Concept Architecture
Concept
Theoretically, Time-to-Event is a quantitative measure representing the elapsed time from a defined starting point until the occurrence of a specified event. It forms the basis of survival analysis and time-to-event modelling, allowing the analysis of outcomes such as disease progression, hospitalisation or death while appropriately accounting for censored observations. In health economics, time-to-event methods are fundamental for estimating transition probabilities, survival, treatment duration and long-term health outcomes within economic evaluation models.
Mathematically, time-to-event is represented as a non-negative random variable described by a probability distribution. Its behaviour is characterised by the survival function, hazard function and cumulative hazard function. These functions estimate the probability of remaining event-free, the instantaneous risk of event occurrence and the cumulative risk over time, respectively.
In practice, time-to-event data are estimated using clinical trial data, registries or observational studies. Survival functions are commonly estimated using the Kaplan?Meier estimator, while hazard functions are modelled using semi-parametric or parametric survival models. Estimated survival curves are incorporated into decision trees, Markov models and discrete event simulations to estimate lifetime costs, quality-adjusted life years and cost-effectiveness.
Purpose
Used to estimate the timing of clinically important events, quantify survival and disease progression, derive transition probabilities and inform health economic models evaluating long-term clinical and economic outcomes.
Mathematical Formulae
Primary Formula
Survival function:
S(t) = P(T > t)
where:
- T = time-to-event random variable
- S(t) = probability that the event has not occurred by time t
Supporting Formulae
Hazard function:
h(t) = f(t)/S(t)
Cumulative hazard:
H(t) = ??? h(u) du
Relationship between survival and cumulative hazard:
S(t) = e?????
Kaplan-Meier estimator:
?(t) = ???�? (1 ? d?/n?)
where:
- d? = number of events at time t?
- n? = number at risk immediately before t?
Related Mathematical Methods
- Survival analysis
- Kaplan-Meier estimation
- Cox proportional hazards regression
- Parametric survival modelling
- Maximum likelihood estimation
- Numerical integration
- Probabilistic sensitivity analysis
Example
A clinical trial follows 500 patients after initiating a new oncology treatment. At five years, the Kaplan-Meier estimate of overall survival is:
S(5) = 0.68
indicating that 68% of patients remain alive five years after treatment initiation. A Weibull survival model is fitted to extrapolate survival beyond the observed follow-up period for incorporation into a cost-effectiveness model estimating lifetime costs and quality-adjusted life years.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| EXP | =EXP(-CumHazard) | Calculate survival from cumulative hazard |
| LN | =-LN(Survival) | Estimate cumulative hazard |
| IF | =IF(Event=1,1,0) | Identify observed events for survival analysis |
| AVERAGE | =AVERAGE(TimeToEvent) | Calculate mean observed event time |
| Solver | Optimise survival model parameters | Fit parametric survival distributions |
VBA (Optional)
Automate fitting of parametric survival models and generation of extrapolated survival curves for health economic analyses.
Sources
- Klein JP, Moeschberger ML. Survival Analysis: Techniques for Censored and Truncated Data. 2nd ed. Springer.
- Collett D. Modelling Survival Data in Medical Research. 3rd ed. Chapman & Hall/CRC.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press; 2006.
- Latimer NR. Survival analysis for economic evaluations alongside clinical trials. Medical Decision Making. 2013.
- NICE. Health Technology Evaluation Manual.
- Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes. 4th ed. Oxford University Press.
Related Concepts (2)
Library
Publications
1
NICE DSU Technical Support Document 16: Adjusting survival time estimates in the presence of treatment switching — Latimer & Abrams, TSD 16 ed., 2014 (NICE Decision Support Unit (University of Sheffield))
Guidance on statistical methods (RPSFTM, IPCW, two-stage) for adjusting overall-survival estimates when patients in a trial switch from the control arm to the experimental treatment, a common problem in oncology economic evaluation.
Frequently Asked Questions (6)
What is time-to-event data?
Outcome data recording the duration until a specific event of interest occurs, such as disease progression or death, possibly censored.
Source: Kalbfleisch & Prentice 2002
Why is time-to-event data common in health economics?
Many outcomes that matter for economic evaluation are naturally measured as the time until something happens, such as how long a patient survives, stays free of progression, or remains out of hospital. Because these durations directly determine how long costs are incurred and health is enjoyed, they feed straight into a model's estimates. Their prevalence, and the fact that many patients are still event-free when a study ends, is why methods for such data are central to the field. Collett (2015) describes their analysis.
Source: Collett 2015
What is censoring in time-to-event data?
Censoring in time-to-event data occurs when the event of interest has not been observed for an individual by the end of their follow-up, so their exact event time is unknown, only that it exceeds the time observed. This happens when a study ends before the event, or a person is lost to follow-up. Right censoring, the most common form, means the event time lies beyond the last observation. Censoring must be handled properly, since ignoring it would bias estimates of the time to event.
Source: Kalbfleisch & Prentice 2002
Why does time-to-event data require special methods?
Time-to-event data require special methods because of censoring: some individuals have not experienced the event by the end of observation, so their event times are unknown, and standard methods that ignore this would discard information or bias results. Survival analysis methods use both the observed event times and the censored times, which contribute the information that the event had not occurred by a certain point. This allows unbiased estimation of the timing and risk of events despite incomplete follow-up, which ordinary analysis cannot do.
Source: Kalbfleisch & Prentice 2002
What methods analyse time-to-event data?
Methods for analysing time-to-event data include the Kaplan-Meier estimator, which estimates the survival curve, the probability of remaining event-free over time, accounting for censoring; the log-rank test, comparing survival between groups; and regression models such as the Cox proportional hazards model, which relates the hazard of the event to covariates. These methods use both observed and censored times to estimate survival, compare groups, and identify factors affecting the timing of events, forming the standard toolkit of survival analysis.
Source: Kalbfleisch & Prentice 2002
How is time-to-event data used in health economics?
In health economics, time-to-event data are central to modelling, since outcomes such as survival and time to progression drive costs and effects. Survival analysis provides the estimates of how long patients remain in health states, which feed transition probabilities and extrapolation in economic models. Because trials have limited follow-up, survival data are often extrapolated beyond the observed period to a lifetime horizon, using parametric survival models, whose assumptions strongly affect results and so are examined carefully.
Source: Kalbfleisch & Prentice 2002
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 7 Oct 2025
Content version: 1.0.0
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- Persistent URI
- https://healtheconomics.wiki/concept/time-to-event
- Term code
- HE-EM-DES-015
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