Imputing Event Times Beyond Censoring

Why Impute Beyond Censoring?

Censoring tells us that an event had not yet occurred by a particular time; it does not reveal exactly when it occurred afterward. In some analyses, it is useful to draw plausible future event times for censored individuals. For example, a completed data set may simplify estimation of a cohort risk at a fixed horizon or a summary of event times. But an imputed time is not a newly observed fact. It is a model-based value whose credibility depends on the information available at censoring and on assumptions about what happens afterward.

Let \(T\) denote the event time, \(C\) the censoring time, \(Y=\min(T,C)\) the observed follow-up time, and \(\Delta=I(T\leq C)\) indicate whether the event was observed. In a competing-risks setting, let \(J\) identify the event cause when an event occurs. For a censored individual, \(Y=C\), \(\Delta=0\), and we know \(T>C\), but generally do not know \(T\) or \(J\). In a multi-state setting, the observed state history up to \(C\) is also relevant: a future transition must be imputed from the state occupied at censoring, conditional on the history that informs subsequent transitions.

The condition \(T>C\) is essential. An imputation must not place an event before the observed censoring time. Further, a future cause or transition must be consistent with the individual’s observed state and history. Imputation is most defensible when the censoring mechanism is independent of the future event process conditional on measured information, and when the model for that future process is credible over the time range being imputed. The earlier material on informative censoring explains why conditioning on relevant predictors matters: if censoring selects people with different future event risks, an unconditional model for those still followed may be unsuitable.

The Conditional Distribution After Censoring

Suppose a person is event-free at censoring time \(c\), with observed covariates and history denoted by \(H(c)\). A time-to-event model supplies a conditional distribution for the remaining waiting time \(T-c\), given \(T>c\) and \(H(c)\). If the model is specified through cause-specific hazards \(\lambda_k(u\mid H(u))\), the total event hazard is the sum over causes, \(\lambda_{\cdot}(u\mid H(u))=\sum_k\lambda_k(u\mid H(u))\). For a person event-free at \(c\), the conditional probability of no event through a later time \(t\) is the ratio of survival probabilities from the model:

This ratio conditions on having survived event-free to \(c\); it is not the unconditional probability of survival from the beginning of follow-up. The conditional probability that cause \(k\) occurs between \(c\) and \(t\) is

Here \(S(u-\mid H)\) is the probability of remaining event-free just before \(u\). The integrand combines the conditional probability of still being event-free at \(u\) with the instantaneous hazard of cause \(k\). The cause-specific probabilities across causes, plus the probability of no event by \(t\), sum to one. These formulas assume a correctly specified event process conditional on the included information; they do not, by themselves, establish that censoring is noninformative or that the model is correct.

When Is Imputation Defensible?

A defensible imputation requires more than a convenient way to fill empty cells. At minimum, the analyst should be able to justify the following conditions for the target population and time range:

• Known survival to censoring: the observed record establishes that the person was event-free and in the recorded state through \(C\). The imputed event time is constrained to be later than \(C\).
• Conditional exchangeability of censoring: given measured covariates and observed history, the future event process among censored people can be represented by the model fitted to comparable people still under observation. Unmeasured predictors of both censoring and future events can undermine this condition.
• A credible event model: the time, cause, and, where relevant, transition distributions are adequately specified conditional on the information used. The model must accommodate the event types and state structure relevant to the estimand.
• Support and positivity: comparable individuals must have a nonzero chance of being observed over the periods needed to estimate the conditional distribution. If follow-up ends for everyone at a particular time, the data alone cannot determine the later event-time distribution.
• An estimand that needs the imputed information: a fixed-horizon risk may not require imputing times beyond that horizon. Imputation should not create detail that the estimand does not need or the data cannot support.

For a multi-state process, imputing one future transition in isolation can produce an impossible history—for example, a transition out of a state the individual never entered. A coherent approach models possible transitions from the current state and, if necessary, continues the process forward, respecting the allowed transitions and the dependence of later transition rates on the history. If event causes are unknown after censoring, the cause must be drawn jointly with the time according to the conditional competing-risk model; assigning times while silently treating the cause as known can distort cause-specific risks.

