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

Writing $d_i$ for difficulty, model1 assumes independent responseswith common dispersion. The estimates are ${\hat{\beta }}_0=9.0827$ and ${\hat{\beta }}_1=-1.4323$.

One extra difficulty level decreases the fitted inverse squared mean response time by $1.4323$. Since $\mu =({\beta }_0+{\beta }_{1}d{)}^{-1/2}$, this means that fitted mean response time increases with difficulty. The negative coefficient is therefore unsurprising; its sign looks counterintuitive only if the inverse-squared link function is ignored. The independence assumption is questionable because every subject contributes eight repeated responses.