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

The Fort Worth observation for ages 75–84 is missing. It is not a zero count and cannot be replaced by one. The missing cell prevents a direct two-city comparison in that age group and leaves an unbalanced age-by-city table. The youngest groups have only one and four cases, so chi-squared asymptotic approximations to tests and uncertainty may be poor there. The open-ended oldest group and broad age bands can conceal different within-group age compositions between cities.

The denominators are population counts rather than reported person-time, and the observation period is not specified. Comparable coverage, case definitions and time periods must therefore be established before interpreting the numbers as comparable incidence measures. The data contain no information on potential confounders such as individual risk factors or differences in ascertainment. These are aggregated observations on women in the listed ages, and any fitted city association is not automatically causal or generalizable beyond that population.

Index city by $g\in \{0,1\}$ and age group by $j$, with the youngest group the reference level. For each of the 15 observed cells, the grouped-binomial logistic regression assumeswith independent cell counts. The response supplied to R is $C_{gj}/N_{gj}$, and the weights specify the binomial numbers of trials $N_{gj}$. They are not 15 arbitrary precision weights: the conditional variance of the proportion is ${\pi }_{gj}(1-{\pi }_{gj})/N_{gj}$. The logit link models log odds, and the two sets of factor coefficients use corner-point constraints.

There are nine parameters: one intercept, seven age contrasts and one city contrast. Hence the residual degrees of freedom are $15-9=6$, not the total population minus nine. The null model has only an intercept and $14$ residual degrees of freedom. Adding age uses seven more parameters and reduces the binomial deviance from $2330.46$ to $232.28$, a drop of $2098.19$. Adding city then uses one parameter and reduces it by $227.12$ to $5.15$. These are sequential likelihood-ratio tests, asymptotically compared with chi-squared distributions on seven and one degrees of freedom. Both provide extremely strong evidence of an effect; the printed age p-value $0.00$ is rounding, not an exactly zero probability. Because the table is incomplete, sequential main-effect sums need not be invariant to the order of fitting.

The intercept estimate $-11.69364$ is the log odds for the youngest Minneapolis–Saint Paul group. It implies baseline odds $e^{-11.69364}$ and a fitted probability about $8.35\times 10^{-6}$. Its standard error is $0.44923$; its reported Wald test tests log odds zero, or probability $1/2$, rather than a scientifically interesting comparison of groups.

The age coefficients are log odds ratios relative to the youngest group within the same city. Their exponentials are approximately $13.86$, $46.82$, $99.03$, $162.23$, $284.38$, $497.07$ and $484.57$ as age increases through the seven other groups. All these comparisons have large positive Wald statistics and small p-values. Thus the fitted probabilities are much higher in the older groups than in the youngest group. The slightly smaller coefficient in the oldest group than in the preceding group does not itself prove a real decline: that claim needs a contrast and its standard error, including the covariance of the two estimates.

The city coefficient $0.85492$ is the log odds ratio for Fort Worth relative to Minneapolis–Saint Paul, adjusted for age. Its standard error $0.05969$ gives $z=14.322$, so the null city effect is strongly rejected by the Wald test, consistent with the sequential likelihood-ratio test. The fitted constant-over-age city odds ratio and an approximate 95% confidence interval areThis is an odds ratio, not generally a probability ratio. The probabilities here are sufficiently small that the two ratios are close. The absence of an age-by-city interaction is an assumption of the fitted additive log-odds model.

The final binomial deviance is $5.1509$ on six degrees of freedom; its approximate upper-tail p-value is $0.525$. It gives no indication of substantial lack of fit against the saturated observed-cell model. The deviance residuals range from $-1.2830$ to $1.0820$, with median zero and no conspicuously large value in the five-number summary. Neither statement proves that every model assumption holds. In particular, the small case counts motivate checking the accuracy of the chi-squared asymptotic approximation, for example by a fitted-model parametric bootstrap.

The dispersion is fixed at one by the binomial distribution; it was not estimated as a free scale parameter. Four Fisher scoring iterations describe the numerical optimization, not four additional parameters. Residual and cellwise observed-versus-fitted comparisons would help assess adequacy and potential overdispersion.

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.

For rare events, $\mathrm{Binomial}(N,\pi )$ is approximately $\mathrm{Poisson}(N\pi )$. The binomial variance is $N\pi (1-\pi )$, close to the Poisson variance $N\pi$, and $\log \{\pi /(1-\pi )\}$ is close to $\log \pi$. Even the largest observed proportion here is below $0.009$. Thus the binomial log-odds model and Poisson log-rate model should give similar fitted counts and tests, as their very similar deviances show.

Both analyses indicate strong age and adjusted city associations, with no apparent large residual lack of fit. The binomial model is natural when each person can contribute at most one case during a fixed period and the denominator counts people at risk. A Poisson count model is natural for event counts with an appropriate exposure or person-time denominator; population size is an approximation to such exposure if the observation periods are comparable. An odds ratio and a rate ratio are different parameters despite being close in this rare-event setting.

The models both assume a common city effect across ages. An age-by-city interaction can be checked by comparing with the saturated model for the 15 observed cells, using the six additional degrees of freedom; the small final deviances offer no substantial evidence against the additive fits in that comparison. This cannot recover the missing city-by-age cell or eliminate unmeasured confounding. Nor is choosing the lower of $5.1509$ and $5.21$ a valid model-selection rule: these deviances are measured against saturated models for different response distributions.