Concept Architecture
Concept
Theoretically, the Hazard Function is the instantaneous rate at which an event occurs among individuals who have survived up to a given time. It represents the underlying risk of experiencing the event at each moment in time and is a fundamental concept in survival analysis. In health economics, the hazard function underpins the modelling of mortality, disease progression and treatment effects, forming the basis of survival models used in cost-effectiveness analyses and health technology assessments.
Mathematically, the hazard function is defined as the instantaneous event rate conditional on survival up to time t. It is derived from the probability density and survival functions and completely characterises the risk process underlying time-to-event data. Different survival distributions produce distinct hazard function shapes, including constant, increasing, decreasing, bathtub-shaped and non-monotonic hazards.
In practice, hazard functions are estimated using parametric survival models, semi-parametric models such as the Cox proportional hazards model or non-parametric methods. Estimated hazard functions are used to compare treatments, estimate transition probabilities, extrapolate long-term survival and calculate life expectancy, quality-adjusted life-years and healthcare costs within health economic models.
Purpose
Used to quantify the instantaneous risk of an event over time, compare treatment effects, estimate survival patterns and support survival modelling in health economic evaluation.
Mathematical Formulae
Primary Formula
h(t) = f(t) / S(t)
where:
- h(t) = hazard function
- f(t) = probability density function
- S(t) = survival function
Supporting Formulae
Relationship with cumulative hazard:
H(t) = ??? h(u) du
Relationship with survival:
S(t) = exp(?H(t))
Probability density:
f(t) = h(t)S(t)
Related Mathematical Methods
- Survival analysis
- Cox proportional hazards model
- Kaplan-Meier estimator
- Nelson-Aalen estimator
- Parametric survival modelling
- Cumulative hazard estimation
- Maximum likelihood estimation
Example
A clinical trial compares two cancer treatments over five years. At three years, patients receiving the standard treatment have a hazard of death of 0.08 per year, whereas patients receiving the new treatment have a hazard of 0.05 per year. These estimated hazard functions are used to construct survival curves that generate lifetime quality-adjusted life-years and incremental cost-effectiveness ratios.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| EXP | =EXP(-B2) | Calculate survival probability from cumulative hazard. |
| LN | =-LN(C2) | Calculate cumulative hazard from survival probabilities. |
| IF | =IF(C2>0,B2/C2,"") | Calculate the hazard from the probability density and survival function. |
| Solver | Minimise negative log-likelihood | Estimate hazard function parameters for parametric survival models. |
VBA (Optional)
Automate estimation, plotting and comparison of hazard functions across competing survival models for economic evaluation.
Sources
- Klein JP, Moeschberger ML. Survival Analysis: Techniques for Censored and Truncated Data.
- Collett D. Modelling Survival Data in Medical Research.
- Cox DR, Oakes D. Analysis of Survival Data.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
- NICE. Health Technology Evaluation Manual.
Related Concepts (2)
Library
Publications
4
NICE DSU Technical Support Document 21: Flexible methods for survival analysis — Rutherford, Lambert, Sweeting, Pennington, Crowther, Abrams & Latimer, TSD 21 ed., 2020 (NICE Decision Support Unit (University of Sheffield))
Guidance extending standard survival analysis to flexible parametric methods — spline-based models, fractional polynomials, mixture and cure models, and relative-survival approaches — for capturing complex hazard functions in economic evaluation.
Survival Analysis: A Self-Learning Text — David G. Kleinbaum & Mitchel Klein, 3rd Edition ed., 2012 (Springer)
A practical introduction to survival data, censoring, Kaplan-Meier methods, Cox regression, proportional hazards and model interpretation.
BookView source →Statistical Models and Methods for Lifetime Data — Jerald F. Lawless, 2nd Edition ed., 2003 (John Wiley & Sons)
An advanced reference on lifetime and event-time data, including censoring, parametric distributions, hazard functions, regression and model assessment.
BookView source →Modelling Survival Data in Medical Research — David Collett, 3rd Edition ed., 2015 (Chapman & Hall / CRC Press)
A medical-research guide to survival modeling, including censored outcomes, parametric models, Cox regression, diagnostics and practical interpretation.
BookView source →
Frequently Asked Questions (6)
What is a hazard function?
A function describing the instantaneous risk of an event at a given point in time, conditional on having survived to that point.
Source: Collett 2015
How does the hazard function differ from the survival function?
The survival function gives the probability of still being event-free by a given time, a quantity that only ever falls, whereas the hazard function gives the instantaneous rate of the event at that moment for those who have survived to it, a rate that can rise, fall, or stay level. One describes accumulated survival, the other the momentary intensity of risk. The two are mathematically linked, since the hazard drives how fast survival declines, but they convey different information. Collett (2015) sets out this relationship.
Source: Collett 2015
What does the hazard function represent?
The hazard function represents the instantaneous event rate at each time among individuals who have survived to that time, so it is a conditional measure of risk rather than a probability over an interval. A high hazard at a time means the event is likely to occur soon for those still at risk then, and a changing hazard shows how risk evolves. The hazard function thus captures the timing of risk, and its shape describes whether risk rises, falls, or stays constant over the survival experience.
Source: Collett 2015
How does the hazard function relate to survival?
The hazard function relates to the survival function 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 hazard, cumulative hazard, and survival function are mathematically linked, each derivable from the others, so specifying the hazard function determines the survival distribution. This connection makes the hazard a fundamental way to characterise how survival declines over time.
Source: Kalbfleisch & Prentice 2002
What shapes can a hazard function take?
A hazard function can take various shapes: constant, as in the exponential distribution; monotonically increasing, as with age-related mortality; monotonically decreasing, as when early risk falls; or non-monotonic, such as bathtub-shaped, high early and late with a low middle, or rising then falling. The shape reflects how risk changes over time and determines which survival distribution fits, so identifying the hazard's shape from the data guides the choice of model in survival analysis.
Source: Collett 2015
Why is the hazard function important in survival analysis?
The hazard function is important because it characterises how the risk of an event changes over time, which underlies the choice of survival model, the interpretation of survival data, and the effects of covariates, as in the Cox model's hazard ratios. Its shape determines the survival distribution, and comparing hazards between groups reveals differences in risk over time. Because the hazard captures the timing of risk in a way survival probabilities alone do not, it is a fundamental tool for modelling and understanding time-to-event data.
Source: Collett 2015
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 21 Oct 2025
Content version: 1.0.0
Canonical Identity
- Persistent URI
- https://healtheconomics.wiki/concept/hazard-function
- Term code
- HE-EM-SM-033
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