Why Censoring Is a Separate Issue
A hazard describes how an event process behaves over time. Censoring describes how long that process is observed. These are related in data analysis, but they are not the same thing: a hazard is defined from the event-time distribution, whereas a censoring assumption concerns whether the observed records let us learn about that distribution.
This distinction matters in competing-risks data. A competing event changes a person’s event history: after it occurs, that person can no longer experience the focal first event. Censoring means that observation ends before the event history is known. The earlier material on competing events establishes why these roles cannot be interchanged. Here the focus is on the additional assumption that censoring does not selectively hide event histories.
Two Hazard Definitions, Neither Defined by Censoring
For a cause \(k\), the cause-specific hazard describes the instantaneous rate of a cause-\(k\) event among people who have not yet had any event. In continuous time, write \(F_k(t)=P(T\le t,J=k)\) and \(S(t)=P(T\ge t)\). At times where the event-time distribution has a density, the cause-specific hazard is
Here \(f_k(t)\) is the cause-\(k\) event density. The denominator concerns event history: anyone who has already experienced any cause is no longer in this event-free group. Whether a person is still being observed does not enter the population definition.
The subdistribution hazard for cause \(k\) uses a different denominator. With \(F_k(t-)\) denoting the probability of a cause-\(k\) event before \(t\), its continuous-time form is
This denominator excludes prior events of cause \(k\), but it does not remove people merely because they experienced a competing event. The subdistribution hazard is not itself a probability. Both definitions concern the event process; neither says that censoring is independent, nor that it is absent.
What Independent Censoring Assumes
A common condition is that, within levels of measured covariates \(Z\), censoring is independent of the event time and cause:
If no covariates are being conditioned on, the corresponding condition is \(C\mathrel{\perp\!\!\!\perp}(T,J)\). Informally, among otherwise comparable people, knowing that observation ends at a particular time should not give information about their unobserved event time or cause. The condition is sometimes described as noninformative censoring. It does not require censoring times to be identical, and it does not mean there are no censored records.
For estimation through time \(t\), the data must also contain people who remain under observation through that time. A standard positivity condition is \(P(C\ge t\mid Z=z)>0\) in the covariate groups for which inference is intended. If nobody can be observed beyond six months, the data cannot identify a hazard at nine months just by invoking independence.
The reason independence matters can be seen by conditioning on remaining observed. Under conditional independence, when \(P(C\ge t\mid Z)>0\),
Thus the people still observed at time \(t\), within a covariate group, are not selectively enriched or depleted in event-free people by censoring. Without this condition, the observed risk set can have a different event-time composition from the group whose hazard is of interest. The same issue arises when learning about a cause-specific event process or a cause-specific cumulative event distribution underlying a subdistribution hazard.
Worked Example: Different Denominators, No Censoring
Consider a hypothetical cohort of 12 people with complete observation through month 10. By month 6, two have experienced cause A and one has experienced cause B. At month 6, one more person experiences cause A. There are no other events before month 6, and no censoring occurs before month 10.
Immediately before month 6, the cause-specific risk set contains \(12-2-1=9\) people: the two prior cause-A events and the prior cause-B event all had some event and are no longer event-free. The cause-A event increment at month 6 is one event out of those nine, or \(1/9=0.1111\) as a discrete conditional event proportion for that time point.
For the subdistribution denominator for cause A, only the two earlier cause-A events are removed. The earlier cause-B event remains in the denominator by definition. There are therefore \(12-2=10\) people in the corresponding event-history denominator, and the cause-A increment is \(1/10=0.1000\).
These are different quantities because their event-history denominators differ, not because censoring was handled differently. With no censoring, there is no censoring assumption to check in this example. In continuous-time data, the corresponding hazards are rates rather than these single-time event proportions.
Administrative Censoring: Often Simple, Not Automatic
Administrative censoring occurs because observation ends at a planned study cutoff, such as a database closing date. If everyone is followed until a common fixed duration \(\tau\), then \(C=\tau\) for all participants. A fixed \(C\) contains no individual variation that could select people according to their event times, and follow-up is complete for event-process quantities through \(\tau\). Nothing in this design, however, reveals events after \(\tau\).
More commonly, participants enter over calendar time and all records close on one calendar date. They then have different follow-up durations. Treating this as benign administrative censoring requires that entry timing and the resulting censoring time do not carry information about future event times and causes, either unconditionally or after conditioning on appropriate covariates. A change in treatment, referral patterns, or patient mix over calendar time can make that assumption implausible if those changes are not accounted for.
Worked Example: A Common Study Cutoff
Suppose a hypothetical study plans to observe every participant for 12 months. At month 8, among 50 participants, three have had cause A and two have had cause B. One additional cause-A event occurs during month 8. No one is lost to follow-up before month 8.
