Signature
C_i = d - e_i; Y_i = min(T_i, C_i); delta_i = (T_i <= C_i)
| Inputs | Definition | Unit |
|---|---|---|
d | Date of the data cut-off on the trial calendar, for example month 36 | months of the trial calendar |
e_i | Date on which patient i entered the trial, on the same calendar as d | months of the trial calendar |
T_i | Time from entry to the event, which is unobserved when it falls after the cut-off | months |
C_i | Potential follow-up of patient i from entry to the data cut-off | months |
|---|---|---|
Y_i | Time from entry to the event or to the cut-off, whichever comes first | months |
delta_i | 1 if the event was observed by the cut-off, 0 if the patient was censored | none (1 or 0) |
Function
Observed follow-up, Kaplan-Meier survival and restricted mean under administrative censoring
Maps each patient's entry date, the shared data cut-off and the event times seen by the cut-off to the observed follow-up and event indicator, then to the Kaplan-Meier survival curve, the restricted mean survival time up to the last observed follow-up and the share of mean survival that has to come from extrapolation beyond it. Staggered entry makes the censoring time differ between patients, so early recruits are observed for longer than late ones. The constant hazard estimated as events divided by person-time is the rate formula of the Adverse Event Rate page (HE-FM-AER-002), and mean survival with an exponential tail has the form of HE-FM-BTH-002 on the Bathtub Hazard page; neither is repeated here.
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Implementations
Excel
Censoring time, observed time and event flag for one patient
With the cut-off in CutOff, the entry month in EntryMonth and the event time in EventTime (left blank when no event was seen by the cut-off), the three formulas return the censoring time in CensorTime, the observed time and the event flag.
=CutOff-EntryMonth; =IF(EventTime="",CensorTime,MIN(EventTime,CensorTime)); =IF(AND(EventTime<>"",EventTime<=CensorTime),1,0)
Assumptions
Common calendar for entry dates and the cut-off
The entry dates and the data cut-off are recorded on the same calendar and in the same unit, and every time after entry is measured from entry or randomisation. Mixing calendar months with months since entry gives censoring times that are too long for late recruits.
No loss to follow-up before the cut-off
Every patient is followed until the event or the cut-off, so all censoring is administrative. Where some patients leave observation earlier, the censoring time is the smaller of the two and the indicator alone cannot show which mechanism applied.
Worked examples
First recruit of the eight-patient arm dies at month 10
In the article's illustrative arm the cut-off falls at month 36. Patient 1 entered at month 0, so the potential follow-up is 36 months; the death at month 10 is observed and the record is a time of 10 with an event indicator of 1.
d = 36; e_i = 0; T_i = 10; C_i = 36; Y_i = 10; delta_i = 1
Last recruit of the eight-patient arm is censored at month 12
Patient 8 entered at month 24, so the censoring time is 36 minus 24, or 12 months, and the patient was alive at the cut-off. The true event time is unknown; the value of 30 months is assumed here for illustration only, and any value above 12 returns the same observed time of 12 with an event indicator of 0.
d = 36; e_i = 24; T_i = 30; C_i = 12; Y_i = 12; delta_i = 0
Common errors
Median of observed times reported as median follow-up
The median of the observed times counts early deaths as short follow-up. In the article's eight-patient arm it is 15 months, whereas the reverse Kaplan-Meier method, which treats censoring as the outcome, gives a median follow-up of 24 months. Both figures are computed here for illustration and are not printed in the article.
Cut-off date used as every patient's follow-up
Taking the calendar time of the cut-off as the follow-up of every patient ignores staggered entry. Patient 8 of the article's arm would be credited with 36 months of observation instead of 12, and the tail of the curve would appear to rest on far more patients than it does. The 36 months is computed here for illustration.
Sources
Censoring at the close of a study and non-informative censoring
Clark TG, Bradburn MJ, Love SB, Altman DG. Survival analysis part I: basic concepts and first analyses. British Journal of Cancer. 2003;89(2):232-238. Section on censoring, which describes censoring when a patient has not had the event by the close of the study or is lost to follow-up, and states that standard methods are valid only if censoring is non-informative; and the section on median follow-up, which recommends the reverse Kaplan-Meier estimator.
Failure and censoring time notation for right-censored data
Kalbfleisch JD, Prentice RL. The Statistical Analysis of Failure Time Data. 2nd ed. Hoboken, NJ: Wiley; 2002. Standard notation for right-censored data, in which the observed time is the smaller of the failure time and the censoring time and an indicator records whether the failure was observed.
Canonical Identity
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