 Solution

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.