Concept Architecture
Concept
Theoretically, Cumulative Hazard is the accumulated risk of experiencing an event from the beginning of follow-up to a specified time. It is a fundamental concept in survival analysis that summarises the total hazard experienced over time and underpins the mathematical relationship between hazard functions and survival probabilities. In health economics, cumulative hazard is used to estimate long-term survival, derive transition probabilities and support decision-analytic modelling.
Mathematically, cumulative hazard is defined as the integral of the hazard function over time. It increases monotonically with time and is directly related to the survival function through an exponential transformation. The cumulative hazard provides an alternative representation of survival data that is particularly useful for model estimation, diagnostic assessment and survival extrapolation.
In practice, cumulative hazard is estimated using non-parametric methods such as the Nelson?Aalen estimator or derived from fitted parametric and semi-parametric survival models. It is routinely used to estimate survival curves, compare treatments, validate survival models and generate transition probabilities for health economic evaluations and health technology assessments.
Purpose
Used to quantify accumulated event risk over time, estimate survival probabilities, derive transition probabilities and support survival modelling and health economic decision analysis.
Mathematical Formulae
Primary Formula
H(t) = ??? h(u) du
Supporting Formulae
Relationship with survival:
H(t) = ?ln(S(t))
S(t) = exp(?H(t))
Nelson?Aalen estimator:
H?(t) = ?(d? � n?)
where:
- H(t) = cumulative hazard
- h(t) = hazard function
- S(t) = survival function
- d? = number of events at time i
- n? = number at risk immediately before time i
Related Mathematical Methods
- Nelson?Aalen estimator
- Kaplan?Meier estimation
- Cox proportional hazards model
- Parametric survival modelling
- Maximum likelihood estimation
Example
A survival model estimates a constant annual hazard of 0.08 over five years.
The cumulative hazard after five years is:
H(5) = 0.08 ? 5 = 0.40
The corresponding survival probability is:
S(5) = exp(?0.40) = 0.670
Thus, the cumulative hazard of 0.40 corresponds to an estimated five-year survival probability of approximately 67%.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| LN | =-LN(B2) | Calculate cumulative hazard from survival probability. |
| EXP | =EXP(-B2) | Convert cumulative hazard into survival probability. |
| SUM | =SUM(C2:C20) | Accumulate interval-specific hazards over follow-up. |
VBA (Optional)
Automate calculation of cumulative hazards and survival probabilities from longitudinal event data for health economic modelling.
Sources
- Nelson W. Hazard Plotting for Incomplete Failure Data. Journal of Quality Technology. 1972;4(1):27?52.
- Aalen OO, Borgan ?, Gjessing HK. Survival and Event History Analysis. Springer.
- 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.
Related Concepts (2)
Library
Publications
1
NICE DSU Technical Support Document 15: Cost-effectiveness modelling using patient-level simulation — Davis, Stevenson, Tappenden & Wailoo, TSD 15 ed., 2014 (NICE Decision Support Unit (University of Sheffield))
Guidance on individual patient-level (microsimulation) cost-effectiveness modelling — when to use it in preference to cohort models, how to structure it, and how to handle the associated computational and uncertainty challenges.
Frequently Asked Questions (6)
What is cumulative hazard?
The total accumulated risk of an event by a given point in time, found by integrating the instantaneous hazard rate up to that point.
Source: Collett 2015
What does a plot of cumulative hazard reveal about risk?
The cumulative hazard adds up the instantaneous risk from the start of follow-up to a given time, so its slope at any point shows how fast risk is accumulating there. A straight line of cumulative hazard against time signals a constant hazard, while a curve that steepens or flattens reveals risk that is rising or falling. Reading its shape therefore exposes how the hazard changes over time, which the survival curve alone conveys less directly. It is a diagnostic as well as a quantity. Klein and Moeschberger (2003) describe it.
Source: Klein & Moeschberger 2003
How is cumulative hazard calculated?
Cumulative hazard is calculated by integrating the instantaneous hazard rate over time from zero to the point of interest, accumulating the risk experienced up to that time. From data, it can be estimated, for example by the Nelson-Aalen estimator, which sums the estimated hazard contributions at each event time. The cumulative hazard increases with time as risk accumulates, and it relates to survival through the formula that the survival probability equals the exponential of the negative cumulative hazard.
Source: Collett 2015
How does cumulative hazard relate to survival?
Cumulative hazard relates to survival through the equation that the survival function equals the exponential of the negative cumulative hazard, so as the cumulative hazard rises, survival falls. This relationship links the two central quantities of survival analysis: the cumulative hazard accumulates risk over time, and survival is the resulting probability of not having had the event. Because of this connection, the cumulative hazard can be obtained from the survival function and vice versa, and it is often used in estimation and diagnostics.
Source: Collett 2015
Why is cumulative hazard useful?
Cumulative hazard is useful because it accumulates the risk of an event over time in a way that connects directly to survival and to the shape of the hazard, and it is convenient for estimation and diagnostics. Its estimator behaves well statistically, and plots of the cumulative hazard help assess the form of the hazard, since, for example, a straight line indicates a constant hazard. The cumulative hazard thus serves both as a descriptive measure of accumulated risk and as a tool in survival analysis.
Source: Kalbfleisch & Prentice 2002
How is cumulative hazard used to check model assumptions?
Cumulative hazard plots are used to check survival model assumptions because the shape of the plotted cumulative hazard reveals the form of the hazard: a straight line through the origin indicates a constant hazard, consistent with an exponential model, while curvature indicates a changing hazard. Comparing cumulative hazards across groups can also check the proportional hazards assumption, since proportional hazards imply a constant ratio. These diagnostic uses make cumulative hazard plots a common tool for assessing which survival model is appropriate.
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/cumulative-hazard
- Term code
- HE-EM-SM-010
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