Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 207 / 1

Let $Z=1$ denote gestation under unrestricted sugar and $Z=0$ gestation under rationing, and let $Y_i(z)$ be person $i$'s potential outcome for later Type 2 diabetes under exposure $z$. A sharp causal null hypothesis isagainst the alternative that the two potential outcomes differ for at least one person. An average-effect formulation instead tests $H_0:\mathbb{E}[Y(1)-Y(0)]=0$ against a nonzero, or scientifically directed positive, average effect.

The birth-period comparison requires consistency, no interference between individuals, and a real discontinuity in sugar exposure at the end of rationing. Near the cutoff, potential diabetes outcomes must otherwise vary continuously with birth date: no simultaneous policy, nutritional, diagnostic, seasonal, or demographic discontinuity may affect them. Birth dates must not be manipulable around the cutoff, and survival and UK Biobank participation must not create differential selection bias. These assumptions make cutoff assignment locally exchangeable and give a regression discontinuity design an exclusion restriction in which birth period affects diabetes through sugar exposure.

First, age and smooth birth-cohort trends affect diabetes risk. Restricting analysis to a narrow bandwidth around September 1953 and fitting flexible trends on both sides reduces this bias. Second, quarter of birth may affect later health through season, maternal infection, or food availability. Compare the same calendar quarters in adjacent years, include season effects, and examine placebo cutoffs. Third, other post-rationing changes in diet, income, healthcare, or early-life conditions may coincide with sugar availability. Measure and adjust those changes where possible, use outcomes they should affect as negative control outcomes, and compare with countries or groups lacking the sugar change. Differential survival or Biobank recruitment is another concern and should be examined with participation data, inverse-probability weighting, and sensitivity analysis.

Cultural abstainers form a negative control group: ending rationing should not materially change their sugar intake. A diabetes discontinuity at the cutoff among consumers but not abstainers supports the proposed sugar pathway, whereas a similar discontinuity in both groups points to cohort confounding. The interaction between cutoff exposure and consumer status, or a difference-in-differences contrast, formalizes this comparison, provided abstention itself is not differentially selected across cohorts.

Nonrepresentativeness does not automatically destroy internal validity: a causal contrast can remain valid among Biobank participants if selection is independent of the joint exposure-outcome process after conditioning on analysis variables. It does create bias if health or affluence affects participation and is also related to birth cohort or diabetes, especially when conditioning on participation opens a collider bias path.

External validity is weaker because affluent, healthy volunteers need not have the same baseline risk or sugar effect as the UK target population. Generalization requires either negligible effect modification by selection-related characteristics or standardization and inverse-probability weighting to the target population using variables measured in both sources. The unweighted estimate should otherwise be described as an effect for Biobank-like participants.