Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 218 / 3

A generalized linear model is overdispersed when the conditional variance exceeds the variance prescribed by its response family and mean, for example $\mathrm{Var}(Y_i\mid X_i)>V({\mu }_i)$ for a unit-dispersion Poisson or binomial model. Unobserved heterogeneity or dependence among repeated observations can cause it. A generalized linear mixed model adds random effects for the corresponding clusters or subjects, thereby representing this heterogeneity and induced dependence.

For player $i$, competition $c$, and rival $r$, the fitted Poisson generalized linear mixed model isA negative-binomial model is commonly obtained by gamma mixing of a multiplicative Poisson mean and has negative-binomial marginals. Here the Gaussian random intercept gives a Poisson-lognormal mixture and correlates observations from the same player.

The player random intercept models stable player-to-player scoring ability and accounts for the six repeated observations on each player. The manager included it because treating those observations as conditionally independent with one common baseline would ignore both heterogeneity and within-player dependence.

The coefficient $0.4385$ says that, holding competition and player effect fixed, the expected scoring rate against S is multiplied byrelative to R. More goals are scored against S, so R appears to be the stronger rival.

The usual residual-deviance chi-squared calibration is unreliable because random effects are estimated and integrated out, changing both the effective degrees of freedom and the null distribution. Use a parametric bootstrap: fit the reported GLMM; compute an observed dispersion statistic such as the Pearson statistic or its ratio to nominal residual degrees of freedom; for each bootstrap replicate draw ten player effects from $N(0,0.1168)$, simulate all 60 Poisson responses from the fitted conditional means, refit the same GLMM, and recompute the statistic. With $B$ replicates, estimate the upper-tail p-value byReject at $5\%$ when $\hat{p}<0.05$, equivalently when $T_{\mathrm{obs}}$ exceeds the empirical $95$th percentile of the bootstrap null distribution.

The fixed-effect correlation pattern changes only moderately after adding the player random intercept: the competition-B/competition-C correlation remains $0.488$, rival S remains essentially orthogonal to those competition contrasts, and the largest changes involve correlations with the intercept. Thus the principal coefficient dependence comes from the fixed-effect coding and design, while player heterogeneity mainly changes the intercept-related uncertainty. The random effect is scientifically useful without materially changing which fixed contrasts are confounded.