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

Vitamin D is a causal determinant of mortality risk if an intervention that changes vitamin-D status changes the distribution of the corresponding potential outcome for mortality. This is a claim about a causal effect, rather than merely an observed association.

First, an observational study can suffer from confounding or reverse causality. Ill health may both lower circulating vitamin D and raise mortality, while lifestyle, socioeconomic status, and comorbidity may affect both variables. Randomization breaks these baseline associations in expectation.

Second, the interventions answer different questions. Supplementation may be too small, too late, too short, or poorly adhered to, and an average effect can be nearly zero when benefit is confined to people with severe deficiency. Such heterogeneous treatment effects can coexist with a strong observational gradient.

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 $G$.

The overall odds ratio is $0.99$ with 95% interval $(0.95,1.02)$, so there is little evidence for an average effect. The estimate changes sharply across residual-vitamin-D strata: it is $0.69$ $(0.59,0.80)$ in the deficient group, $0.94$ $(0.89,0.99)$ in the insufficient group, and close to one in the two higher groups. The pattern is consistent with effect modification: raising vitamin D may reduce mortality among deficient people but offer little benefit once concentration is adequate. The claim still depends on the Mendelian randomization assumptions and should account for the fact that several subgroup estimates were examined.

One check is to use measured risk factors as negative control outcomes. A valid instrument should not predict traits that cannot plausibly be downstream consequences of vitamin D; persistent associations would expose horizontal pleiotropy or population structure.

A second check is to repeat the analysis with separate biologically motivated variants or gene-region scores and compare their ratio estimates. Agreement across instruments with distinct biological pathways supports the common vitamin-D mechanism, whereas excess between-instrument heterogeneity suggests direct effects. The same data can also support sensitivity analyses that adjust for measured pleiotropic pathways.