= Solution
For rare events, $\operatorname{Binomial}(N,\pi)$ is approximately $\operatorname{Poisson}(N\pi)$. The binomial <variance> is $N\pi(1-\pi)$, close to the Poisson <variance> $N\pi$, and $\log\{\pi/(1-\pi)\}$ is close to $\log\pi$. Even the largest observed proportion here is below $0.009$. Thus the binomial log-odds model and Poisson log-rate model should give similar fitted counts and tests, as their very similar deviances show.
\b[Both analyses indicate strong age and adjusted city associations, with no apparent large residual lack of fit.] The binomial model is natural when each person can contribute at most one case during a fixed period and the denominator counts people at risk. A Poisson count model is natural for event counts with an appropriate exposure or person-time denominator; population size is an approximation to such exposure if the observation periods are comparable. An <odds ratio> and a rate ratio are different parameters despite being close in this rare-event setting.
The models both assume a common city effect across ages. An age-by-city <interaction> can be checked by comparing with the saturated model for the 15 observed cells, using the six additional <degrees of freedom>; the small final deviances offer no substantial evidence against the additive fits in that comparison. This cannot recover the missing city-by-age cell or eliminate unmeasured <confounding>. Nor is choosing the lower of $5.1509$ and $5.21$ a valid model-selection rule: these deviances are measured against saturated models for different response distributions.
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