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Write $t_{ij}$ for stirring rate of observation $i$ in furnace $j$. The independent random-intercept and random-slope model isErrors and random effects are independent. The separate R terms (1 | furnace) and (0 + stir | furnace) impose independent random intercepts and random slopes; (1 + stir | furnace) would instead estimate their covariance as well. The estimates areThe corresponding estimated standard deviations are $6.9813$, $0.1147$, and $6.0249$; they are not additional parameters.

For a furnace with predictor vector $t_j$, the marginal formulation, obtained by integrating the Gaussian random effects, isindependently across furnaces. In particular,In stacked notation $Y=X\beta +Zb+\epsilon$, $b\sim N(0,D)$, giving the Gaussian linear mixed model marginal law $N(X\beta ,V)$ with $V=ZD{Z}^T+{\sigma }^{2}I$.

The residual-versus-fitted plot on page 17 has residuals of both signs over the fitted range, with no convincing smooth curvature or clear fan shape. Its quantile-quantile plot is roughly straight in the central region, with some tail departures and a few large negative residuals. There is no decisive visible violation, but investigate those observations. These plots mainly address conditional observation errors; they do not validate the distribution of furnace random effects or independence within furnaces. With only three furnaces, normality and the variance of the random effects are especially difficult to assess.