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> contains
$$
G\longrightarrow X\longleftarrow U\longrightarrow Y.
$$
Although $G$ 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 $G$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 207 / 1 / 1 / 4 / Solution

Residual exposure stratification subtracts the genetically predicted component of an exposure before forming exposure strata. Under a suitable additive first-stage model, this avoids directly conditioning on the component caused by the instrument and can reduce the resulting <collider bias>.

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 Incoming links: Residual exposure stratification