Why Follow-Up Time Changes the Comparison
In a platform or umbrella trial, participants may enter in different calendar periods, while the analysis uses a common data-cutoff date. Someone who enrolled early may have many months in which an event could be recorded; someone enrolled recently may have only a short observation window. A person with no recorded event after two months is not known to be event-free at six months. Treating the two observations as equivalent can distort an outcome comparison.
The key distinction is between the time an event occurs and the time observation ends. Unequal follow-up is not automatically a source of bias: time-to-event methods can use each participant’s observed time if censoring is appropriately handled. But those methods do not correct changes in outcome measurement, changes in prognosis across calendar time, or gaps in which an event could not be detected. Those problems require separate assessment.
For participant \(i\), let \(A_i\) denote treatment assignment and let \(E_i\) denote calendar entry date. With a data cutoff date \(D\), administrative follow-up is limited by the time from \(E_i\) to \(D\). Other reasons observation may end include withdrawal, loss to follow-up, or a participant reaching a prespecified study end. The analysis must use a consistent time origin and a clearly specified event definition in every period.
Choose the Outcome Target Before Choosing the Method
A time-to-event target describes the distribution of \(T\), such as the probability of remaining event-free beyond time \(t\). For treatment group \(a\), write this survival probability as \(S_a(t)=P(T^a>t)\), where \(T^a\) is the potential event time under assignment \(a\). A fixed-horizon target might instead be the risk of an event by \(\tau\), \(1-S_a(\tau)\), or the difference in those risks between groups. The horizon \(\tau\) should be clinically meaningful and supported by adequate follow-up in the populations being compared.
The time-to-event approach retains information from participants whose observation ends before the event, provided the censoring assumptions are credible. The fixed-horizon approach is direct when everyone’s event status through \(\tau\) is known. It is not valid to code a person censored before \(\tau\) as event-free through \(\tau\), or to compare only those with complete follow-up without considering why follow-up is incomplete.
Calendar time may need to be among the conditioning variables. For example, enrollment period can be associated both with administrative follow-up length and with changes in patient prognosis or care. Including calendar period in a censoring model or stratified analysis can address some measured differences, but it cannot create information where there is no follow-up or no overlap between treatment groups and periods. The earlier material on covariate overlap and period adjustment is relevant to that separate comparability question.
Kaplan–Meier Estimation With Unequal Follow-Up
The Kaplan–Meier estimator estimates survival by multiplying estimated probabilities of surviving each observed event time. At event time \(t_j\), let \(n_j\) be the number still at risk just before \(t_j\), and let \(d_j\) be the number of events at that time. The estimated conditional probability of surviving past that event time is \(1-d_j/n_j\). Censoring removes a person from later risk sets but is not counted as an event.
This product has a simple justification. Survival through \(t\) can be written as the product of conditional probabilities of surviving each event time up to \(t\). If, within the relevant analysis group or conditional strata, censoring is independent of event time, those still under observation at \(t_j\) can represent the event experience of the group still at risk. The observed event fraction \(d_j/n_j\) then estimates the conditional event probability at that time. Multiplying the estimated conditional survival probabilities gives the product-limit estimate. If censoring is informative, this risk-set argument fails.
Worked Example: Calculating Survival With Censoring
Suppose a hypothetical comparison follows six participants in each of two groups from randomization. In Group A, events occur at months 2, 4, and 6; censoring occurs at months 3, 5, and 8. In Group B, events occur at months 3 and 5; censoring occurs at months 4, 6, 7, and 8. Assume for illustration that censoring is independent of event time within group and calendar period, and that event ascertainment is the same in both groups.
