 Solution

Write $r=1,\dots ,16$ for rats and $j$ for repeated observations. With diet 1 as the reference level in a regression factor, the random-intercept linear mixed model iswith the two families of normal distributions independent and the covariates treated as fixed. The reported estimates areThe corresponding estimated standard deviations are $36.577$ and $8.176$ grams. Individual realized random intercepts are not parameters in this marginal model, and their predictions are not supplied by this summary. For distinct measurements on the same rat, $\mathrm{Cov}(Y_{rj},Y_{rk})={\tau }^2$, while the marginal variance is ${\tau }^2+{\sigma }^2$; different rats are independent.

The estimated mean weight rises by $0.58568$ grams per day, adjusting for diet and the rat's random intercept. This common slope is imposed across all diets: there is no time-by-diet interaction term, so these estimates do not compare diet-specific growth slopes. The intercept refers to time zero for diet 1 and a zero rat effect.