Cox–Snell residual ($r_i={\hat{H}}_i(x_i)$)

A Cox–Snell residual is the fitted cumulative hazard function evaluated at an observed event or censoring time. Under a correct model, complete transformed event times are unit exponential; censored transformed times retain their event indicators. A logarithmic Kaplan–Meier estimator survivor plot should be close to a line of slope minus one.

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.

For an observed transformed time $Y=H(\min (T,C))$ and event indicator $\Delta$, the correction $Y^{\ast }=Y+1-\Delta$ has mean one under independent censoring. Its complement $1-Y^{\ast }=\Delta -Y$ is the corresponding martingale residual.