One Ending to Follow-Up, Several Different Facts
When a participant’s visits stop, the record does not automatically tell us why they stopped or what happened afterward. The participant may have reached the planned end of the study while still event-free, become unreachable, or experienced an event that makes later measurements impossible or meaningless. These are different observation histories, even if each leaves the data set with no later biomarker values.
The key question is not simply, “When did follow-up stop?” It is, “What is known about the participant’s event status and outcome after that time?” That distinction matters because a time-to-event analysis can use a known event-free period up to a censoring time, whereas a longitudinal analysis may face unobserved measurements, and a terminal event may mean the later measurement is not defined at all.
This definition concerns an event-time outcome. A participant can stop attending biomarker visits yet still have their event status ascertained through a registry. In that case, longitudinal follow-up has ended, but event follow-up may continue. Conversely, if event ascertainment also ends, the last known event-free time may be the event-time censoring time. The two observation processes should not be collapsed into one date without checking what was actually observed.
Three Ways Follow-Up Can End
Administrative censoring occurs when observation ends for a planned reason, such as the study’s fixed closing date. If a participant is known to be event-free through that date, their event time is right-censored there. The record contains a known event-free interval, not an unknown event status during that interval.
Loss to follow-up occurs when the study can no longer obtain information from a participant through its usual follow-up process. If the participant was last confirmed event-free at time \(C_i\), the event time may be right-censored at \(C_i\). But the record does not establish that the participant remained event-free after \(C_i\). The participant’s later event status is unknown unless it is obtained from another source. Loss to follow-up describes a follow-up problem; right censoring describes what is known about the event time. Loss to follow-up can produce right censoring, but the terms do not mean the same thing.
A terminal event is an event after which the longitudinal outcome of interest cannot be observed as the same outcome. Death is a common example when the outcome is a living person’s blood pressure. After death, blood pressure is not merely missing for that person at a scheduled visit; it is not defined for that person at that time. A terminal event may also be the time-to-event outcome itself. In that case it is an observed event, not a censored observation. The interpretation depends on the chosen outcome and event definition.
| Situation | What is known after the last visit? | Implication |
|---|---|---|
| Study closes as planned; participant is confirmed event-free at closure | Event-free through the closure time; later status is not observed | Right-censored at the closure time for the event-time outcome |
| Participant is lost; last contact confirms no event | Event-free through last confirmation; later status is unknown | May be right-censored at last confirmation, subject to censoring assumptions |
| Participant dies before a scheduled biomarker visit | The later biomarker value is not defined for that participant | Terminal event for that biomarker outcome; not ordinary missingness |
| Visits stop, but an external source continues to ascertain events | Biomarker values stop; event status may remain known | Longitudinal dropout need not censor event-time follow-up |
Worked Examples: Reading Follow-Up Timelines
Worked Example: Visit Dropout With Continued Event Ascertainment
Consider three hypothetical participants in a 12-month study. Participant A stops attending biomarker visits after month 5, but a registry records a hospitalization at month 9. Participant B is last confirmed free of hospitalization at month 5 and cannot be contacted afterward; no registry information is available. Participant C remains under observation and is confirmed free of hospitalization through the planned study close at month 12. Classify each participant’s record for the biomarker and hospitalization outcomes.
Participant A. Biomarker follow-up ends after month 5, so later biomarker values are unobserved. However, hospitalization is ascertained at month 9. For the hospitalization time, \(\tilde T_A=9\) months and \(\Delta_A=1\): the event is observed. The visit dropout did not censor the event time because event ascertainment continued.
Participant B. Biomarker follow-up ends at month 5, and hospitalization status is not known afterward. For the hospitalization time, the record supports right censoring at month 5: \(\tilde T_B=5\) months and \(\Delta_B=0\), provided the participant was confirmed event-free through that time. It would be incorrect to say the participant remained event-free through month 12.
Participant C. The biomarker record continues through month 12, and hospitalization is not observed by the planned close. For the event-time outcome, \(\tilde T_C=12\) months and \(\Delta_C=0\). This is administrative right censoring. The known event-free period extends through month 12, unlike Participant B’s known period, which ends at month 5.
Thus, “stopped visits” alone does not determine the event-time record. The relevant question is how long event status was actually ascertained.
Worked Example: Likelihood Contributions for an Event and Two Censored Times
Suppose, hypothetically, that event times follow an exponential distribution with hazard \(\lambda=0.10\) per month. Three participants are observed: one has the event at month 4, one is confirmed event-free through month 6, and one through month 3. Assume censoring is independent of event time under the model, and assume event status is known through each stated censoring time. What is the event-time likelihood contribution, omitting the censoring-process terms?
For an exponential event time, the density is \(f(t)=\lambda e^{-\lambda t}\) and the survival function is \(S(t)=P(T>t)=e^{-\lambda t}\). An observed event contributes its density; a right-censored observation contributes its survival probability through the censoring time. Therefore,
Substituting \(\lambda=0.10\) per month gives
The event contribution includes the density at month 4. The other two contributions say that the event time exceeds 6 months and 3 months, respectively. The likelihood value has units of inverse months because one observation contributes an event-time density; it is not itself a probability that the model is true. If the month-6 participant had actually been lost at month 3, the data would support \(S(3)\), not \(S(6)\).
