Model1 is the <simple linear regression>
$$
Y_{ij}=\alpha+\beta X_{ij}+\varepsilon_{ij},
\qquad
\varepsilon_{ij}\stackrel{\mathrm{iid}}\sim N(0,\tau^2).
$$
Model2 is a <random-intercept linear mixed model>
$$
Y_{ij}=\alpha+\beta X_{ij}+b_i+\varepsilon_{ij},
\qquad
b_i\stackrel{\mathrm{iid}}\sim N(0,\sigma^2),
\quad
\varepsilon_{ij}\stackrel{\mathrm{iid}}\sim N(0,\tau^2),
$$
with the random intercepts and errors mutually independent.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 218 / 5 / a / Solution

{wiki}

Simple linear regression uses one predictor and an intercept, $Y_i=\alpha+\beta x_i+\varepsilon_i$.


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