Concept Architecture
Concept
Theoretically, Constant Hazard is the assumption that the instantaneous risk of an event remains unchanged over time. It is a fundamental concept in survival analysis and reliability theory and forms the basis of the exponential survival model. In health economics, the constant hazard assumption is used when event risks are believed to remain stable throughout the period of analysis or when simple survival extrapolation is appropriate.
Mathematically, a constant hazard is represented by a hazard function that does not vary with time. Under this assumption, survival declines exponentially, and the hazard rate uniquely determines the survival distribution. The exponential distribution is the only continuous probability distribution with a constant hazard function and is frequently used as a baseline parametric survival model.
In practice, the constant hazard assumption is evaluated by comparing fitted exponential models with observed survival data and alternative parametric distributions. It is applied in decision-analytic models, Markov models and health technology assessments when empirical evidence supports a stable event rate or when model simplicity is justified.
Purpose
Used to model situations in which the instantaneous event risk remains unchanged over time, providing a simple framework for survival estimation, extrapolation and health economic decision modelling.
Mathematical Formulae
Primary Formula
h(t) = ?
Supporting Formulae
Survival function:
S(t) = exp(??t)
Probability density function:
f(t) = ?exp(??t)
Cumulative hazard:
H(t) = ?t
where:
- ? = constant hazard rate
- h(t) = hazard function
- S(t) = survival function
- H(t) = cumulative hazard
Related Mathematical Methods
- Exponential survival model
- Parametric survival analysis
- Maximum likelihood estimation
- Kaplan?Meier estimation
- Cox proportional hazards model
Example
A chronic disease has an annual mortality hazard of 0.08 that is assumed to remain constant throughout follow-up.
The probability of surviving five years is:
S(5) = exp(?0.08 ? 5)
S(5) = exp(?0.40) = 0.670
The estimated five-year survival probability is approximately 67.0%.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| EXP | =EXP(-B2*A2) | Calculate survival probability assuming a constant hazard. |
| LN | =-LN(B2)/A2 | Estimate the constant hazard rate from observed survival. |
| POWER | =EXP(-B2*5) | Project long-term survival under an exponential model. |
VBA (Optional)
Automate estimation of constant hazard models and generate survival projections for health economic analyses.
Sources
- Collett D. Modelling Survival Data in Medical Research. CRC Press.
- Klein JP, Moeschberger ML. Survival Analysis: Techniques for Censored and Truncated Data. Springer.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press.
- Kalbfleisch JD, Prentice RL. The Statistical Analysis of Failure Time Data. Wiley.
- NICE. Health Technology Evaluation Manual.
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 a constant hazard?
A survival pattern where the instantaneous risk of an event stays the same at every point in time, matching an exponential distribution.
Source: Collett 2015
What is the memoryless property of a constant hazard?
A constant hazard means the risk of the event is the same at every moment, so a patient's chance of the event in the next interval does not depend on how long they have already survived. This gives the constant-hazard, or exponential, model a memoryless character, since the future looks the same regardless of elapsed time. It is a strong assumption, realistic only where risk genuinely does not change with time or age, and it is often too simple for chronic disease. Collett (2015) describes this property.
Source: Collett 2015
What distribution corresponds to a constant hazard?
A constant hazard corresponds to the exponential distribution, the parametric survival distribution whose hazard is constant over time. Under the exponential model, survival declines at a steady proportional rate, and the distribution is defined by a single rate parameter equal to the constant hazard. Because it has only one parameter and a constant hazard, the exponential distribution is the simplest survival model, but its assumption of unchanging risk limits its applicability where the hazard actually varies with time.
Source: Collett 2015
What does a constant hazard imply?
A constant hazard implies that the risk of the event is the same regardless of how long an individual has survived, so the process is memoryless: past survival does not change future risk. This means survival declines exponentially, and the expected remaining time is the same at any point. While mathematically convenient, this implication is often unrealistic, since real risks usually change over time, rising with age or disease progression or falling after an initial period, so a constant hazard is a strong and frequently inappropriate assumption.
Source: Collett 2015
When is a constant hazard a reasonable assumption?
A constant hazard is a reasonable assumption when the risk of the event genuinely does not change much over the relevant period, which may hold over short intervals or for some processes without ageing or progression effects. It can serve as a simple approximation where the hazard is roughly stable. However, over longer periods or where risk clearly changes, the constant-hazard assumption is inappropriate, so it is checked against the data, and if the hazard varies, a distribution allowing a changing hazard is used instead.
Source: Kalbfleisch & Prentice 2002
How is the constant-hazard assumption checked?
The constant-hazard assumption is checked by examining whether the hazard appears stable over time, for instance by plotting an estimate of the hazard against time to see if it is roughly flat, or by checking whether the exponential model fits the survival data well compared with distributions allowing a changing hazard. A cumulative hazard that is roughly linear in time also indicates a constant hazard. If these checks show the hazard varies, the exponential model is rejected in favour of a more flexible distribution.
Source: Collett 2015
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 20 Oct 2025
Content version: 1.0.0
Canonical Identity
- Persistent URI
- https://healtheconomics.wiki/concept/constant-hazard
- Term code
- HE-EM-SM-007
Stable URI · Machine-readable · Resolvable · CC BY 4.0