Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 37 / 3

Under the proposed exposure-only model, suppose a unit of infectious exposure has the same probability of producing an ascertained onset in either cohort. Conditional on the total of $108$ onsets, the allocation is a binomial distribution with younger-cohort probabilityThus the expected cohort counts are $36$ and $72$, respectively, whereas the observed allocation is reversed. The Pearson chi-squared goodness-of-fit test givesWith one degree of freedom its approximate p-value is $2.0\times 10^{-13}$; the exact upper binomial distribution tail $\mathrm{Pr}\{\mathrm{Bin}(108,1/3)\ge 72\}$ is approximately $1.56\times 10^{-12}$. These are exceptionally incompatible with proportional allocation. Equivalently, the observed onset count per exposure unit is four times larger in the younger cohort:Under the stated common-risk exposure model, the observed counts are not consistent with the early exposure period as their sole explanation. This does not identify the source of the discrepancy, or prove a particular later exposure caused it. The inference also requires comparable ascertainment and a common fraction of exposure-derived cases manifesting by the observation date. Age-independent incubation periods alone do not guarantee that fraction when the two cohorts' exposure profiles within the decade differ; decade totals omit that timing information. The reasoning concerns the hypothetical historical BSE/vCJD model, not an estimate of current disease risk.

Two distinct issues are:

Thus the comparison must account for uncertainty in both its exposure denominator and its observed-disease numerator; a very small sampling p-value does not remove those model uncertainties.

Using the upper stated scrapie-test-positive fraction, its value among slaughtered sheep is $2000/4000000=0.0005$. The stipulated conditional BSE positivity then gives marginal test-positive probabilitybecause the other stratum contributes zero. If the sample consists of independently tested, representative animals, the count $D$ has a binomial distribution with parameters $n,q$. ThereforeThe two results areThe Poisson distribution approximation uses mean counts $0.5$ and $5$, respectively. If fewer than 2000 animals belong to the test-positive stratum, these probabilities of finding nothing are higher; the calculation at 2000 is not a guaranteed detection probability.

There is also a sampling-design distinction. If exactly $K$ positive animals in a fixed annual population of $N$ are known and the sample is without replacement, use the hypergeometric distribution instead:The given expected prevalence does not assert an exactly fixed $K$, so the independent prevalence model is the natural interpretation of the requested calculation.

Surveillance should be calibrated to the prevalence of concern, $q_{\ast }=1/200000=5\times 10^{-6}$, rather than only to the higher rate used in the preceding calculation. With $50000$ representative independent tests,Thus 50,000 tests give only about 22% detection probability at the concerning rate; a negative survey is quite likely, with probability about $78\%$. It costs $50000\times \pounds 40=\pounds 2$ million. Even $500000$ tests, costing $\pounds 20$ million, give about $91.8\%$ detection probability at $q_{\ast }$.

For a target $95\%$ detection probability, solveThe cost is about $\pounds 24$ million. The zero-event binomial upper confidence bound gives the same perspective: after zero positives in $50000$ tests, the one-sided $95\%$ upper bound isroughly one in $16700$, well above the concerning rate. A 50,000-animal survey is insufficient to rule out that prevalence with high confidence. A practical design should specify its desired detection probability, diagnostic sensitivity, representativeness and dependence between sampled animals. Targeting the higher-risk scrapie-positive stratum may improve efficiency if testing that stratum is feasible and its contribution to the slaughter population is estimated; its result then needs the appropriate population weighting. A cost alone does not establish adequate surveillance.