Why Censoring Assumptions Matter
The Aalen–Johansen estimator uses people who remain under observation and free of every event type to estimate event increments. Its risk sets are therefore informative only when those still observed represent the event-free population with respect to their future event experience. If people leave observation because their risk of an event is related to their censoring, the observed risk sets can become systematically unrepresentative.
This condition is different from requiring everyone to have complete follow-up. Independent censoring allows some participants’ outcomes to become unknown before the study horizon. What matters is not that no one is censored, but that censoring does not select people according to their event time or cause, either overall or after an appropriate adjustment for measured factors.
Here \(J\) identifies mutually exclusive competing causes, and a censored observation has no observed event cause. For a fixed horizon, censoring after that horizon does not prevent observing whether an event occurred by the horizon. Censoring before it does. Estimation at a time \(t\) also requires positivity: there must be a positive probability of remaining under observation long enough to contribute information at \(t\). If everyone is censored before \(t\), the data do not identify the event risk there.
How the Condition Supports Aalen–Johansen
At an event time \(t\), let \(Y(t)\) be the number observed and event-free immediately before \(t\), and let \(d_k(t)\) be the number of observed events from cause \(k\) at \(t\). The observed cause-specific event increment is \(d_k(t)/Y(t)\). For this increment to estimate the event transition in the target event-free population, the observed risk set must not be selected on future event outcomes.
The key justification can be seen by considering a short interval after \(t\). Among people who have had no event by \(t\), the probability of a cause-\(k\) event in that interval is the event transition of interest. The observed risk set also requires that a person has not yet been censored. If \(C\) is independent of \((T,J)\), conditioning on being uncensored through \(t\) does not change that event transition:
In discrete time, write \(q_k(t)=P(T=t,J=k\mid T\geq t)\). Under independent censoring, the probability of a cause-\(k\) event at \(t\), conditional on being event-free and still observed just before \(t\), is also \(q_k(t)\). The probability of remaining event-free through that transition is \(1-\sum_k q_k(t)\). The cumulative incidence \(F_k(t)\), the probability of having experienced cause \(k\) by time \(t\), accumulates the cause-\(k\) transition probability among those who have remained event-free up to each transition:
where \(S(u-)\) is the probability of having experienced no event before \(u\). Aalen–Johansen replaces the unknown transitions with the observed risk-set increments \(d_k(u)/Y(u)\), and the corresponding event-free survival transitions with \(1-\sum_k d_k(u)/Y(u)\). This substitution explains why censoring that changes who remains in a risk set can distort the estimate: the observed increments may no longer represent the target event transitions.
Worked Example: Independent Censoring in a Small Risk Set
Suppose six hypothetical participants are followed for two competing causes. Their observed outcomes are: cause 1 at time 2; censoring at time 3; cause 2 at time 4; cause 1 at time 5; cause 2 at time 6; and cause 1 at time 7. Assume, for this example, that censoring is independent of event time and cause, and estimate the cumulative incidences through time 5.
At time 2, \(Y(2)=6\) and \(d_1(2)=1\). The cause 1 increment is \(1/6\), so \(\widehat F_1(2)=1/6\), and the estimated event-free probability becomes \(1-1/6=5/6\). The censoring at time 3 removes one participant from later risk sets but does not itself change the event-free estimate.
At time 4, \(Y(4)=4\), since one participant had an event at time 2 and one was censored at time 3. There is one cause 2 event, so the cause 2 increment is \(1/4\). The cumulative incidence increment for cause 2 is the event-free probability just before time 4 times this transition:
After time 4, the event-free estimate is \((5/6)(1-1/4)=5/8\). At time 5, \(Y(5)=3\) and one cause 1 event occurs. Therefore,
The calculation uses the censoring time to construct the risk sets; it does not treat censoring as an event. The independence assumption is what allows the observed fractions to stand in for the event transitions in the target population. The small data set cannot establish that assumption.
Informative Censoring Can Distort the Estimate
Censoring is informative for this estimation problem when being censored carries information about a person’s unobserved event time or cause, after accounting for the variables used in the analysis. For example, people with worsening health might be more likely to leave follow-up and also more likely to experience an event soon. The remaining risk set would then tend to contain people at lower event risk. The direction and size of bias depend on how censoring and event outcomes are related; informative censoring does not always produce underestimation.
Worked Example: High-Risk Participants Censored Before the Horizon
Consider a hypothetical cohort of 100 high-risk and 100 low-risk participants, with a common event horizon. Suppose 80 high-risk participants are censored before the horizon. Among those still observed, 16 high-risk and 20 low-risk participants experience cause 1; 4 high-risk and 20 low-risk participants experience cause 2; and the remaining 60 experience neither event by the horizon. Assume the 80 censored high-risk participants would all have experienced cause 1 by the horizon if their outcomes had been observed.
