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Joined 2026-09-21 Message
Incoming links: Covariate balance
Past exam of the mathematics course of the University of Cambridge 2012 iii Paper 41 1 a iii Solution Created 2026-10-03 Updated 2026-10-07
Measure baseline covariates before the intervention can change them. These measurements describe the recruited population, permit assessment of covariate balance, and provide prognostic predictors that can improve statistical efficiency. In particular, including prognostic baseline covariates in a linear regression reduces unexplained outcome variation without conditioning on a treatment consequence.
Past exam of the mathematics course of the University of Cambridge 2021 iii Paper 221 4 a Solution 2026-09-28
Restricting to pupils who already attended a Catholic middle school improves covariate balance: the treatment-group differences in family income, urban residence, and prior mathematics score are all much smaller. This makes severe extrapolation and measured confounding less prominent.
The restriction reduces the sample from to , so estimates are less precise. It also changes the target population to Catholic-middle-school pupils, reducing external validity for all United States pupils.
Past exam of the mathematics course of the University of Cambridge 2023 iii Paper 207 1 1 2 Solution 2026-09-28
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 and 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:
- Check covariate balance across birthday-event groups, including age, household size, calendar time, local prevalence, and pre-event infection or testing. Repeat the analysis at placebo dates and for negative control outcomes.
- Measure a first-stage proxy for gatherings, such as mobility, restaurant visits, or contact reports, and verify instrument relevance. An event-study around the birthday can test for pre-existing trends and for an effect concentrated after, rather than before, the birthday.
These checks cannot prove independence or exclusion, but failures directly falsify implications of those assumptions.
Past exam of the mathematics course of the University of Cambridge 2026 iii Paper 207 1 d Solution Created 2026-09-24 Updated 2026-09-25
Using 50 controls per UTC can reduce sampling variance by averaging over more control outcomes. The gain has sharply diminishing returns because controls matched to the same UTC are not 50 independent treated-control contrasts.
The disadvantage is poorer covariate balance: the 50th-nearest school will usually be much less comparable than the fifth-nearest school, increasing residual confounding and changing the target population. A caliper or variable matching ratio should therefore prevent distant matches even if many controls are available.