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

For student $i$ in school $j$, "lme1" is the random-intercept linear mixed modelwhereindependently. The estimates areThe fixed intercept is the population-average expected post-study score for an untreated reference student with PTHK zero. The random intercept $b_j$ is school $j$'s deviation from that population intercept.

Treatment was assigned by school, so observations within one school are clustered data and plausibly correlated. A random intercept models shared unobserved school characteristics and prevents the effective amount of treatment-level information from being exaggerated by treating all students as independent.

The fitted school standard deviation is $0.209$, so the estimated between-school variation is nonzero. Accounting for it raises the TV and SC standard errors from about $0.065$ in "lm1" to about $0.105$ in "lme1", reflecting that only 28 independently randomized schools identify those effects.

Conditionally on the school effect, the response variance is ${\hat{\sigma }}^2=1.6531$. Marginally,and two distinct students in the same school have estimated covariance $0.0437$.

There is no single coefficient estimate for the random effect because the 28 school deviations are modeled as draws from a mean-zero distribution. The summary reports the estimated distributional variance; individual empirical Bayes predictions can be extracted separately.

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.