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

The first approach specifies marginal means, variances and an exchangeable correlation for each student's repeated counts. It is a moment model suitable for a generalized estimating equation; equality of the mean and variance alone does not specify a full Poisson joint distribution. Different students supply independent sampling clusters, while the three counts within a student are dependent.

The second approach is a Poisson generalized linear mixed model with a positive shared random multiplier $U_i=e^{b_i}$. Conditional on it, the three counts are independent Poisson variables; integrating over it gives a fully specified dependent joint law. The gamma random-intercept Poisson model uses Gamma shape ${\tau }^2/\theta$ and rate $\tau /\theta$, so $\mathbb{E}{U}_i=\tau$ and $\mathrm{Var}(U_i)=\theta$. Treat the covariates as fixed, with the same multiplier distribution across covariate/treatment groups, as required by this model.

Let $a_{ij}=\exp ({\beta }_0+{\beta }^{\mathsf{T}}{x}_{ij})$ and $m_{ij}=\mathbb{E}{Y}_{ij}$. The law of total expectation, law of total variance and law of total covariance giveThus each marginal is a negative binomial distribution by the Poisson-gamma mixture, rather than generally a Poisson variable. Put $\kappa =\theta /{\tau }^2$. Its variance is $m_{ij}+\kappa m_{ij}^2$, and the marginal correlation isIt is positive but usually depends on the two marginal means, so it need not be the common exchangeable correlation proposed in the first approach. The first method focuses on population mean effects with a chosen working correlation matrix; the second models latent student heterogeneity and separates conditional Poisson variation from additional marginal variation. Correctly specified likelihood inference for the second uses more distributional assumptions, while the first can retain valid mean inference despite a wrong working covariance by using a cluster-level sandwich variance.

For the marginal mean model, $e^{{\beta }_0}$ is the expected term count for a student with all recorded covariates at their reference or zero values. If the treatment indicator is one for the new therapy and zero for the reference treatment, then $e^{{\beta }_1}$ is the ratio of population mean counts under the two treatments, holding the remaining covariates fixed. Values below one mean a reduced expected count; $100(1-e^{{\beta }_1})\%$ is the corresponding percentage reduction.

For the random-intercept model, $e^{{\beta }_0}$ is the conditional reference count at $b_i=0$, or multiplier $U_i=1$; the conditional baseline for student $i$ is $U_i{e}^{{\beta }_0}$. It is not automatically the mean-population baseline, which is $\tau e^{{\beta }_0}$. The conditional treatment rate ratio for students with the same multiplier is $e^{{\beta }_1}$. Because the multiplier has the same distribution in the treatment groups and the mean uses a logarithmic link function, that is also the marginal treatment rate ratio. This is the property that marginal and conditional slopes agree for an independent log-link random intercept.

The parameter $\theta$ is the variance of the positive student multiplier, not a Poisson sampling variance or a treatment coefficient. Greater $\theta$ at fixed $\tau$ means greater heterogeneity in underlying propensity and larger between-term covariance. Its extra contribution to the count variance is $\theta a_{ij}^2$. If $\tau =1$ is imposed, $\theta$ is also the relative multiplier variance; otherwise the relative heterogeneity is $\theta /{\tau }^2$.

There is a necessary identifiability qualification: if the multiplier mean $\tau$ is also unrestricted, scale identifiability in a gamma random-intercept Poisson model shows that replacingleaves the response law unchanged. Consequently only ${\beta }_0+\log \tau$ and $\theta /{\tau }^2$, together with the slopes, are identified from these counts without a scale convention. Fixing the multiplier mean, usually to one, is needed to interpret the conditional intercept and heterogeneity scale separately.

Integrating the conditional mean in the gamma random-intercept Poisson model givesand henceThe marginal model's logarithmic mean family is therefore correctly specified, with a shifted intercept and identical slopes. Under the usual independent-cluster and estimating-equation regularity conditions, fitting that marginal mean consistently estimatesThus the treatment and other slopes estimate the intended marginal log rate ratios; indeed they also coincide with the conditional slopes under the stated common random-effect distribution. The intercept consistently estimates the marginal baseline, but not the conditional $b_i=0$ intercept unless $\tau =1$. If the first investigator interprets the intercept as a conditional latent-student baseline without recognizing this shift, that interpretation is incorrect.

The assumed Poisson marginal variance and constant correlation generally fail under this random-effect law. They need not invalidate mean-coefficient consistency for a suitable generalized estimating equation, because the expected residual vector is still zero at the true marginal mean. They do invalidate naive standard errors based only on that assumed covariance; robust inference requires the correction in part (iv). The simple unchanged-slope result relies on the random multiplier distribution not changing with covariates, rather than on a universal equivalence of marginal and conditional regression models.

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.