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

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.