changed by the instrument without measuring every confounder.

Conditionally on $X$, a valid instrumental variable $Z$ must satisfy three core conditions.

Together with consistency of potential outcomes and positivity in causal inference, these assumptions make variation in $A$ induced by $Z$ causally interpretable. In the displayed graph, relevance is the edge $Z\to A$, independence is the absence of a path from $Z$ to $U$ after conditioning on $X$, and exclusion is the absence of a direct $Z\to Y$ edge.

The plots test the plausibility of the instrumental-variable independence and exclusion restriction assumptions. Both candidate instruments are strongly associated with 25(OH)D, supporting instrument relevance. The focused instrument has estimates near zero for the other measured traits. The polygenic instrument is strongly associated with LDL cholesterol and triglycerides, suggesting horizontal pleiotropy: it may influence mortality through lipid pathways that do not pass through vitamin D.

I would therefore prefer the focused instrument. Its use of fewer biologically understood variants may reduce precision, but its cleaner associations make the causal assumptions more credible.

Let $G$ be the genetic instrument, $X$ vitamin-D concentration, $U$ an unmeasured cause of vitamin D and mortality, and $Y$ mortality. The causal directed acyclic graph containsAlthough $G$ and $U$ are marginally independent, $X$ is a collider. Conditioning on $X$ opens the path $G\leftrightarrow U\to Y$, violating instrumental-variable independence within the resulting strata. Residual exposure stratification instead removes the component of $X$ predicted by $G$ before stratification; under the additive first-stage model, the stratifying variable is no longer caused by $G$.

In potential outcome notation, a valid instrumental variable $Z$ must satisfy:

In the graph, $Z_1$ has open noncausal paths to $Y$ that do not pass through $A$, includingEquivalently, these paths remain after deleting the causal edge $A\to Y$. Thus $Z_1$ is associated with potential outcomes through the latent variables $C$ and $S$, violating instrumental-variable independence; it is not a valid marginal instrument.

The assumptions are plausible only approximately. First, birthday timing can correlate with age, household composition, season, holidays, local epidemic phase, testing, or health-care use. Any such common cause of $Z$ and $Y$ violates instrumental-variable independence. Second, a birthday may alter contacts, deliveries, travel, or testing even without the intended party exposure; those pathways violate the exclusion restriction. The instrument may also be weak where restrictions or low local prevalence suppress gatherings.

Two useful assessments are:

These checks cannot prove independence or exclusion, but failures directly falsify implications of those assumptions.

Let $Z$ indicate a recent birthday event, $A$ indicate a social gathering, and $Y$ denote subsequent household COVID-19 infection. The standard instrumental variable conditions are:

The usual consistency in causal inference, well-defined exposure, and absence of relevant interference in causal inference are also needed to interpret the result causally.

The instrumental variable graph iswith no arrow $Z\to Y$ and no common cause of $Z$ with $A$ or $Y$.

In potential outcome notation, a valid instrument requires: