 Cox–Snell residual

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.