Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 218 / 3

Let $T_i>0$ be record time and let $x_i=(1,c_i,d_i{)}^T$ contain standardized climb and distance. Model 1 is the normal linear modelModel 2 applies the same model after a logarithmic transformation:so $T_i$ is conditionally log-normal. Model 3 is a Gamma regression with logarithmic link:

Model 1's residuals have a systematic curved pattern and a spread that grows strongly with fitted time. This indicates an incorrect linear mean on the original scale and heteroscedasticity; several observations are also influential or outlying. The constant-variance assumption is therefore implausible.

Logging time greatly stabilizes the spread and removes most of the mean pattern, so model 2 is much more compatible with constant conditional variance and linearity. A few conspicuous residuals remain. A residual-versus-fitted plot alone does not check independence or fully establish normality.

For a likelihood $L(\theta )$ with $k$ estimated parameters, the Akaike information criterion iswhere $\hat{\theta }$ is the maximum-likelihood estimator. Among likelihoods for the same observed response and reference measure, smaller AIC estimates smaller expected out-of-sample Kullback-Leibler divergence up to a model-independent constant.

The printed values appear to favor model 2 because $-12.529<336.381$. They are not directly comparable: model 2 reports the Gaussian likelihood of $Z_i=\log T_i$, whereas model 3 reports a density for $T_i$. The change-of-variables formula for a probability density givesso the transformed model's AIC on the original response scale isBecause the standardized predictors have zero sample means and the ordinary-least-squares residuals sum to zero,Thereforewhich is slightly worse than model 3's $336.381$.

The ordinary least squares equations for model 2 areFor the Gamma generalized linear model, $V(\mu )={\mu }^2$ and $d\mu /d\eta =\mu$, so its score equation under the logarithmic link isWhen $T_i$ is close to ${\mu }_i$,by the first-order Taylor expansion of the exponential function. The Gamma score equations then become the model-2 normal equations, so their coefficient estimates are close.

Model 3 directly specifies the conditional mean and variance of the positive response on its observed scale. Its coefficients give multiplicative effects on mean record time, its prediction intervals concern time itself, and its likelihood can be compared directly with other original-scale models.

Model 2 instead models the mean of $\log T$. Back-transformation does not give the mean time without a retransformation bias correction; for a log-normal model, $\mathbb{E}(T\mid x)=\exp (x^T\alpha +{\tau }^2/2)$. The Gamma model also represents the increasing original-scale variance seen in the diagnostic plot, so it is preferable here.