Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 13K / b / Solution

The estimates maximize the grouped-binomial likelihood, numerically obtained by iteratively reweighted least squares. At the fitted probabilities, the estimated covariance matrix is the inverse observed information, approximatelyThe Std. Error column is the square root of its diagonal. Each z value is Estimate/Std. Error, and the two-sided Wald test p-value isfor the null hypothesis that the corresponding coefficient is zero.

Thus ${\hat{\beta }}_0=1.6855$ is the fitted log odds for a nonmalignant patient at site A. The site coefficients compare B and C with A, while ${\hat{\beta }}_M=-0.9048$ compares malignant with nonmalignant tumours after adjustment for site. In odds-ratio form,At level $0.05$, the B-versus-A contrast is not significant ($p=0.0624$), the C-versus-A contrast is not significant ($p=0.2610$), and malignancy has a significant negative association with survival ($p=0.0175$). The intercept test merely compares the baseline survival probability with $1/2$.

The null deviance $11.693$ compares the intercept-only fit with the saturated model and has $6-1=5$ residual degrees of freedom. The residual deviance $0.85048$ compares the four-parameter fitted model with the saturated model and has $6-4=2$ degrees of freedom. Finally, the Akaike information criterion is