Cox–Snell residual survival diagnostic (${\hat{S}}_r(r)\approx e^{-r}$)

For a correct fitted survival law, complete Cox–Snell residuals are approximately unit exponential. Residuals computed at observed censored times retain their censoring indicators. Fit a Kaplan–Meier estimator to the transformed times and indicators and compare it with $e^{-r}$, or compare their Nelson–Aalen estimator cumulative hazard with $r$. Treating censored transformed times as complete exponential observations gives a distorted Q–Q check.

If a common Kaplan–Meier estimator $\hat{S}$ is fitted and the same data are transformed to $u_i=-\log \hat{S}(x_i)$ with censoring indicators retained, event ordering and risk sets are preserved under compatible tie conventions. At a transformed event time $u_j$, the residual product-limit estimate equals $\hat{S}(x_j)=e^{-u_j}$. Consequently agreement of the pooled residual curve with the exponential target is largely built into the construction. Separate group curves or a model fitted with covariates can reveal differences concealed by the pooled fit. Residuals become infinite if the fitted survivor reaches zero.

// Target: survival-analysis.bigb