Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 32 / 7

Begin by defining the scientific target: prediction for a new patient, or estimation of adjusted covariate effects. Check event definitions, follow-up origins, right-censoring codes and any delayed entry. The analysis assumes independent patients and censoring that is noninformative conditional on modeled covariates. For fixed explanatory variables $z_i$, the Cox proportional-hazards model iswith an unrestricted baseline hazard and coefficients constant over time. Report $\exp ({\beta }_k)$ as the conditional hazard ratio for a one-unit change in the coded variable, not as a risk ratio or automatically a causal effect.

For untied event times, fit the coefficients by maximizing the Cox partial likelihoodThe baseline cancels within each event risk set. Use an appropriate tied-event method, such as an Efron approximation or an exact method for a genuinely discrete event scale; the Breslow approximation for tied event times is another explicit approximation. Numerical score/information methods fit the model, and inverse information gives conventional covariance estimates for independent subjects. Report coefficient estimates, hazard ratios, uncertainty and the coding that makes their interpretation meaningful. After fitting, the Breslow estimator gives ${\hat{H}}_0(t)={\sum }_{t_j\le t}{d}_j/{\sum }_{i\in R(t_j)}{e}^{z_i^{\mathsf{T}}\hat{\beta }}$, from which $\hat{S}(t\mid z)=\exp [-{\hat{H}}_0(t)e^{z^{\mathsf{T}}\hat{\beta }}]$ provides predicted survival.

Adequacy is broader than significance of coefficients. Inspect influential observations, deviance residuals and data errors. Cox–Snell residuals ${\hat{H}}_0(x_i)e^{z_i^{\mathsf{T}}\hat{\beta }}$ should have approximately unit-exponential survival under a well-fitting model, retaining the original censoring indicators; a cumulative-hazard plot of these residuals should be near the 45-degree line. This is an overall diagnostic, not a substitute for the functional-form and proportionality checks below. For prediction, assess held-out discrimination and survival prediction calibration at prespecified time horizons with methods accounting for censoring. A high concordance index alone does not establish good calibration or correct hazard structure.

Use bootstrap or held-out cross-validation that repeats the complete modeling procedure, including imputation, variable selection and tuning. Validation of only the final fitted coefficients understates overfitting from earlier decisions. Report any substantive lack of fit rather than presenting a single time-independent hazard ratio when the data do not support that representation.

Choose candidate variables from subject-matter knowledge, measurement quality and the target question before screening on outcomes. For an adjusted-effect analysis, retain prespecified confounders and essential design variables even when their individual $P$ values are large. For prediction, consider reliable predictors available at the intended prediction time; do not include future information. Handle missingness transparently, using a justified multiple imputation strategy where appropriate rather than changing the analyzed population unpredictably as variables enter.

Encode a categorical variable with $K-1$ indicators relative to a stated reference category, and assess it as a group. Check multicollinearity, sparse categories and plausible interactions. The relevant information is chiefly the number and distribution of events, not merely the number of enrolled subjects: hundreds of correlated or rarely varying predictors cannot be supported by a small event count.

Compare prespecified nested models by a likelihood-ratio test based on the partial log-likelihood and the change in parameter dimension. Penalized methods such as ridge regression or Lasso regression applied to the Cox likelihood can stabilize many-variable models; choose tuning by suitable validation and account for mandatory variables. The Akaike information criterion or a limited, documented model-selection procedure can help balance fit and complexity. Unrestricted stepwise searches and repeated univariable screening invite unstable choices, omit joint confounding effects, and make naive post-selection intervals misleading. Prefer a defensible, validated covariate set to a collection chosen only for small $P$ values.

The model is linear in each coded covariate on the log-hazard scale, not necessarily in the raw scientific variable. For a continuous variable $z$, begin with plots and a scientifically plausible range of forms. Compare $\beta z$ with a prespecified transformation such as $\beta \log z$ when $z>0$, or use a restricted cubic spline or fractional polynomial to represent a flexible smooth effect. Do not treat arbitrary integer category codes as a linear quantitative measurement without justification.

Plot martingale residuals from a suitable model against $z$ with a smooth trend. A residual pattern can suggest a missing or misspecified effect; these residuals are asymmetric, so the smooth relationship is more informative than judging normality. A model omitting $z$ helps reveal its overall shape; plots after including it help assess remaining misspecification. Partial-residual displays or fitted effect curves with uncertainty give complementary information.

Assess nonlinearity with a joint likelihood-ratio or score test for the nonlinear terms in a spline extension. The models containing only $z$ and only $\log z$ are generally nonnested, so a difference in their log-likelihoods is not automatically chi-squared. They can be compared by a justified nonnested criterion or validation, or embedded in a model containing both $z$ and $\log z$ and tested by dropping one term. Such an encompassing model may be highly collinear, and its coefficients should not be interpreted in isolation. If $z$ can be zero or negative, $\log z$ is undefined; choose an appropriate form rather than silently adding an arbitrary offset. Transformation choice is a question about the entire effect curve and its supported range, not only one coefficient's significance. Recheck time constancy after revising the functional form, since misspecification can mimic nonproportionality.

At each event time, a Schoenfeld residual is the observed event covariate minus its risk-set weighted mean:Under a correct constant-coefficient model, these event-time residuals have no systematic time trend. Plot scaled Schoenfeld residuals against time or a prespecified transformation such as log time, with smooth curves and uncertainty bands. Depending on the plotting convention, adding the fitted coefficient produces an estimate of ${\beta }_k(t)$; a horizontal curve then supports a constant coefficient, while a clear trend suggests violation. Examine sparse late follow-up with its wider uncertainty.

Formal tests can use residual-time association or a proportional-hazards time interaction extension ${\beta }_k(t)={\beta }_k+{\gamma }_{k}g(t)$, testing ${\gamma }_k=0$ jointly for a multi-parameter factor and globally across covariates. Evaluate $z_{k}g(t)$ at each current event/risk-set time: multiplying a baseline variable by that person's eventual observed follow-up time would use outcome information and would not be the intended time-dependent model. A nonsignificant test with limited information does not establish proportionality.

For categorical groups, approximately parallel empirical log-minus-log survival curves provide another check, because proportional hazards imply $\log [-\log S(t\mid z)]=z^{\mathsf{T}}\beta +\log H_0(t)$. Use observed group curves, not fitted proportional-hazards curves that are parallel by construction, and recognize possible confounding by other variables. If a violation is real, consider stratifying on a categorical nuisance variable, adding an explicitly time-varying effect, or choosing another survival-model family. Stratification allows separate baseline hazards but sacrifices a single estimated hazard ratio for that stratification variable. Do not force a constant coefficient merely to preserve the original model label; describe the time pattern or limit the interpretation of the constant summary.