Past exam of the mathematical and Computational Biology course of the University of Cambridge / 2025 / Paper 1 / 4 / g / Solution

The Poisson generalized linear mixed model assumes conditionally Poisson distributed counts with a log link. The fitted variance components are both zero, so this fit lies on a boundary and reduces to ordinary Poisson regression. It estimatesThus the fitted annual increase is about 6.6%, with approximate 95% confidence interval for the rate ratio $(e^{0.001},e^{0.128})\simeq (1.001,1.137)$. The age evidence is much weaker than in the Gaussian fit: the quoted p-value is $0.047$. Zero estimated group variances do not prove that real bird and year differences are absent.

The Poisson distribution forces conditional variance to equal the conditional expected value. Here fitted means are around two to three hatchlings, whereas the Gaussian model estimated a residual variance of only $0.188$. This suggests checking underdispersion and the bounded nature of hatching counts, rather than assuming a Poisson response solves the diagnostic problems. Inspect simulated count residuals and predictive distributions; if the number of eggs is observed, fit a binomial mixed model and check that model's dispersion too.

The displayed Gaussian and Poisson Akaike information criterion values should not simply be ranked as if they used the same likelihood convention: one evaluates a continuous density and the other a probability mass. Compare properly specified count models on the same observed response scale, with compatible likelihoods and predictive checks. A boundary fit, doubtful dispersion assumptions, and possible nonlinear age effects all warrant further work before a strong biological conclusion.