The study’s 12-month administrative cutoff does not censor anyone before month 8. Just before the new event, the cause-specific denominator is \(50-3-2=45\), and the month-8 cause-A event proportion is \(1/45=0.02222\). The subdistribution denominator for cause A is \(50-3=47\), because the two competing events are retained in that denominator. Its corresponding increment is \(1/47=0.02128\).
The planned cutoff is later than the event time being analyzed, so it does not affect these calculations. It does limit the period the study can inform: a cause-specific hazard after month 12 is not identified by this follow-up. The example assumes events are recorded correctly and participants have the stated observation through month 8; it does not need an independence assumption for censoring before month 8 because none occurs in that period.
Worked Example: When Censoring Selects Higher-Risk People
Imagine 100 hypothetical participants: 20 are in a high-risk group and would all experience the event of interest at month 6; the other 80 would remain event-free through month 6. Suppose all 20 high-risk participants are lost to follow-up at month 5, while all 80 lower-risk participants remain observed through month 6.
In the underlying group, the month-6 event proportion is \(20/100=0.2000\). In the observed records, there are zero recorded month-6 events among the 80 people still followed. A calculation using only those observed at month 6 gives \(0/80=0\), which does not represent the underlying 0.2000 event proportion.
Here censoring is informative: group membership predicts both censoring and event time. The missing event histories are not interchangeable with the histories of the 80 observed participants. No amount of careful interpretation of the hazard definition repairs this selection. The analysis needs a defensible adjustment—for example, measured covariates that explain the risk and censoring differences—or must acknowledge that the target quantity is not identified from these records under the stated information.
How to Check the Assumption and What to Do Instead
Independent censoring cannot generally be proven from observed data: the event times of censored participants are unknown. It is a design and modeling assumption, assessed using study knowledge and available variables. Useful checks include:
- Compare censoring patterns across measured prognostic groups and calendar periods.
- Ask why follow-up ended. A planned study closure differs from withdrawal because health worsened or a participant entered hospice.
- Check whether factors related to both censoring and event risk have been measured and included in the analysis.
- Confirm that follow-up support extends through the times being reported; independence does not create observations where none exist.
If censoring depends on measured covariates, analyses can sometimes condition on those covariates or adjust for the probability of remaining observed, provided the relevant model and positivity assumptions are credible. If censoring depends on unmeasured event risk, the observed data alone generally cannot establish the missing event histories. Sensitivity analyses can show how conclusions change under plausible departures from independence.
Complete follow-up avoids this particular censoring problem over the observed period, but may be impractical and still cannot support inference beyond the last observed time. Independent censoring permits standard analyses to use incomplete follow-up under an assumption. Modeling censoring or using inverse-probability weighting can address some measured dependence, but these approaches do not automatically fix unmeasured dependence or a lack of follow-up support. These are alternatives with assumptions, not substitutes for stating the assumptions.
Practical Interpretation
When reporting an analysis, keep three statements separate: which hazard is being discussed; how competing events enter its definition; and what assumption is made about censoring. For example, a cause-specific hazard is defined among those still event-free, while a subdistribution hazard uses a denominator that retains competing events. Either quantity may be estimated from right-censored data only under suitable assumptions about how observation ends and with adequate follow-up support.
This separation prevents a common overstatement. A hazard’s definition does not guarantee that an estimate from a particular study is unbiased or even identified. That conclusion also depends on how participants were followed, why censoring occurred, what covariates were measured, and the time range over which people could be observed.
Check Your Understanding
Answer each question, then compare with the brief solutions below.
- Does the definition of a cause-specific hazard include an assumption that censoring is independent? Explain.
- In a cohort of 40, two people have already had cause A and three have had cause B just before time \(t\). What are the cause-specific and subdistribution denominators for cause A, assuming complete observation and no other events?
- Why can a common administrative study cutoff be uninformative censoring through the cutoff but provide no information about hazards after it?
- If people with worsening health are more likely to leave follow-up and worsening health predicts an event, what assumption is threatened?
- Can observed data prove that censoring is independent of event time and cause? Give one reason for your answer.
Solutions. (1) No. The hazard is defined from the event-time and cause distribution; independent censoring is an additional assumption for inference from censored observations. (2) The cause-specific denominator is \(40-2-3=35\); the subdistribution denominator for cause A is \(40-2=38\), retaining prior cause-B events. (3) Through the cutoff, follow-up is complete by design; after it, no participants are observed, so there is no support for estimating later hazards. (4) Conditional independent censoring is threatened unless measured factors adequately account for the dependence. (5) Generally no: event times and causes after censoring are unobserved, so independence cannot be directly verified from those records.