Why the Right-Censoring Contribution Has This Form
The likelihood contribution follows directly from what the record reveals. If an event is observed at time \(t\), the density of the event time at that value is \(f(t)\). If the record is right-censored at time \(c\), the observed fact is \(T>c\), whose probability is the survival function \(S(c)\). Writing the two cases together gives the event-time contribution
When \(\Delta_i=1\), the exponents select \(f(\tilde T_i)\); when \(\Delta_i=0\), they select \(S(\tilde T_i)\). This expression is the event-time likelihood contribution when the censoring mechanism can be ignored for the event-time model—for example, under an appropriate independent-censoring assumption, possibly conditional on measured covariates. If censoring depends on event risk in ways not accounted for, the usual event-time analysis may be biased. A record labelled “censored” does not by itself establish that this assumption is reasonable.
This derivation also shows why an event after the censoring time cannot be treated as though its time were known. For a participant censored at \(c\), the data contribute information that \(T>c\), not a particular value of \(T\) after \(c\).
Worked Example: A Terminal Event and a Longitudinal Mean
In a hypothetical cohort of 100 participants, 20 die before the 12-month visit. Among the 80 participants alive at month 12, the mean measured biomarker is 72 units. Can 72 units be reported as the cohort’s 12-month mean biomarker?
The arithmetic mean among the observed survivors is
This correctly describes the 80 participants alive and measured at month 12, assuming their values were recorded as stated. It does not give a mean for all 100 participants at month 12. For the 20 participants who died, a living-person biomarker value at month 12 is not defined. Calling those values “missing” and silently treating the survivor mean as the full-cohort mean changes the target being described.
A study could instead define an outcome that includes death, such as a composite ranking or an outcome assessed only while alive, but that requires an explicit definition. It could also report the survivor mean alongside mortality. It cannot recover an all-participant mean of a post-death biomarker without defining a different, hypothetical outcome and making additional assumptions.
Diagnostics, Comparisons, and Limits
Begin with a participant-level timeline. Mark the last observed longitudinal measurement, the last date event-free status was confirmed, the event date if observed, and the planned study end. These dates may differ. Check the protocol and data sources to determine whether event surveillance continued after visits stopped. A single database field called “withdrawal date” may not contain enough information to classify both processes.
Useful checks include tabulating reasons and times for visit discontinuation, comparing event ascertainment after visit dropout with ascertainment among retained participants, and checking whether last-known event-free dates are recorded consistently. A record in which an event is known to have occurred after the stated censoring date signals a date or coding inconsistency. A record with no confirmation of event-free status at the proposed censoring time does not justify asserting survival through that time.
Administrative censoring and loss to follow-up can both appear as right-censored observations in an event-time data set, but their practical implications differ. Administrative censoring often follows a common study end; loss to follow-up can occur at participant-specific times and may be related to health or event risk. Neither label alone proves that censoring is independent of the event time. Terminal events differ more fundamentally for longitudinal outcomes: after the event, the outcome may cease to exist rather than merely go unrecorded.
A common mistake is to treat every absent value after a participant’s last visit as ordinary missing data and every last visit date as an event-time censoring date. The correction is to ask two separate questions: Was the longitudinal outcome still defined? and Was event-free status known through the proposed censoring time? If the outcome was defined but not measured, it is unobserved. If it was not defined after a terminal event, it is not ordinary missingness. If event status was known only through a last contact, right censoring can be recorded there, but later event status remains unknown.
These distinctions improve interpretation, but they do not solve every analysis problem. Right-censoring methods require assumptions about the censoring process; longitudinal analyses require a clearly stated target when outcomes cease to exist after a terminal event. The observed data alone cannot always reveal whether loss to follow-up was related to unobserved health or event risk.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A participant attends visits through month 4, then stops attending. A registry records no event through month 10. What is known about the event time and the longitudinal outcome after month 4?
- A participant is confirmed event-free at the planned study close of month 18. State the event-time observation.
- Why is a biomarker value after death not always appropriately described as an ordinary missing value?
- Suppose a participant has an observed event at month 7. What is the event-time likelihood contribution: \(f(7)\) or \(S(7)\)? Explain.
Solutions. (1) Event status is known through month 10, so the event time is right-censored at 10 months if no event occurred by then. Biomarker values after month 4 are unobserved; the registry information does not supply them. (2) \(\tilde T=18\) months and \(\Delta=0\): event-free through the administrative close, with later event time unobserved. (3) Death can make a living-person biomarker undefined, rather than defined but unrecorded; the outcome and estimand need explicit definition. (4) The contribution is \(f(7)\), because the event time is observed at month 7; \(S(7)\) would represent being event-free beyond month 7.