Concept Architecture
Concept
Theoretically, the Survival Function is the probability that an individual remains free of a specified event beyond a given point in time. It is a fundamental function in survival analysis and reliability theory that characterises the distribution of time-to-event data. In health economics, the survival function underpins estimation of life expectancy, disease progression and long-term treatment effectiveness for economic evaluation.
Mathematically, the survival function is defined as the probability that the event time exceeds a specified time point. It is directly related to the probability density function, cumulative distribution function, hazard function and cumulative hazard function. Survival functions may be estimated non-parametrically using the Kaplan?Meier estimator or modelled parametrically using distributions such as the Weibull, Gompertz or log-normal distributions.
In practice, the survival function is estimated from censored patient-level data obtained from clinical trials, observational studies and disease registries. The resulting survival curves are used to estimate median survival, restricted mean survival time, transition probabilities and lifetime health outcomes in health economic models.
Purpose
Used to estimate the probability of surviving beyond specified time points, describe time-to-event outcomes, compare treatments and provide survival inputs for health economic decision models.
Mathematical Formulae
Primary Formula
S(t) = P(T > t)
Where:
S(t) = survival probability at time t
T = time to event
t = specified time
Supporting Formulae
Relationship with the cumulative distribution function:
S(t) = 1 ? F(t)
Relationship with the hazard function:
S(t) = exp(?H(t))
Relationship with the cumulative hazard:
H(t) = ?ln(S(t))
Related Mathematical Methods
- Kaplan?Meier estimator
- Nelson?Aalen estimator
- Cox proportional hazards regression
- Parametric survival modelling
- Restricted mean survival time
- Flexible parametric survival modelling
Example
A survival model estimates that the probability of remaining alive two years after treatment is 0.82.
Therefore:
S(2) = 0.82
The corresponding cumulative hazard is:
H(2) = ?ln(0.82) = 0.198
This indicates an estimated cumulative hazard of approximately 0.20 by two years.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| EXP | =EXP(-B2) | Converts cumulative hazards into survival probabilities. |
| LN | =-LN(B2) | Calculates cumulative hazards from survival probabilities. |
| IF | =IF(A2<=$F$1,B2,"") | Returns survival probabilities up to a specified follow-up time. |
| MATCH | =MATCH(0.5,B2:B100,-1) | Identifies the point at which survival falls below 50% for median survival estimation. |
VBA (Optional)
VBA can automate estimation of survival functions, update survival tables and generate publication-quality survival curves from patient-level datasets.
Sources
Kalbfleisch JD, Prentice RL. The Statistical Analysis of Failure Time Data. 2nd ed. Wiley; 2002.
Klein JP, Moeschberger ML. Survival Analysis: Techniques for Censored and Truncated Data. 2nd ed. Springer; 2003.
Collett D. Modelling Survival Data in Medical Research. 3rd ed. Chapman & Hall/CRC; 2015.
Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press; 2006.
NICE. Health Technology Evaluations: The Manual. National Institute for Health and Care Excellence; 2022.
Related Concepts (2)
Library
Publications
1
NICE DSU Technical Support Document 14: Survival analysis for economic evaluations alongside clinical trials – extrapolation with patient-level data — Nicholas R. Latimer, TSD 14 ed., 2013 (NICE Decision Support Unit (University of Sheffield))
The reference guidance on survival analysis for economic evaluation: fitting standard parametric models (exponential, Weibull, Gompertz, log-logistic, log-normal) to censored trial data and extrapolating to estimate lifetime survival benefit, with a process guide for model selection and justification.
Frequently Asked Questions (6)
What is the survival function?
A function giving the probability that a person survives without the event of interest beyond a specified point in time.
Source: Collett 2015
How does the survival function behave over time?
A survival function starts at one at the beginning of follow-up, when everyone is event-free, and never rises thereafter, falling as events occur and approaching zero over long enough time if the event is certain eventually. Its value at any time is the probability of remaining event-free beyond that point. This steadily declining shape, bounded between one and zero, is what makes it a convenient summary of survival experience. It can never increase, since an event once occurred cannot be undone. Collett (2015) describes these properties.
Source: Collett 2015
What are the properties of the survival function?
The survival function equals one at time zero, since all individuals are event-free at the start, is non-increasing over time, since the probability of remaining event-free can only fall as events occur, and approaches zero as time increases if all individuals eventually experience the event. It gives, at each time, the probability of surviving beyond that time. These properties, starting at one and declining monotonically, define the survival function and distinguish it from the cumulative distribution of event times, which is its complement.
Source: Collett 2015
How does the survival function relate to the hazard?
The survival function relates to the hazard through the cumulative hazard: survival equals the exponential of the negative cumulative hazard, which is the integral of the hazard over time. So a higher hazard accumulates more cumulative hazard and lowers survival faster. The survival function, hazard, and cumulative hazard are mathematically linked, each derivable from the others, so specifying any one determines the others. This connection means the shape of the hazard governs how the survival function declines.
Source: Kalbfleisch & Prentice 2002
How is the survival function estimated?
The survival function is estimated non-parametrically by the Kaplan-Meier estimator, which gives a step-function estimate from the observed events and censoring without assuming a distribution, or parametrically by fitting a distribution, such as the Weibull or log-normal, and using its survival function, which gives a smooth curve that can be extrapolated. Both handle censoring. The choice depends on whether a data-driven description within the observed period or a smooth, extendable curve for extrapolation is needed.
Source: Collett 2015
Why is the survival function central to survival analysis?
The survival function is central because it directly answers the question of how likely individuals are to remain event-free over time, summarising the survival experience, and because many analyses and summaries derive from it, such as median survival, restricted mean survival time as its area, and comparisons between groups. It connects to the hazard and cumulative hazard, and it is what is estimated, plotted, and extrapolated. As the probability of surviving over time, the survival function is the primary object of interest in time-to-event analysis.
Source: Collett 2015
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 24 Oct 2025
Content version: 1.0.0
Canonical Identity
- Persistent URI
- https://healtheconomics.wiki/concept/survival-function
- Term code
- HE-EM-SM-084
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