 Solution

For rare events, $\mathrm{Binomial}(N,\pi )$ is approximately $\mathrm{Poisson}(N\pi )$. 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.

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.