 Solution

The deviance entries are $-2\ell$ for these fitted models. The completed values areThe likelihood-ratio test compares a model with one age coefficient against its restriction ${\beta }_1=0$. Its small p-value, approximately $0.000529$, supports a positive age association under the assumed Gaussian linear mixed model. Refitting with maximum likelihood estimation is essential here: restricted maximum likelihood values from models with different fixed-effect design matrices cannot be compared by this likelihood-ratio test.