Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 207 / 1 (Statistics in Medical Practice)

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 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.

Political environment may be an effect modifier of the first stage: if red-county households hold larger birthday gatherings or use fewer mitigations, a truly causal contact mechanism predicts a larger birthday-associated infection increase there than in otherwise comparable blue counties. The comparison is therefore a mechanism check for heterogeneous treatment effects.

Its interpretation requires comparable epidemic timing, baseline prevalence, demographics, urbanicity, testing, reporting, and public-health rules across the compared counties, or adequate adjustment for them. It also assumes political classification changes gathering behavior without creating a different direct birthday-to-testing or birthday-to-infection pathway. Because voting category is ecological rather than individual, interpreting the pattern as individual behavior additionally risks the ecological fallacy.

First, compare counties only after matching, weighting, or regression adjustment for baseline prevalence, calendar date, population density, age structure, household size, income, testing intensity, and public-health restrictions. Use county-clustered uncertainty to respect within-county dependence.

Second, replace the coarse red/blue split by continuous vote share and estimate a prespecified birthday-event-by-vote-share interaction. A continuous analysis retains information, permits a dose-response check, and avoids sensitivity to an arbitrary 50% cutoff. Reporting subgroup sample sizes and correcting for multiple subgroup searches would further reduce selective interpretation.

A useful comparison divides counties into periods with strict and lenient limits on private gatherings. The split is worthwhile because the policy should alter the size or frequency of birthday gatherings, providing an independent check on the proposed first-stage mechanism.

If gatherings causally raise infection risk and restrictions reduce birthday contacts, the birthday-event association should be smaller under strict restrictions and larger under lenient restrictions. A graded pattern across restriction intensity would be stronger evidence than a single binary contrast.

The analysis assumes that restriction status is not merely a proxy for local epidemic severity, testing, voluntary caution, vaccination, or other determinants of infection, and that it does not change the direct effect of birthday timing on outcome ascertainment. It also assumes comparable compliance within each policy category and no differential migration or reporting.

Assess these assumptions by balancing or adjusting for pre-policy prevalence, testing, vaccination, mobility, demographics, and calendar time; inspect infection and testing trends before policy changes; and use mobility or contact data to confirm that restrictions actually weaken the birthday-to-gathering first stage. Placebo outcomes and dates provide additional negative control outcomes.