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: $r_i={\hat{H}}_i(Y_i)$. Its reference probability distribution comes from the probability integral transform. If the true event time $T_i$ has a proper continuous survival function $S_i$, then $H_i=-\log S_i$ satisfiesFor example, an exponential distribution with rate $\lambda$ has $H(t)=\lambda t$, so rescaling its event time by $\lambda$ produces a unit-rate exponential distribution.

With right censoring, $Y_i=\min (T_i,C_i)$ and the transformed observation is $H_i(Y_i)$ together with its original event indicator ${\delta }_i$. Under independent censoring, a Kaplan–Meier estimator applied to these transformed observations should approximately follow $e^{-r}$; its logarithm should approximately follow a straight line of slope minus one. Censored residuals must remain censored in this comparison.

The exact exponential identity uses the true model, whereas ${\hat{H}}_i$ is estimated from data: fitted residuals are approximate diagnostics and need not be independent exact exponentials. Atoms in a discrete event-time probability distribution, or a cure fraction with positive probability of no finite event, invalidate the unmodified proper-continuous reference law. Such models require diagnostics respecting their jumps or surviving mass.

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

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.