= Solution
Retain the <random-slope linear mixed model> of part (e), but replace the within-laboratory independent error covariance by <continuous-time autoregressive residual correlation>:
$$
\operatorname{Cov}(\varepsilon_{ij},\varepsilon_{kj})
=\sigma^2e^{-\kappa|t_{ij}-t_{kj}|},\qquad\kappa>0.
$$
Errors from different laboratories remain independent and are independent of the <random slopes>. The resulting marginal <covariance> within laboratory $j$ is
$$
\tau^2t_{ij}t_{kj}+\sigma^2e^{-\kappa|t_{ij}-t_{kj}|}.
$$
The exponential decay applies to the conditional errors, not the entire marginal correlation, which also contains the shared <random slope>.
Use `nlme`, since `lme4::lmer` does not fit this general residual correlation structure. With a data frame containing the observed variables, suitable code is
``
library(nlme)
dat <- data.frame(size = size, days = days, lab = lab)
dat <- dat[order(dat$lab, dat$days), ]
tumour_cor <- lme(size ~ days,
random = ~ 0 + days | lab,
correlation = corCAR1(form = ~ days | lab),
data = dat, method = "REML")
``
Here `corCAR1` estimates $\rho=e^{-\kappa}\in(0,1)$, the error correlation one day apart; correlation at separation $\Delta$ is $\rho^{|\Delta|}$. This works for noninteger or unequally spaced days. At equally spaced integer days, `corAR1(form = ~ days | lab)` gives the corresponding discrete-time construction when its parameter is positive. Distinct times within each error series are required: separate tumours with repeated day values should have separate series identifiers or an additional independent-error component, rather than being assigned perfectly correlated errors. The <continuous-time autoregressive residual correlation> documentation describes the grouping and continuous-time conventions: https://stat.ethz.ch/R-manual/R-patched/library/nlme/html/corCAR1.html. The fixed `days` term here follows the fitted output rather than the inconsistent command in part (e).
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