Concept Architecture
Concept
Theoretically, Competing Risks arise when an individual is simultaneously at risk of experiencing multiple mutually exclusive events, where the occurrence of one event prevents the occurrence of the others. In health economics, competing risks most commonly occur when death from causes unrelated to the disease of interest precludes the occurrence of disease-specific outcomes. Explicit modelling of competing risks is essential for obtaining unbiased estimates of event probabilities, survival and long-term health outcomes.
Mathematically, Competing Risks are represented within a multiple-failure survival framework in which each event type has its own cause-specific hazard or subdistribution hazard. The overall hazard is the sum of the cause-specific hazards, while cumulative incidence functions estimate the probability of experiencing each competing event over time. This framework ensures that competing events are correctly accounted for when estimating event probabilities.
In practice, Competing Risks are incorporated into health economic models using cause-specific hazards or cumulative incidence functions derived from clinical trials, observational studies or disease registries. They are routinely applied in state-transition models, microsimulation and individual patient simulation to model disease progression, mortality and other mutually exclusive clinical events. Failure to account for competing risks can lead to overestimation of disease incidence, survival benefits and cost effectiveness.
Purpose
Used to model mutually exclusive clinical events that prevent the occurrence of one another, ensuring accurate estimation of survival, event probabilities and long-term cost effectiveness.
Mathematical Formulae
Primary Formula
Overall hazard function:
h(t) = ????? h?(t)
where:
- h(t) is the overall hazard at time t
- h?(t) is the cause-specific hazard for event k
- K is the number of competing event types.
Supporting Formulae
Overall survival function:
S(t) = exp(???? ????? h?(u) du)
Cumulative incidence function for event k:
F?(t) = ??? S(u) h?(u) du
Related Mathematical Methods
- Survival analysis
- Cause-specific hazard modelling
- Cumulative incidence analysis
- Fine and Gray subdistribution hazard model
- Parametric survival modelling
- State-transition modelling
- Individual patient simulation
Example
A lifetime model for patients with advanced chronic kidney disease considers two competing causes of death:
- Death due to kidney disease: annual hazard = 0.08
- Death due to unrelated causes: annual hazard = 0.03
The overall mortality hazard is:
h(t) = 0.08 + 0.03 = 0.11
The corresponding annual survival probability is:
S(1) = e??�?? = 0.896
or 89.6%.
The model separately estimates the cumulative probability of each cause of death while ensuring that a patient who dies from one cause cannot subsequently experience the other event.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| SUM | =SUM(B2:C2) | Calculate the overall hazard from competing cause-specific hazards. |
| EXP | =EXP(-D2) | Calculate survival probability from the overall hazard. |
| IF | =IF(RAND()<Cause1Prob,""Cause 1"",IF(RAND()<Cause2Prob,""Cause 2"",""Alive"")) | Simulate mutually exclusive competing events in patient-level models. |
| XLOOKUP | =XLOOKUP(Stage,HazardTable[Stage],HazardTable[Cause1Hazard]) | Retrieve cause-specific hazards by disease stage. |
| INDEX | =INDEX(HazardTable[BackgroundHazard],MATCH(Age,AgeTable[Age],0)) | Retrieve background mortality hazards for competing-risk calculations. |
VBA (Optional)
Automate simulation of competing clinical events by sampling mutually exclusive outcomes from multiple cause-specific hazard functions during each model cycle.
Sources
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press; 2006.
- Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes. 4th ed. Oxford University Press; 2015.
- Kalbfleisch JD, Prentice RL. The Statistical Analysis of Failure Time Data. 2nd ed. Wiley; 2002.
- Fine JP, Gray RJ. A proportional hazards model for the subdistribution of a competing risk. Journal of the American Statistical Association. 1999;94(446):496?509.
- National Institute for Health and Care Excellence (NICE). Health Technology Evaluation Manual. Latest edition.
Related Concepts (2)
Library
Publications
1
Decision Modelling for Health Economic Evaluation — Briggs, Claxton & Sculpher, 1st Edition ed., 2006 (Oxford University Press)
Foundational textbook on decision-analytic modelling for economic evaluation, covering decision trees, Markov models, handling parameter and structural uncertainty, probabilistic sensitivity analysis, and value of information. Volume 1 in the Handbooks in Health Economic Evaluation series.
BookView source →
Frequently Asked Questions (6)
What are competing risks?
A situation where a person is at risk of more than one type of event, so one event's occurrence prevents another from ever happening.
Source: Kalbfleisch & Prentice 2002
What is an example of competing risks?
Competing risks occur when a person faces several possible events and the first to happen rules out the others. A patient awaiting a transplant may receive the organ, die while waiting, or become too unwell to proceed, and whichever comes first ends the possibility of the rest. Treating one such event in isolation, as if the others could not intervene, overstates its chance. Recognising that the events compete is what makes their analysis different from a single-outcome one. Putter and colleagues (2007) describe competing risks.
Source: Putter et al. 2007
Why do competing risks require special analysis?
Competing risks require special analysis because standard survival methods, which treat other events as censoring, can mislead when those events are competing rather than independent censoring. Treating a competing event as if the person could still experience the event of interest overstates the latter's risk, since in reality the competing event has removed them. Proper methods, such as the cumulative incidence function, account for competing events correctly, giving unbiased estimates of the probability of each event in the presence of the others.
Source: Kalbfleisch & Prentice 2002
How are competing risks handled statistically?
Competing risks are handled using methods that account for the fact that one event precludes others, notably the cumulative incidence function, which estimates the probability of each event type over time in the presence of the competing events, rather than treating them as independent censoring. Cause-specific hazard models and subdistribution hazard models are also used. These methods give correct estimates of the risk of each event, avoiding the overestimation that arises from naively censoring competing events in standard survival analysis.
Source: Kalbfleisch & Prentice 2002
How do competing risks arise in health economic models?
In health economic models, competing risks arise when patients face several mutually exclusive events, such as death from the disease and death from other causes, or different clinical events that preclude one another. The model must ensure that experiencing one event removes the patient from risk of the others, so that probabilities are not double-counted. Representing competing events correctly, so that their combined probability is handled consistently, is important for the model to estimate outcomes accurately.
Source: Kalbfleisch & Prentice 2002
How do competing risks relate to cause-specific mortality?
Competing risks relate closely to cause-specific mortality, since deaths from different causes are competing events: a person can die of only one cause, so death from one precludes the others. Cause-specific mortality rates are the cause-specific hazards in a competing-risks framework, and combining them requires competing-risks methods to give the probability of dying from each cause. Recognising deaths by cause as competing risks ensures that overall and cause-specific mortality are handled consistently rather than as independent processes.
Source: Chiang 1984
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 9 Oct 2025
Content version: 1.0.0
Canonical Identity
- Persistent URI
- https://healtheconomics.wiki/concept/competing-risks
- Term code
- HE-EM-MP-006
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