Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 36 / 4 / iv / Solution

Use a generalized estimating equation with any well-behaved positive-definite working correlation matrix. Let $\gamma$ collect the marginal intercept and slopes, let $m_i(\gamma )$ be the vector of three correct marginal means, and put $D_i=\partial m_i/\partial {\gamma }^{\mathsf{T}}$. Write the working covariance as $V_i=A_i^{1/2}{R}_i(\alpha )A_i^{1/2}$, where the proposed Poisson variance gives $A_i=\mathrm{diag}(m_i)$; working independence is also allowed. SolveAt the true mean, each cluster residual has expectation zero, so the estimating equation is unbiased regardless of whether $V_i$ equals the actual covariance. With independent students, sufficient covariate variation and regularity as $m\to \infty$, this gives consistent mean coefficients.

Replace the model-based covariance by the cluster-level sandwich covariance matrix. Evaluated at the fitted values, defineThenThe residual outer products estimate the actual within-student variability, including correlations and extra-Poisson variance absent from the working model. Use these standard errors for asymptotic Wald tests and confidence intervals; exponentiating the treatment coefficient and its interval gives the mean-count rate ratio and interval. Treating the $3m$ observations as independent when forming the sandwich meat would miss the within-student dependence. The effective asymptotic replication is the number of students, so a small number of clusters requires further finite-sample care rather than assuming that three measurements per student repair the large-sample approximation.