Concept Architecture
Concept
Theoretically, a Time-Varying Hazard is a hazard function whose value changes over time rather than remaining constant throughout follow-up. It represents the changing instantaneous risk of experiencing an event as time progresses and reflects the dynamic nature of many biological and clinical processes. Time-varying hazards are fundamental to survival analysis because most diseases and treatments produce risks that evolve over time rather than remaining fixed.
Mathematically, a time-varying hazard is represented by a hazard function h(t) that explicitly depends on time. The hazard may increase, decrease or follow more complex patterns depending on the underlying survival distribution or regression model. Time-varying hazards are estimated using non-parametric, semi-parametric or parametric survival models, with parameters commonly estimated by maximum likelihood or partial likelihood methods.
In practice, time-varying hazards are estimated from censored time-to-event data using survival models such as the Weibull, Gompertz, flexible parametric and Cox models. They are routinely applied in health economics to estimate disease progression, extrapolate long-term survival and generate transition probabilities for decision-analytic models.
Purpose
Used to model changing event risks over time, characterise disease progression, improve survival estimation and support long-term extrapolation in health economic evaluation.
Mathematical Formulae
Primary Formula
h(t) = f(t) / S(t)
Where:
h(t) = hazard function at time t
f(t) = probability density function
S(t) = survival function
Supporting Formulae
Cumulative hazard:
H(t) = ??? h(u) du
Relationship with the survival function:
S(t) = exp(?H(t))
Instantaneous hazard ratio:
HR(t) = h?(t) / h?(t)
Related Mathematical Methods
- Cox proportional hazards regression
- Parametric survival modelling
- Weibull survival modelling
- Gompertz survival modelling
- Flexible parametric survival modelling
- Kaplan?Meier estimation
- Nelson?Aalen estimation
Example
Following major surgery, the hazard of postoperative mortality is highest during the first month and declines thereafter. A Weibull survival model estimates a decreasing hazard over time with a shape parameter less than one, allowing the survival model to represent the changing risk experienced by patients during recovery.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| EXP | =EXP(-B2) | Converts cumulative hazards into survival probabilities. |
| LN | =-LN(B2) | Derives cumulative hazards from survival estimates. |
| IF | =IF(A2>0,C2/B2,"") | Calculates interval-specific hazard estimates from observed data. |
| SUMPRODUCT | =SUMPRODUCT(HazardRange,IntervalWidthRange) | Approximates cumulative hazard over follow-up. |
VBA (Optional)
VBA can automate estimation of time-varying hazards, cumulative hazards and survival projections from patient-level survival 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 (7)
What is a time varying hazard?
A hazard rate that changes in magnitude over follow-up, unlike a constant hazard that stays the same at every point.
Source: Collett 2015
What is a time-varying hazard?
A time-varying hazard is a hazard rate that changes in magnitude over follow-up, so the instantaneous risk of the event differs at different times, unlike a constant hazard that stays the same at every point. The hazard may rise, fall, or turn over time. Most real survival processes have time-varying hazards, since risk usually changes with age, disease progression, or time since an event, so distributions and models must be able to represent a hazard that varies over time.
Source: Collett 2015
What changing event rate is a time-varying hazard?
A time-varying hazard is an event rate that changes as time passes, rather than remaining constant. The instantaneous risk of the event may rise, fall or follow a more complex path over the follow-up period. Many clinical situations show such patterns, for instance a high early risk after surgery that later declines. Representing this requires survival models whose hazard is allowed to vary with time, unlike the constant-rate exponential form. Recognising a changing hazard matters, because assuming a constant rate can distort estimates and extrapolations.
Source: Collett 2015
How does a time-varying hazard differ from a constant hazard?
A time-varying hazard changes over follow-up, with the risk of the event differing at different times, whereas a constant hazard, as in the exponential distribution, stays the same at every point, giving a memoryless process. A time-varying hazard may increase, decrease, or turn, reflecting how risk evolves, while a constant hazard implies unchanging risk. Because real risks usually change over time, distributions with time-varying hazards, such as the Weibull, Gompertz, or log-normal, are often needed rather than the constant-hazard exponential.
Source: Collett 2015
What patterns can a time-varying hazard take?
A time-varying hazard can increase monotonically, as with age-related mortality; decrease monotonically, as when early risk falls; rise to a peak then decline, a unimodal pattern; or follow more complex shapes such as a bathtub, high early and late with a low middle. Each pattern reflects how risk changes over time and determines which survival distribution fits. Identifying the pattern of the time-varying hazard from the data guides the choice of model in survival analysis.
Source: Collett 2015
How is a time-varying hazard modelled?
A time-varying hazard is modelled using survival distributions or models whose hazard can change over time: parametric distributions such as the Weibull, Gompertz, log-normal, or log-logistic, each allowing particular shapes; flexible parametric models using splines for complex shapes; or piecewise models with different hazards in segments. The choice depends on the hazard pattern, whether monotonic, turning, or complex. Modelling the time-varying hazard appropriately is important, since the assumed shape affects both fit and extrapolation of survival.
Source: Kalbfleisch & Prentice 2002
Why does a time-varying hazard matter?
A time-varying hazard matters because assuming a constant hazard when the risk actually changes misrepresents the survival process, biasing fit and especially extrapolation, since the projected survival follows the assumed hazard. Real risks usually vary, rising with age or falling after an initial period, so capturing the time-varying hazard is necessary for a valid model. Recognising how the hazard changes guides the choice of distribution and affects estimates of long-term and mean survival, making the hazard's variation over time a central consideration.
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/time-varying-hazard
- Term code
- HE-EM-SM-088
Stable URI · Machine-readable · Resolvable · CC BY 4.0