Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 206 / 6 / e / Solution

The generalized additive mixed model fitswhere $u_j$ denotes the furnace random intercept and $f$ is a centered penalized cubic regression spline. Its wiggly components have a mixed-model representation; there is no explicit furnace-specific random slope in this formula.

The plotted smooth gives no evidence that a nonparametric fixed effect is needed. It is essentially a straight rising line, and s(stir,1) indicates effective degrees of freedom close to one. The wider bands at extreme stirring rates reflect uncertainty, not detected nonlinearity. A linear fixed stirring effect is adequate on this evidence.

If the three supplied Akaike information criterion values are on a common valid likelihood basis, choose strength_model3, the fixed linear slope plus random-intercept model: its AIC $206.6408$ is smallest, versus $208.3243$ for the additional random slope and $210.3425$ for the smooth. The differences are $1.6835$ and $3.7017$, respectively, so especially the random-slope comparison represents modest support rather than decisive evidence. Retaining the population slope is consistent with part (c); this choice removes only its random variation.

There is a likelihood-basis caveat in the literal printed commands. lmer fits by REML by default, whereas the Gaussian gamm call defaults to ML; direct AIC on these objects need not compare the same type of likelihood. Moreover the earlier ML ANOVA prints $210.22$ for strength_model, whereas this table prints $208.3243$, so these numbers should not silently be treated as one identical ML fit. To obtain a defensible common comparison, fit all candidates to the same observations by ML:The supplied ranking favours the simpler random-intercept model; the displayed original calls alone do not certify that ranking as a consistent ML comparison. This distinction is particularly relevant when the smooth changes the fixed-effect space. The Gaussian gamm ML default and mixed-model representation are documented at stat.ethz.ch/R-manual/R-devel/library/mgcv/html/gamm.html.