Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 46 / 1

Treating the two-year observation period as approximately complete, divide total person-time at risk by two to estimate the average number of children being observed. Dividing again by the number of families givesThus the estimate is about 1.1 eligible children per family. This is a person-time-based approximation for children contributing to the study, not a count of older children or an exact reconstruction of family composition if follow-up or eligibility changed.

The control attendance rate is approximately $0.0442$ per child-year. Assuming that rate is roughly constant over the first five years, expected attendance count is $5(0.0442)\simeq 0.221$. A crude rare-event approximation gives about a 22% chance of at least one attendance.

A constant-rate Poisson process instead givesThus roughly one child in five is the appropriate order of magnitude. An attendance rate counts recurrent attendances, whereas a first-attendance risk counts children. The aggregate rate alone does not identify that risk without assumptions about recurrence and age-specific incidence; the Poisson conversion makes those assumptions explicit.

Let $Y_I,Y_C$ be attendance totals and $T_I,T_C$ the observed person-time at risk. Under independent Poisson distributions for the totals, the estimated incidence rate ratio is $\hat{R}=(Y_I/T_I)/(Y_C/T_C)$. Treating the exposures as known, the supplied large-count approximation, or the delta method, givesIndependence adds the log-count variances, while fixed exposure terms add no variance. Form the normal confidence interval on the log scale and exponentiate:This specifies the calculation without evaluating it. Individual randomization makes a child the assignment unit, but an independent Poisson count model remains an additional assumption about outcomes, not a consequence of randomization alone.

This is a family-level cluster-randomized trial, but there are only about 1.1 contributing children per family. Most families therefore contribute a single eligible child, leaving little opportunity for within-family intraclass correlation to add much uncertainty. The equal-size exchangeable approximation has design effect $1+(m-1)\rho$, with cluster size $m$ and correlation $\rho$; when $m$ is close to one, this is close to one. A multilevel Poisson regression can consequently give nearly the same standard error as the independent-child working analysis.

Agreement is compatible with limited within-family dependence and little change from covariate adjustment; it does not prove all dependence absent. Mean family size alone does not determine the exact design effect for highly unequal cluster sizes.

Larger differences would be expected with larger families and stronger within-family correlation, especially with unequal family sizes. Pronounced between-family heterogeneity, recurrent attendances concentrated in particular children, or important imbalanced covariates can also create overdispersion or change the adjusted comparison.

One possibility is increased recognition and care seeking: consultation may make caregivers more alert to minor injuries or more inclined to seek professional care, increasing recorded attendance without increasing injury frequency. This is the distinction in care-seeking effects on a recorded injury outcome.

A second possibility is reduced vigilance or greater exposure after feeling protected: reliance on supplied equipment may lead caregivers to relax other precautions. This is risk compensation in a prevention trial. Both are hypotheses to investigate; an elevated attendance incidence rate ratio does not establish which mechanism occurred.