For Group A, the risk set has 6 participants at month 2, so the survival factor is \(5/6\). One participant is censored at month 3, leaving 4 at risk at month 4; the factor there is \(3/4\). After censoring at month 5, 2 remain at risk at month 6, giving a factor of \(1/2\). Thus:
For Group B, the factors at the event times are \(5/6\) at month 3 and \(3/4\) at month 5. Censoring at month 4 occurs after the first event and before the second, and censoring at month 6 is after the last event. Therefore:
The estimated six-month event-free proportions are 31.25% and 62.50% in this small hypothetical example. Their difference is not, by itself, evidence of a treatment effect: the example has very few participants, and a valid comparison would also require an appropriate study design, uncertainty estimates, and credible comparability and censoring assumptions. The arithmetic illustrates how censoring changes who remains in each risk set; it does not turn censoring into an event.
Administrative Censoring and Calendar Periods
Administrative censoring arises when observation is stopped at a planned data cutoff or study end. If enrollment dates differ, follow-up lengths differ even if every participant is followed continuously until that cutoff. A common cutoff therefore does not imply a common follow-up duration. The cutoff is often predictable from enrollment date; the important question is whether the censoring mechanism is independent of future event time after accounting for relevant calendar-period and participant information.
Worked Example: Reading Follow-Up at a Data Cutoff
Consider two hypothetical participants with a six-month event horizon. Participant 1 is randomized on January 1 and has no event by a July 1 data cutoff. Participant 2 is randomized on May 1 and also has no event by July 1. For simplicity, use calendar months as the follow-up unit.
Participant 1 has approximately six months of observation and can be classified as having no observed event through the six-month horizon, assuming complete ascertainment. Participant 2 has only approximately two months of observation. Their record is \(Y=2\) months and \(\delta=0\), not a known six-month event-free outcome. The absence of a recorded event by July 1 establishes only that no event was observed during those two months.
A Kaplan–Meier analysis can use Participant 2’s observed two months of follow-up as a censored observation, subject to its censoring assumptions. A direct six-month risk comparison cannot count that participant as event-free at six months. If the recent entrants have not yet had an opportunity to reach month six, the six-month target is not yet fully observed for them.
Calendar-related concerns go beyond duration. If recent participants receive different care, have a different risk profile, or have less complete event ascertainment, then calendar period can affect both the outcome process and the observation process. Adjustment for period may address measured differences only when there is adequate overlap and the model is appropriate. If one treatment is observed only in early periods and another only in later periods, a statistical model cannot reliably separate treatment from calendar change without additional assumptions.
A further distinction is between right censoring and incomplete or delayed ascertainment. If visits occur on a schedule and an event is known only to have occurred between visits, the event time may be interval-censored rather than exactly observed. If events are missed more often in one period, the problem is differential measurement, not merely shorter follow-up. Such data require methods and sensitivity analyses suited to the actual ascertainment process; treating every last contact as ordinary independent censoring is not a remedy.
Fixed-Horizon Analysis and Alternatives
When all participants in the target cohort have complete ascertainment through a prespecified horizon \(\tau\), compare the proportions with an event by \(\tau\). If some participants have not reached \(\tau\), either wait for adequate follow-up, define a target cohort with complete follow-up, or use a method that explicitly handles censoring under stated assumptions. Restricting to early entrants can yield a valid fixed-horizon description for that restricted population, but it changes the target population and may reduce relevance to later entrants.
Worked Example: A Common Fixed Horizon
Suppose a hypothetical trial has 50 participants per group in a cohort selected so that every participant has complete six-month ascertainment. By six months, 8 of 50 participants in Group A and 14 of 50 in Group B have experienced the event. Because every participant’s six-month status is known, the estimated risks are:
The estimated risk difference, Group A minus Group B, is \(0.16-0.28=-0.12\), or 12 fewer events per 100 participants by six months in Group A in this hypothetical cohort. This is a descriptive comparison; causal interpretation additionally depends on the treatment assignment and target-population assumptions. If participants censored before six months had been included and counted as non-events, this calculation would no longer estimate the six-month risks.
