Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 206 / 1 / e / Solution

The fitted formula and coefficient table describe a random-slope linear mixed model:with random effects independent of the errors. There is a common fixed intercept, a population slope ${\beta }_1$, and a laboratory random slope deviation $b_j$. In tumour1, the three laboratory slopes are unrelated unknown fixed coefficients. Here they are modelled as draws from a common normal distribution; estimated laboratory deviations are shrunk toward zero. After integrating out $b_j$, observations in laboratory $j$ have covariance ${\tau }^2{t}_{ij}{t}_{kj}+{\sigma }^2{1}_{\{i=k\}}$, so they are marginally dependent even when their conditional errors are independent.

The displayed command contains a source inconsistency: size ~ 1 + (0 + days | lab) would impose zero population slope and could not produce the displayed fixed days coefficient. The output corresponds to size ~ days + (0 + days | lab). Its estimates are ${\hat{\beta }}_0=0.6464$, ${\hat{\tau }}^2=17.08$ and ${\hat{\sigma }}^2=227.76$. The estimated daily increase averaged over the laboratory population isbecause $\mathbb{E}{b}_j=0$.

A random slope is appropriate if these laboratories can reasonably be viewed as exchangeable representatives of a larger population and the goal includes population inference or predictions for other laboratories. If these particular three laboratories are the whole set of interest, or have systematic differences incompatible with exchangeability, laboratory-specific fixed slopes are preferable. Only three groups give weak information about between-laboratory variance; the mixed model does not itself resolve possible nonlinear growth. Its population mean is distinct from a realized finite-group average ${\beta }_1+(b_A+b_B+b_C)/3$.