Plausible and Implausible Imputations Can Change the Answer

The following hypothetical example shows why an imputation rule can materially alter a competing-risk estimate. Suppose 10 individuals are followed toward year 10. Four have an observed event of interest by year 10, four have an observed competing event by year 10, and two are censored event-free at year 4. A fitted conditional model assigns each censored individual a probability \(0.30\) of the event of interest before year 10. For this illustration, consider the expected cumulative incidence estimate. The four observed events of interest contribute 4; each censored person contributes an expected \(0.30\) to the numerator, while the observed competing events contribute no event of interest.

Now compare two extreme, deliberately artificial imputations. If both censored people are assigned a cause 1 event before year 10, the completed-data fraction becomes \((4+2)/10=0.600\). If both are assigned a competing event, or an event of interest after year 10, it becomes \(4/10=0.400\). Neither assignment is supported merely because it produces a completed data set. A defensible imputation draws times and causes from a conditional model consistent with the observed histories; it does not choose whichever future produces a preferred estimate.

Even when a completed-data point estimate seems stable, uncertainty must reflect that the future outcomes were not observed. A single imputed data set treats its invented values as if they were known and generally understates uncertainty. With multiple imputation, repeat the imputation under the specified model and combine the estimates and their variances. In a standard multiple-imputation calculation, if \(Q_l\) is the estimate and \(U_l\) its variance from imputed data set \(l\), then \(\bar Q=m^{-1}\sum_l Q_l\), \(\bar U=m^{-1}\sum_l U_l\), and \(B=(m-1)^{-1}\sum_l(Q_l-\bar Q)^2\). A commonly used total variance is \(\bar U+(1+1/m)B\), under the usual multiple-imputation conditions. The between-imputation component \(B\) reflects uncertainty about the missing outcomes; it does not account for every form of model misspecification.

Diagnostics, Alternatives, and Limits

Before using imputed times, check whether the model produces plausible distributions conditional on censoring time, covariates, and state history. Useful checks include comparing predicted and observed event patterns over follow-up where outcomes are observed, checking whether predicted cause probabilities add to one with the no-event probability, and examining whether simulated multi-state paths obey the allowed transition structure. Also inspect how far imputed times extend beyond observed follow-up. A model that fits well within observed follow-up may still extrapolate poorly afterward.

A warning sign is an imputed distribution that changes sharply for small modelling choices, puts substantial probability in sparsely observed periods, or predicts event rates inconsistent with observed patterns among comparable individuals. Extreme or implausible predicted times may reveal an unsuitable tail model, omitted predictors, or a target time beyond the support of the data. If censoring is informative even after conditioning on measured history, ordinary model-based imputation does not automatically correct the problem. The later material on sensitivity analysis addresses how to examine departures from censoring assumptions; here, the central point is that an imputed result is conditional on those assumptions.

These approaches answer related but not identical practical needs. Imputation can be convenient for a target that uses future event timing or a complete trajectory, while weighting-based approaches estimate population quantities without asserting a particular event time for each censored person. None is assumption-free. When follow-up provides little or no information about the event-time tail, the tail is not identified by the observed data alone; a precise-looking imputed answer can therefore conceal substantial dependence on modelling assumptions.

Common Pitfall

Wrong reasoning: “The person was censored at year 4, so I will assign an event at year 5, or use the average observed event time. The imputed values are only a small part of the data, so they cannot change the conclusion.” This ignores the known constraint \(T>4\), the person’s conditional risk and state history, the competing causes, and the possibility that the imputed values have substantial influence on the estimate.

Correction: Specify the conditional future event process first, check that the censoring and model assumptions are reasonable for the target, and draw each future time and cause from the distribution conditional on the observed history. Compare results across plausible model specifications when conclusions depend on extrapolation, and include between-imputation uncertainty. Imputation makes assumptions operational; it does not make unobserved outcomes observed.