 Solution

The marginal Akaike information criterion is $\mathrm{AIC}=-2\ell ({\hat{\vartheta }}_{\mathrm{ML}})+2k$, where $k$ counts estimated population parameters. Here they are ${\beta }_0,{\beta }_1,{\sigma }^2,{\tau }^2$, so $k=4$. The realized incubator slopes have been integrated out of the likelihood function; they are not four additional free population coefficients. From the marginal Gaussian log-likelihood, the expression isAll hats here denote the ordinary maximum likelihood estimation fit.

The displayed value $14041.1$ is instead a restricted maximum likelihood criterion. Thus it cannot simply be substituted for the ordinary maximum-likelihood deviance. The logLik method defaults to the fitting convention. If one applies AIC(fly.model) directly to the displayed REML object, the default restricted log-likelihood and the same parameter count give the software's restricted-likelihood-based value $14041.1+8=14049.1$. This number should be labelled as such; it is not the ordinary marginal AIC derived above. To obtain the latter, refit and calculateThe output does not contain that numerical answer. In particular, restricted criteria must not be used to compare models with different fixed-effect designs. A conditional AIC targeting predictions for existing groups is yet another criterion and does not follow by counting each latent slope as an ordinary parameter.