For response $j$ from subject $s$, model2 is the <generalized linear mixed model>
$$
Y_{sj}\mid b_s\sim\operatorname{IG}(\mu_{sj},\lambda),
\qquad
\frac1{\mu_{sj}^2}=\beta_0+\beta_1d_{sj}+b_s,
\qquad
b_s\overset{\mathrm{iid}}\sim N(0,\tau^2),
$$
with conditional independence given the <random intercepts>. The fitted values are $\widehat\beta_0=8.9966$, $\widehat\beta_1=-1.2317$, $\widehat\tau^2=1.7533$, and fitted residual dispersion $0.7591$.

The random intercept models persistent between-subject differences and the resulting within-subject dependence among repeated measurements. That is the main feature absent from model1.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 218 / 1 / d / Solution

{wiki}

A random intercept gives every group a latent additive shift, inducing dependence among observations from the same group.


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 Incoming links: Random intercept