At the event horizon, the observed risk set contains \(20+100=120\) participants. The cause 1 increment is \(36/120\), so the Aalen–Johansen estimate for cause 1 is \(0.300\). In this single transition, the cause 2 increment is \(24/120=0.200\), leaving \(0.500\) event-free.
With complete outcome information, the hypothetical cause 1 risk would be \((80+16+20)/200=116/200=0.580\). The observed estimate of \(0.300\) is far lower because most of the high-risk participants who would experience cause 1 were removed before the horizon. The example is deliberately constructed so that the unobserved outcomes are known; in a real study, those outcomes are precisely what censoring hides.
Conditional Independence and Pooled Estimates
A censoring mechanism can be independent within levels of a measured prognostic factor but still make a pooled, unadjusted Aalen–Johansen estimate misleading. If censoring rates differ across groups with different event risks, censoring changes the groups’ proportions in the pooled risk set. The pooled event increment then reflects the altered mixture, not necessarily the mixture in the target population.
Worked Example: Why Adjustment for a Prognostic Group Can Matter
Suppose a hypothetical target population has 100 high-risk and 100 low-risk participants. By a common event time, the cause 1 risks are \(0.80\) in the high-risk group and \(0.20\) in the low-risk group. In each group, censoring is independent of event outcomes within that group, but 80 high-risk participants are censored before the event time and no low-risk participants are censored. Among the 20 observed high-risk participants, 16 have cause 1 and 4 have cause 2. Among the 100 observed low-risk participants, 20 have cause 1, 20 have cause 2, and 60 have no event by that time.
The pooled observed cause 1 increment is \(36/120=0.300\). The target-population cause 1 risk, however, weights the group risks by their proportions in the original population:
Estimating within each risk group gives \(16/20=0.80\) and \(20/100=0.20\); standardizing those estimates to the original equal group proportions gives \(0.50\). This example does not show that conditional independent censoring is sufficient for every analysis. It shows that the conditioning variables must be respected in the estimation: pooling the observed risk sets without adjustment changes the group weights.
Assumptions, Diagnostics, and Alternatives
The standard marginal Aalen–Johansen estimator requires that censoring be independent of event time and cause in the target population, along with enough follow-up to estimate the time range of interest. If censoring is independent only after conditioning on measured covariates, the analysis must use those covariates appropriately—for example, through suitable strata or standardization. In either case, the assumption is about the event outcomes that would have been observed, not merely about whether censoring times appear unrelated to recorded events.
Independent censoring cannot generally be confirmed using observed outcomes alone: the event time and cause after censoring are missing. Diagnostics can reveal reasons for concern, but they cannot prove the condition. Useful checks include examining censoring patterns over time, comparing censoring across measured prognostic groups, and reviewing whether reasons for loss to follow-up plausibly relate to health or event risk. A large difference in censoring across groups is a warning when those groups also have different event risks.
If censoring is plausibly independent only conditional on measured factors, stratified estimation or standardization may be appropriate. Inverse-probability-of-censoring weighting is another approach: it reweights observed participants to represent those with similar measured characteristics who were censored. Such weighting requires a suitable censoring model, conditional independence given the modelled factors, and positivity; it is not a remedy for unmeasured informative censoring by itself. Complete follow-up avoids loss to follow-up before the horizon, but it is not required for estimation and may be impractical. Sensitivity analyses can show how conclusions change under plausible assumptions about unobserved outcomes.
The practical distinction is important: complete follow-up means outcomes are observed through the horizon, whereas independent censoring means partial follow-up can still support estimation because censoring does not selectively remove people according to their future event experience. When the latter condition is doubtful, the Aalen–Johansen calculation may remain mechanically available, but its interpretation as an estimate of the target cumulative incidence is no longer justified without further assumptions or adjustment.
Check Your Understanding
Answer each question, then compare with the solutions.
- State the marginal independent-censoring condition using \(T\), \(J\), and \(C\).
- Does independent censoring require that nobody be censored before the study horizon?
- A study’s pooled estimate is \(0.30\), while group-specific risks are \(0.80\) and \(0.20\) in two equally sized target groups. What is the standardized risk?
- Why can comparing censoring rates across measured groups not prove independent censoring?
Solutions. (1) Marginally, \(C\) is independent of the event outcome \((T,J)\). (2) No. It permits censoring before the horizon, provided the censoring condition holds and there is positive probability of remaining observed through the times being estimated. (3) The standardized risk is \(0.5(0.80)+0.5(0.20)=0.50\). (4) Such comparisons use observed characteristics and can identify warning signs, but outcomes after censoring are missing; therefore they cannot reveal whether censoring depends on those unobserved event times or causes.