Concept Architecture
Concept
Theoretically, the Hazard Rate is the instantaneous rate at which an event occurs among individuals who remain event-free immediately before a given time. It quantifies the risk of failure, death or disease progression at each instant, conditional on survival up to that time. In health economics, the hazard rate is fundamental to survival analysis, informing estimates of treatment effectiveness, disease progression and long-term health outcomes used in decision-analytic models.
Mathematically, the hazard rate is defined as the limiting probability of an event occurring within an infinitesimally small time interval divided by the interval length, conditional on survival until the start of the interval. It is mathematically equivalent to the hazard function and is related directly to the probability density, survival function and cumulative hazard. The shape of the hazard rate determines the behaviour of survival models and varies according to the underlying probability distribution.
In practice, hazard rates are estimated using non-parametric, semi-parametric and parametric survival models. Estimated hazard rates are used to compare treatments through hazard ratios, derive transition probabilities for state-transition models, extrapolate long-term survival and estimate quality-adjusted life-years and healthcare costs in economic evaluations.
Purpose
Used to quantify the instantaneous risk of an event, compare treatment effects, estimate transition risks and support survival modelling in health economic evaluation.
Mathematical Formulae
Primary Formula
h(t) = f(t) / S(t)
where:
- h(t) = hazard rate
- 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))
Hazard ratio:
HR = h?(t) / h?(t)
Related Mathematical Methods
- Survival analysis
- Cox proportional hazards model
- Hazard ratio estimation
- Kaplan-Meier estimator
- Nelson-Aalen estimator
- Parametric survival modelling
- Maximum likelihood estimation
Example
A randomised oncology trial reports an estimated hazard rate of 0.04 deaths per patient-year for the intervention group and 0.06 deaths per patient-year for the comparator group. The resulting hazard ratio is HR = 0.04 � 0.06 = 0.67, indicating a 33% lower instantaneous risk of death for patients receiving the intervention. These estimates are subsequently used to project long-term survival within a cost-effectiveness model.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| IF | =IF(C2>0,B2/C2,"") | Calculate the hazard rate from the probability density and survival function. |
| EXP | =EXP(-B2) | Convert cumulative hazard into survival probability. |
| LN | =-LN(C2) | Calculate cumulative hazard from estimated survival probabilities. |
| Solver | Minimise negative log-likelihood | Estimate hazard rate parameters for parametric survival models. |
VBA (Optional)
Automate estimation and comparison of hazard rates across treatment groups and generate survival projections for economic evaluation.
Sources
- Cox DR, Oakes D. Analysis of Survival Data.
- Klein JP, Moeschberger ML. Survival Analysis: Techniques for Censored and Truncated Data.
- Collett D. Modelling Survival Data in Medical Research.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
- NICE. Health Technology Evaluation Manual.
Related Concepts (2)
Library
Publications
1
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 →
Tools & Resources
1
survHE — Survival Analysis for Health Economic Evaluation (R package) — Gianluca Baio, R package ed., 2023 (CRAN)
An R package for fitting and comparing parametric survival models for health economic evaluation, including Bayesian estimation, and for extrapolating time-to-event data to inform cost-effectiveness models.
Software (R package)View source →
Frequently Asked Questions (6)
What is a hazard rate?
The instantaneous probability of an event at a specific point in time, given survival without the event up to then, on a continuous scale.
Source: Collett 2015
Why can a hazard rate exceed one?
A hazard rate measures events per unit of time rather than a chance between zero and one, so it can take any positive value, including numbers above one. A rate of two events per person-year, for instance, is perfectly meaningful and does not imply an impossible probability, because it counts expected occurrences over time rather than a likelihood. This is why a rate must be converted, not equated, when a probability over a period is needed. The distinction matters whenever rates feed a model. Collett (2015) explains it.
Source: Collett 2015
How does a hazard rate differ from a probability?
A hazard rate differs from a probability in that it is an instantaneous, continuous measure of the event rate at a moment among those at risk, and can exceed one, whereas a probability is the chance of the event over a specific interval and lies between zero and one. A rate must be converted to a probability for a given interval using an exponential relationship. Confusing the two leads to errors, so the distinction between the instantaneous hazard rate and an interval probability is important in modelling.
Source: Collett 2015
How is a hazard rate estimated?
A hazard rate is estimated from survival data by methods that relate events to the number at risk over time, such as smoothing the increments of the Nelson-Aalen cumulative hazard estimator, or by fitting a parametric survival distribution whose hazard is then evaluated. Non-parametrically, the hazard is harder to estimate directly than the cumulative hazard, so it is often obtained by smoothing. The estimated hazard rate over time reveals the shape of the risk, guiding the choice of survival model.
Source: Kalbfleisch & Prentice 2002
How does the hazard rate relate to the hazard function?
The hazard rate is the value of the hazard function at a particular time, so the hazard function is the hazard rate expressed as a function of time. The hazard function describes how the hazard rate varies across the whole survival experience, while the hazard rate refers to its value at a specific moment. The two terms are closely related and often used interchangeably, both denoting the instantaneous risk of the event conditional on survival, with the function giving the pattern over time.
Source: Collett 2015
Why does the hazard rate matter in survival modelling?
The hazard rate matters because its pattern over time characterises the survival process and determines which distribution fits, and because comparing hazard rates between groups, as through hazard ratios, quantifies differences in risk. Whether the hazard rate is constant, rising, falling, or turning guides model choice and extrapolation, since the assumed hazard shape strongly affects projected survival. The hazard rate also underlies the conversion between rates and probabilities needed for discrete-time models, so it is central to survival analysis and modelling.
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-rate
- Term code
- HE-EM-SM-034
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