Signature
A_tail = S_tau / h; P_tail = A_tail / (RMST_tau + A_tail)
| Inputs | Definition | Unit |
|---|---|---|
S_tau | Estimated survival at tau, where the observed curve ends | probability |
h | Hazard applied from tau onwards, above zero | events per patient-month |
RMST_tau | Restricted mean survival time from 0 to tau, from HE-FM-ADMC-003 | months |
A_tail | Mean event-free time added after tau under the constant hazard h | months |
|---|---|---|
P_tail | Proportion of mean survival that comes from the extrapolated tail | proportion |
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.
Try this function
Implementations
Excel
Tail area and tail share in two cells
With survival at the last follow-up in SurvTau, the tail hazard in TailHazard and the restricted mean in RMST, the first formula returns the tail area in TailArea and the second its share of mean survival.
=SurvTau/TailHazard; =TailArea/(RMST+TailArea)
Assumptions
Constant hazard for life after the last follow-up
The hazard stays at h from tau onwards, so survival after tau is S_tau multiplied by exp(-h(t minus tau)). The share is a property of that assumption, and the clinical plausibility of the hazard has to be judged separately.
Tail starts at the last observed follow-up
tau is the last observed follow-up time measured from entry, month 34 in the article, not the calendar time of the cut-off, month 36. The result is undiscounted.
Worked examples
Tail share of 48% at a hazard of 0.02 per month
With 0.4 alive at month 34, a restricted mean of 21.7 months and a hazard of 0.02 per month afterwards, the tail adds 20 months and supplies about 48% of the mean of 41.7 months.
S_tau = 0.4; h = 0.02; RMST_tau = 21.7; A_tail = 20; P_tail = 0.4796
Tail share of 39% at the trial hazard of 4 deaths in 136 months
The trial estimate of the hazard is 4 deaths in 136 patient-months, about 0.029412 per month. The tail then adds 13.6 months and supplies about 39% of the mean of 35.3 months.
S_tau = 0.4; h = 0.029412; RMST_tau = 21.7; A_tail = 13.60; P_tail = 0.3853
Tail share of 32% at a hazard of 0.04 per month
At 0.04 per month the tail adds 10 months and supplies about 32% of the mean of 31.7 months.
S_tau = 0.4; h = 0.04; RMST_tau = 21.7; A_tail = 10; P_tail = 0.3155
Common errors
Share censored read as the share extrapolated
Half of the article's patients were censored, but survival at month 34 is 0.40 and the tail supplies between about 32% and 48% of the mean depending on the hazard. Maturity is judged from survival at the end of follow-up and the hazard assumed after it, not from the proportion censored.
Tail hazard taken from the earliest recruits alone
After month 24 the article's curve rests on one of the earliest recruits. If prognosis improved during recruitment, a tail hazard fitted to that part of the curve describes the earliest cohort and misstates the share of mean survival for the arm as a whole.
Sources
Extrapolation of censored survival and the influence of the extrapolated period
Latimer N. NICE DSU Technical Support Document 14: Survival analysis for economic evaluations alongside clinical trials, extrapolation with patient-level data. Sheffield: Decision Support Unit, ScHARR, University of Sheffield; 2011, last updated March 2013. Section 2, on mean survival as the area under the curve and the need for extrapolation unless data are complete; section 2.1, the exponential model with a constant hazard; and recommendation 11, which suggests reporting results based only on observed data where censoring is substantial, to show the influence of the extrapolated period.
Administrative censoring and extrapolation in DSU TSD 21
Rutherford MJ, Lambert PC, Sweeting MJ, Pennington B, Crowther MJ, Abrams KR, Latimer NR. NICE DSU Technical Support Document 21: Flexible methods for survival analysis. Sheffield: Decision Support Unit, ScHARR, University of Sheffield; 2020 (updated March 2022). Section 1, which notes that trials are usually subject to a large amount of administrative censoring with short follow-up relative to patients' lifespans, and section 6.6, Figure 26, which compares extrapolations of the same simulated trial with administrative censoring at 3 and 5 years.
Canonical Identity
Stable URI · Machine-readable · Resolvable · CC BY 4.0