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

Expanding the exponent of the Inverse Gaussian distribution givesThus the exponential dispersion family representationhasandThe standard cumulant identities yieldHence the variance function is $V(\mu )={\mu }^3$, and the canonical link function is $g(\mu )=\theta (\mu )=-1/(2{\mu }^2)$.

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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.

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The Pearson residual iswhere the moment estimate of dispersion isIf the fitted model is adequate and the deviance residual sum is close to the Pearson statistic, then

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For response $j$ from subject $s$, model2 is the generalized linear mixed modelwith conditional independence given the random intercepts. The fitted values are ${\hat{\beta }}_0=8.9966$, ${\hat{\beta }}_1=-1.2317$, ${\hat{\tau }}^2=1.7533$, 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.

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Testwith a likelihood-ratio test. Model1 has three likelihood parameters and model2 has four, so their AIC values giveThe statistic isUsing a naive ${\chi }_1^2$ reference gives $p=0.0341$. Because the null variance lies on the boundary, the standard asymptotic reference is the mixture $\frac{1}{2}{\chi }_0^2+\frac{1}{2}{\chi }_1^2$, giving $p=0.0170$. Either calibration rejects at the five-percent level and supports a subject random effect.

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