Residual Diagnostics Under Right Censoring

Why Censoring Changes a Residual Plot

A survival residual is often intended to put event times on a common scale so that their distribution can be checked against a known target. But in right-censored data, some event times are not observed. Replacing each such time with its censoring time does not give an observed event-time residual: it gives a lower bound on that residual. A diagnostic that ignores this distinction can make a model look wrong—or hide a real problem—because it treats incomplete information as exact data.

The earlier material on Event-Time Probability Integral Transform Residuals motivates transforming fitted event-time distributions to a common reference scale. Here the central issue is how to retain the censoring information after that transformation. In a joint model, use the fitted event-time distribution under the same conditioning convention as the residual being checked—for example, conditional on the specified covariates, longitudinal history, and latent effects when those are part of the residual definition.

The cumulative-hazard transformation is useful because, under a correctly specified continuous event-time distribution with no unmodelled cure fraction, the true residual \(\Lambda_i(T_i^*)\) has a unit exponential distribution. Thus its survival function is \(S_R(r)=\exp(-r)\) for \(r\geq0\), and its cumulative hazard is \(r\). In practice, replacing the true cumulative hazard with an estimate makes the diagnostic approximate; the reference distribution does not remove estimation uncertainty.

The construction also relies on a suitable censoring assumption: conditional on the variables used for the event-time model and residual, censoring must be independent of the event time. If censoring depends on relevant information that has not been conditioned on or modelled, a residual plot can reflect that dependence as well as event-model misfit. The fitted cumulative hazard must be nondecreasing, as cumulative hazards are; it need not be strictly increasing for the censoring-bound inequality.

Keep the Bound, Not a Fictional Event

For an observed event, the transformed event time is known. For a censored observation, only the fact that the event had not occurred by \(Y_i\) is known. Since \(\widehat{\Lambda}_i\) is nondecreasing and \(T_i^*>Y_i\) for a censored subject, it follows that

If the fitted cumulative hazard is strictly increasing from \(Y_i\) to \(T_i^*\), the inequality is strict. If it is flat there, the transformed values can be equal, even though the event occurs later. Therefore a censored point plotted at \(c_i^R\) must be marked as censored—not presented as an observed event residual.

The naive alternative is to plot every censored subject at \(c_i^R\) as if an event occurred there, or to calculate a mean from those values and the observed event residuals. That approach pulls censored observations toward smaller residual values: it substitutes a lower bound for an unknown value. Omitting censored subjects is not a general fix either. It discards information and can select an unrepresentative subset, particularly when censoring patterns are related to covariates or follow-up.

Kaplan–Meier Diagnostics on the Residual Scale

A direct censoring-aware summary retains both the transformed time and the event indicator. For each subject, record \(W_i=\widehat{\Lambda}_i(Y_i)\) and \(\delta_i\). An event contributes an observed residual at \(W_i\); a censored subject contributes a right-censored residual at \(W_i\). Under the assumptions above, a Kaplan–Meier estimate of residual survival can be compared with the exponential reference \(S_R(r)=\exp(-r)\).

At each distinct residual time \(r_j\), let \(n_j\) be the number at risk just before that time and \(d_j\) the number of observed residual events there. The Kaplan–Meier estimate is the product of the estimated conditional probabilities of surviving each event time:

Censored observations reduce the later risk sets but do not create downward steps in the estimated survival curve. Under a well-calibrated residual model, the curve should broadly track \(\exp(-r)\), allowing for sampling variability. A plot of \(-\log\widehat{S}_R(r)\) can instead be compared with the straight line \(r\), wherever the estimated survival is positive. This is a diagnostic comparison, not a formal proof of model adequacy.

A pooled Kaplan–Meier curve is appropriate when the transformed residuals have the common reference distribution and residual censoring is independent of residual event time. Conditional independence of the original event and censoring times given model variables supports this construction when the transformation and conditioning are applied consistently. If that independence is doubtful, or if the diagnostic concerns a subgroup or a particular covariate pattern, investigate the conditioning and censoring mechanism rather than assuming that a pooled curve settles the issue.

Randomized Residuals and Other Comparisons

A different censoring-aware construction imputes a residual beyond each censoring bound using the fitted conditional distribution. Under the unit exponential reference, a censored residual can be drawn as \(c_i^R+E_i\), where \(E_i\sim\operatorname{Exp}(1)\). Repeating the draw produces randomized complete residual sets that can be compared with an exponential reference. The procedure uses the bound correctly, but its result includes Monte Carlo variation and depends on the fitted tail beyond censoring.

The Kaplan–Meier residual plot and randomized residuals answer related but different practical needs. The former estimates the residual survival distribution while retaining censoring indicators. The latter creates complete-looking residual samples by adding model-based information beyond each bound. A plot of observed event residuals alone is simpler but generally not a valid distributional check when censoring is present. Martingale and deviance residuals provide other summaries of survival-model fit, but they do not turn censoring times into observed event times or replace the need to respect censoring in a distributional residual plot.

Diagnostics, Failure Modes, and Limits

Before interpreting a residual diagnostic, check what the residual conditions on and whether censoring is plausibly independent given those variables. Confirm that event indicators and times have been coded consistently, that every censored residual is represented as censored, and that transformed times are calculated using the same fitted cumulative hazard used to define the residual. If the fitted event distribution has a cure fraction or does not describe the full event-time distribution, the unit-exponential target needs to be modified; it should not be assumed automatically.

Patterns can guide investigation without identifying a unique cause. A systematic departure of the Kaplan–Meier curve from \(\exp(-r)\) may signal event-model misfit, but it can also arise from an inadequate conditioning set, dependent censoring, or estimation error. Examine the number at risk and uncertainty before drawing conclusions from the tail. Residual plots based on the same fitted data can also appear more favorable than out-of-sample checks; parameter estimation and repeated model fitting affect the reference variation. Do not label every visible discrepancy evidence of a specific hazard or association error.

Common mistake: “The subject was censored at residual 1.00, so the event residual equals 1.00.” This confuses a lower bound with an observed value. The correction is to record a censoring time of 1.00 and retain the censoring indicator; the latent residual is at least 1.00, with strict inequality only when the fitted cumulative hazard increases strictly over the relevant interval. Another mistake is to delete censored subjects and compare the remaining event residuals with the full-sample exponential reference. That comparison ignores the information and selection introduced by censoring.