Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 218 / 4 / d / Solution

Model "lme2" replaces $b_j$ by a random intercept and random PTHK slope:with a fitted bivariate normal covariance matrix for $(b_{0j},b_{1j})$. This adds a random-slope variance and an intercept-slope covariance.

The likelihood-ratio statistic isAn ordinary chi-squared reference is unreliable because the null random-slope variance is on the boundary and its correlation is unidentified there; the reported correlation of one also signals a nearly singular fit. A valid practical test is a parametric bootstrap: simulate many datasets from fitted "lme1", refit both models by maximum likelihood to each, recompute the likelihood-ratio statistic, and estimate the p-value by the fraction at least $8.4$. The AIC favors "lme2" slightly, whereas its BIC is larger, so the descriptive criteria do not agree.