| Approach | What it estimates or uses | Key requirement and limitation |
|---|---|---|
| Complete fixed-horizon risk | Event risk by a specified time \(\tau\) | Event status through \(\tau\) must be known for the target cohort; restricting to complete cases can change the target population. |
| Kaplan–Meier survival | Survival probabilities over follow-up time, accounting for right censoring | Requires independent censoring within the analysis groups or conditional strata and follow-up support at the times reported. |
| Regression time-to-event model | A modeled association, such as a hazard ratio, possibly adjusted for measured covariates | Requires censoring assumptions and a suitable model; a hazard ratio is not a fixed-horizon risk difference and proportional hazards may not hold. |
| Restricted mean survival time | Expected event-free time accumulated up to a prespecified \(\tau\) | Requires credible survival estimates over the interval through \(\tau\); it does not solve inadequate follow-up or informative censoring. |
The Kaplan–Meier curve is useful for displaying when estimated event-free survival changes and how risk sets shrink. A regression model can account for measured covariates, including calendar period when appropriate, but adjustment does not fix unmeasured temporal differences or lack of overlap. Restricted mean survival time, the area under the survival curve from 0 to \(\tau\), can provide an interpretable contrast in average event-free time through \(\tau\), particularly when a single hazard ratio is not a suitable summary. All these approaches depend on having adequate observation and a defensible censoring model for the stated target.
Diagnostics, Failure Modes, and a Practical Sequence
Before estimating a treatment contrast, inspect the pattern of observation as well as the event data. Summarize follow-up duration and censoring reasons by treatment group and calendar period; display numbers at risk beneath survival curves; and check whether the requested horizon is supported by meaningful numbers still under observation in each group. A tail estimate based on very few people can be highly unstable even if software returns a number. Compare event ascertainment schedules, missingness, and censoring reasons across periods.
A practical analysis sequence is:
State the event, time origin, population, treatment contrast, and either the time-to-event summary or the fixed horizon \(\tau\).
Tabulate follow-up and censoring by treatment and calendar period, distinguish administrative censoring from other losses, and check whether event ascertainment changed.
Check risk sets through the target horizon and assess whether censoring is plausibly independent after conditioning on measured predictors and calendar period.
Use a fixed-horizon risk when status is known through \(\tau\), or a censoring-aware time-to-event analysis when its assumptions are credible. State the population and time range the result represents.
Common mistake: “Both participants had no recorded event, so both count as event-free at six months.” This is wrong when one participant was observed for only two months. The correction is to treat that record as right-censored at two months for a suitable time-to-event analysis, or to obtain six-month status before including it in a direct six-month risk calculation. A second common error is to interpret a Kaplan–Meier curve as repairing calendar-period differences in care or measurement. It handles unequal observed time only under its censoring assumptions; it does not remove other period-related differences.
When assumptions are doubtful, report the concern and examine how conclusions change under plausible alternatives, such as different follow-up windows or censoring assumptions. If no participants in a relevant group remain under observation near the chosen horizon, do not present an extrapolated estimate as if it were supported by observed follow-up. The limitations are substantive: a method cannot recover unobserved outcomes without assumptions, and an adjusted analysis cannot identify a contrast unsupported by the data.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A participant has no event recorded by month 3, when follow-up ends, and the chosen horizon is month 6. What is known about the participant’s six-month event status?
- At an event time, 20 participants are at risk and 4 experience the event. What is the Kaplan–Meier survival factor at that time?
- Why can a common administrative data cutoff produce unequal follow-up lengths?
- Every participant in a selected cohort has complete follow-up to six months. What type of outcome comparison can be made directly, and what does it estimate?
- What does Kaplan–Meier estimation fail to fix if event ascertainment becomes less complete in a later calendar period?
Solutions. 1. Only that no event was observed through month 3; the participant is right-censored at month 3, so six-month status is unknown. 2. The factor is \(1-4/20=0.80\), the estimated conditional probability of surviving that event time among those at risk. 3. Participants who enter earlier have more time between entry and the same cutoff than later entrants. 4. Compare the proportions with an event by six months, estimating the six-month risks in that selected cohort (and, for a difference, their risk difference). 5. It does not correct differential event measurement; the problem is not ordinary right censoring and needs an ascertainment-aware analysis or sensitivity assessment.