Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 206 / 3

The initial Poisson regression contains altitude, temperature, and their interaction term, with a logarithmic link function. Its interaction Wald test has $p=0.521927$, suggesting removal of that term while initially preserving both main effects. The resulting additive model has a negligible temperature effect ($p=0.874$), suggesting a second simplification to altitude alone. The nested analysis of deviance for nested generalized linear models gives the same provisional decisions: deviance increases are $83.326-82.916=0.410$ and $83.351-83.326=0.025$, each for one removed coefficient, with approximate ${\chi }_1^2$ upper-tail probabilities $0.522$ and $0.874$. The Akaike information criterion also decreases at each step.

For independent island counts the final proposed model iswith fitted relationThis is an exponential mean relation on the count scale, not a linear relation between count and altitude. One additional metre multiplies the fitted expected count by $e^{0.01898}\approx 1.01916$. The null model's insignificance of individual main effects in the interaction fit does not require discarding every predictor: correlated design columns can make those conditional tests imprecise. These reductions concern relative mean complexity under the working Poisson regression; the large residual deviance in all three models warns that their variance or mean assumptions still need investigation.

For independent Poisson distributions, ignoring the factorial term only when taking likelihood differences,The saturated likelihood maximizes each mean separately at ${\tilde{\mu }}_i=y_i$, including its limiting value at zero. Therefore the Poisson deviance iswhere $0\log (0/{\hat{\mu }}_i)$ is defined to be zero and ${\hat{\mu }}_i=e^{3.24926+0.01898a_i}$. The expression is twice a saturated-model likelihood ratio difference, not twice a likelihood ratio itself.

With sufficiently large fitted counts and regularity, $D$ is approximately ${\chi }_{30-2}^2$ under a correctly specified Poisson regression. Here $D=83.351$ on 28 residual degrees of freedom, with upper-tail probability $2.0935\times 10^{-7}$. The Poisson model fits poorly on this diagnostic. Its deviance is nearly three times its degrees of freedom, consistent with overdispersion, although a wrong mean function, dependence, or influential islands could also cause the discrepancy. The smaller Akaike information criterion than its competitors does not make this an adequate absolute fit.

The Pearson dispersion estimator is $\hat{\varphi }=3.099079$, well above the Poisson value one. A Quasi-Poisson regression retains the logarithmic mean model but usesThis is a quasi-likelihood specification; it does not by itself identify a count probability distribution. For a common $\varphi$, the quasi-score equation is the Poisson score divided by $\varphi$, so its roots and fitted means are unchanged. ThusCoefficient standard errors are multiplied by $\sqrt{\hat{\varphi }}$. The altitude standard error becomes $0.00447\sqrt{3.099079}\approx 0.0078691$, giving the requested approximate normal confidence intervalIt excludes zero, so retain altitude on this approximate 5% test. Its test statistic is about $2.412$ after allowing for overdispersion, rather than the Poisson value $4.245$. Rescaling uncertainty cannot repair a systematically misspecified mean; the diagnostics in part (e) motivate reconsidering that mean as well.

Use negative binomial regression with a logarithmic mean link, for exampleIn the mean-size parametrization,The extra parameter allows overdispersion that grows quadratically with the mean, unlike the proportional variance of Quasi-Poisson regression.

The printed assertion needs qualification: a negative-binomial model with fixed size is a generalized linear model. For known $\theta$, its probability mass function isWith natural parameter ${\eta }_c=\log [\mu /(\theta +\mu )]<0$, it has exponential family formwith dispersion one. A logarithmic link is permitted even though it is not the canonical link function. This is the fixed-size negative binomial generalized linear model.

What prevents a single ordinary fixed-family GLM fit in glm.nb is joint estimation of the unknown size $\theta$: it changes the cumulant, carrier, and variance function, rather than merely multiplying a fixed variance function by a scalar dispersion. The routine alternates a GLM fit at fixed $\theta$ with size-parameter likelihood updates. Thus the intended distinction is between that enlarged fitting problem and a single fixed-family GLM, not a claim that negative-binomial regression can never be a GLM. The package documentation explicitly supports both conventions: stat.ethz.ch/R-manual/R-patched/library/MASS/html/glm.nb.html and stat.ethz.ch/R-manual/R-patched/library/MASS/html/negative.binomial.html.

The residual smooth rises from negative values, arches above zero, then falls at high fitted values. This suggests a nonlinear altitude effect on the log mean. Since the fitted log mean in the linear model increases with altitude, the horizontal fitted-value axis is also an increasing rescaling of altitude. A generalized additive model with Quasi-Poisson regression is therefore a reasonable candidate:The centering constraint identifies the intercept. Here bs="cr" uses a penalized cubic regression spline. The plotted count residuals are deviance residuals, with the Q-Q panel using leverage-standardized versions. The quantile-quantile plot also shows tail discrepancies, but normality of count residuals is not a distributional assumption of this model: its main requirements are an adequate mean, variance relation, and independence. Investigate the labelled islands and possible excess variation as well.

The estimated smooth in the second plot increases fairly rapidly at low altitude, flattens around altitude 20 to 25, and is nearly level at the upper end. The pointwise bands widen where there are fewer observations. A penalized spline can describe a rise followed by a plateau without forcing a global parabola. A quadratic log mean has slope ${\beta }_1+2{\beta }_{2}a$, so negative curvature eventually forces a decline and its curvature is constant everywhere. A cubic regression spline allows different curvature over different ranges, with a penalty controlling unnecessary wiggles and natural linear tails outside its boundary knots. The displayed smooth supports this greater flexibility, but it is not by itself a formal rejection of every quadratic model. Select complexity and compare predictive performance using cross-validation, then reassess residuals and the Pearson dispersion estimator.