Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 41 / 4 / c

The second analysis is an exposure-adjusted Poisson regression for case counts:Cell counts are independent, and their conditional variance equals their conditional mean. The population term is an offset with coefficient fixed at one. Thus the fitted case rate is ${\lambda }_{gj}/N_{gj}=\exp ({\alpha }_P+a_{P,j}+{\gamma }_{P}g)$; treating $\log N_{gj}$ as a free covariate would fit a different model. Age multipliers and the adjusted city rate ratio are exponentials of the corresponding regression coefficients.

The fitted model again has nine parameters and six residual degrees of freedom. Sequential additions of age and city reduce the Poisson deviance by $2095.56$ on seven degrees of freedom and $226.52$ on one, respectively. The residual deviance is $5.21$, close to the six residual degrees of freedom. The city likelihood-ratio test is overwhelmingly significant, as in the binomial analysis. The table provides no Poisson coefficient estimates, so an exact Poisson city rate ratio cannot be read from it.