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

For observation $j$ of bird $i$ in year $t(i,j)$, a Gaussian linear mixed model with crossed random intercepts iswhere the $u_i$, $v_t$, and ${\epsilon }_{ij}$ are mutually independent random variables. Here $a_{ij}$ is age, ${\beta }_0$ is the population intercept at age zero, and ${\beta }_1$ is the change in conditional expected value per year of age. The $u_i$ capture persistent between-bird variation, the $v_t$ capture shared annual conditions, and the regression residuals capture remaining observation-level variation. Two observations from the same bird share covariance ${\tau }_b^2$; two from the same year share ${\tau }_y^2$, with both contributions present if both group labels agree. The model also assumes the random effects are independent of the age covariate.

Conditional predictions for observed birds and years add their fitted random intercepts to the fixed part of the Gaussian linear mixed model:These are predicted expected values, so they need not be integers. The first is likely more reliable: age three is within the displayed age range, whereas age seven requires extrapolation beyond ages one to six. A quantitative comparison would need the prediction intervals, including uncertainty in both fixed coefficients and random effects; the displayed coefficients alone do not supply them.

The deviance entries are $-2\ell$ for these fitted models. The completed values areThe likelihood-ratio test compares a model with one age coefficient against its restriction ${\beta }_1=0$. Its small p-value, approximately $0.000529$, supports a positive age association under the assumed Gaussian linear mixed model. Refitting with maximum likelihood estimation is essential here: restricted maximum likelihood values from models with different fixed-effect design matrices cannot be compared by this likelihood-ratio test.

Yes. The bootstrap confidence interval for the age coefficient, $(0.057060,0.193584)$, excludes zero and supports the same positive association. A well-implemented parametric bootstrap can give better finite-sample calibration than the asymptotic likelihood-ratio test, especially with only eleven year groups and estimated variance components. Its advantage depends on simulating the full fitted Gaussian linear mixed model and refitting correctly; resampling individual records as if they were independent would discard the grouping. A bootstrap confidence interval is also not literally the same procedure as a bootstrap-calibrated test.

The bird-level standard deviation is appreciable, with interval $(0.436,0.720)$ hatchlings, whereas the year-level estimate is much less precise, with interval $(0.0195,0.333)$. The residual standard deviation is approximately $0.381$–$0.493$. Boundary fits in 23 of the 1000 replicates indicate that at least one fitted variance component sometimes reaches zero; the convergence warning should also be investigated before trusting the reported intervals. Neither the asymptotic test nor a model-based bootstrap repairs incorrect response-distribution or mean-structure assumptions.

“After accounting for bird identity and year using random intercepts, hatching success increased by an estimated 0.123 hatchlings per additional year of age (95% bootstrap confidence interval 0.057–0.194).”

This describes an observational association under the fitted Gaussian linear mixed model, rather than a causal age effect. The diagnostic assessment below is needed before treating this inferential statement as reliable.

The regression diagnostics raise concerns about the mean and variance assumptions. The residual-versus-fitted smoother is curved rather than horizontal, and the scale-location plot suggests changing residual spread, contrary to homoscedasticity. Diagonal bands are expected when an integer-valued response is fitted by a continuous model, since each fixed response value gives $r=y-\hat{y}$. They are not, by themselves, proof of a failed mean model. The residual Q-Q plot is broadly linear but has tail departures and unusual observations, so exact normality is questionable.

The Cook's distance plot identifies observations worth checking for recording errors and influence on the age estimate. A rough $4/n\simeq 0.023$ screening threshold flags several points, but it is not an automatic deletion rule. Repeat the fit with and without influential observations as a sensitivity analysis and report material changes.

The bird random intercept Q-Q plot has a pronounced upper-tail departure; the year plot is closer to a line, although eleven groups give little power to assess normality. These plotted effects are estimated, shrunken conditional mode of Gaussian random effects, not independent observed draws from the latent normal distribution, so their appearance requires care.

The analysis is therefore provisional. Check a nonlinear age effect, the changing residual variance, dependence beyond the fitted random intercepts, and a response model appropriate for counts. If clutch size is known, a binomial distribution model for hatched eggs out of available eggs may be more meaningful. A bootstrap from the existing Gaussian model cannot validate the assumptions questioned by these plots. Distinguishing within-bird ageing from between-bird differences would also strengthen interpretation.

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.