Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 40 / 3 / ii / Solution Created 2026-10-03 Updated 2026-10-14
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Four distinct objections, each with a corresponding improvement, are as follows.
  • Different patients imply different baseline risks. Raw death rates confound performance with case mix, including urgency, disease severity and comorbidity. Use a prespecified, validated risk-adjusted provider comparison, standardizing to comparable patients or comparing observed deaths with . Check overlap and model calibration rather than assuming adjustment removes all confounding.
  • Small annual volumes produce unstable rankings. A few deaths can move a surgeon far up or down a league table; zero deaths need not indicate zero risk. Display counts and suitably calibrated confidence intervals for binomial proportions, preferably in funnel plots against volume. Pool suitable periods or use a hierarchical Bayesian model to stabilize estimates, while allowing genuine time changes.
  • Many simultaneous comparisons generate chance outliers. With independent null comparisons, the probability of at least one false positive is , and ranks amplify selection of extremes. Use multiple testing adjustments or simultaneous control limits and seek confirmation before interpreting an apparent outlier. Dependence through the national comparator must also be accounted for.
  • The endpoint and unit of attribution can mislead. In-hospital death depends on discharge and transfer policy, and care is delivered by a team; follow-up after discharge is omitted. Use an independently audited fixed-horizon endpoint, such as a prespecified 30-day mortality outcome, linked across hospitals and to death registrations, and report an appropriate team or institutional comparison alongside any surgeon-level estimate. Reliable attribution and consistent inclusion criteria reduce incentives to manipulate coding or case selection.
A confidence interval containing the national rate does not demonstrate equivalence, and a raw ranking does not establish a causal performance difference. The improved presentation should convey precision and uncertainty as well as the estimated outcome.
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