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

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.