Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 13K (Statistical Modelling)

Let $Y_i$ be the number of survivors among the $n_i$ patients in row $i$. The fitted grouped-binomial logistic regression iswithSite A and nonmalignant tumours are the reference levels. There is no site-by-malignancy interaction, so the malignancy log-odds effect is assumed to be the same at all three sites. In the R call, survive/total supplies the observed proportions and weights = total supplies the binomial denominators $n_i$.

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

Changing the reference level in a regression factor from A to B does not change the statistical model, fitted probabilities, likelihood, or deviance. If the first parametrization isthen the second isNumerically,

The A-versus-B test is the same test as the former B-versus-A test, with its sign reversed, so its p-value remains $0.0624$. The malignancy coefficient is unchanged in this additive model, so its p-value remains $0.0175$. The intercept now represents site B rather than site A, so its test changes.

The siteC coefficient also changes meaning: in fit1 it tests C versus A, whereas in fit2 it tests C versus B. Its standard error and p-value therefore need not agree. The displayed tables show that C versus A is not significant ($p=0.2610$), while C versus B is significant ($p=0.0160$). There is no contradiction: these are different pairwise hypotheses.

The reduced model fit3 retains malignancy but omits both site indicators. The analysis of deviance for nested generalized linear models therefore testsagainst the additive model with a site effect. The likelihood-ratio statistic is the reduction in deviance,The full model adds two parameters, so under $H_0$ this is approximately ${\chi }_2^2$. Sincebut is less than the $0.99$ quantile $9.2103$, the p-value is between $0.01$ and $0.05$; in fact, because ${\chi }_2^2$ has survival function $e^{-x/2}$,We reject the no-site-effect null at the $5\%$ level and find evidence that survival differs by site after adjustment for malignancy.