Draft for review · v0.1.0HealthEconomics.wiki · Interactive demonstrator
A patient who dies of another cause can no longer have the stroke, relapse or fracture a treatment is meant to prevent. Change the hazards below and see how ignoring that fact inflates risk, overstates the benefit of prevention and breaks transition probabilities.
Competing risks are events that each end a patient's time at risk, such as cardiovascular and other-cause death, so the first to occur rules out the rest. The explorer follows the chain a model builder works through.
All numbers are illustrative. The first scenario is the worked example on the concept page; the others vary one feature so you can see what drives the error.
The shaded areas show where the cohort has gone by each year. The dashed line is one minus Kaplan-Meier with the competing event treated as censoring: the risk in a hypothetical population where the competing event cannot happen.
| Year | 1 − KM, Stroke | CIF, Stroke | CIF, Other-cause death | Event free | Sum of three |
|---|
Selected cohort, rounded to whole people.
The same hazard ratio, the same cost per event. Only the way risk is calculated differs.
| Quantity at the horizon | 1 − Kaplan-Meier (naive) | Cumulative incidence |
|---|
A cohort state-transition model needs one probability per exit per cycle. This uses the hazards without treatment and the time horizon above.
Correct for competing exits
Separate conversion, one exit at a time
The first formula applies to a state whose exits lead to states not left again within the cycle (Jones, Epstein and García-Mochón).
| Per cycle | Correct | Separate |
|---|
Each statement describes a choice an analyst might make. Decide whether it is sound. Most errors with competing risks overstate the risk of the event of interest, and so overstate the benefit of preventing it.
Summarised from the published concept page. Formulas assume constant hazards unless stated.
The cause-specific hazard is the instantaneous rate of an event among people who have not yet had any of the events. The cumulative incidence is the probability of having had that event by time t, and depends on every hazard.
Look for: a curve that levels off at h₁/(h₁+h₂) rather than approaching 1.
Censoring the competing event estimates risk in a population where it cannot happen. This biases incidence upward whether or not the events are independent. In a heart failure case study, five-year cardiovascular death was 43.0% by one minus Kaplan-Meier and 36.8% by cumulative incidence.
Look for: event-specific risks that add up to more than the all-event risk, or even more than 1.
A censored patient could still have the event later, which is why independent censoring can be assumed. A patient who has had a competing event cannot.
Look for: whether the person can still have the event after leaving observation.
A cause-specific hazard model (Cox, competing events censored) describes the rate among people still free of all events. The Fine and Gray subdistribution model keeps people with a competing event in the risk set and describes the effect on cumulative incidence. In heart failure data, cancer had a cause-specific hazard ratio of 0.96 for cardiac death but a subdistribution hazard ratio of 0.82.
Look for: a decision model that tracks people through states needs cause-specific hazards.
Evidence often arrives as probabilities rather than rates. Converting them one exit at a time fails with three transitions: in a heart transplant example it projected 23 people still waiting at three years against an observed 9. The ISPOR-SMDM report suggests estimating the probability of death first and then a conditional distribution of causes, when cause of death matters.
Look for: exit probabilities derived jointly, not one at a time.
Either sample a time for each possible event and take the earliest, or sample one time to the next event and then its type. The first is easier to fit to data; the second estimates timing jointly and is preferred by the ISPOR-SMDM report where feasible. Degeling and colleagues recommend time first, type second, though in their case study the choice did not change outcomes substantially.
Look for: how the next event, and its type, are chosen.
A hazard rate no longer fixes risk on its own. The probability of each event depends on every competing hazard, and the probabilities of all mutually exclusive events must add up to the probability of having had any event.
Content source: Baines, D. (2026). "Competing Risks." Health Economics Concepts. London: Intelligent Parameters Ltd. Version 1.0.2. https://healtheconomics.wiki/concept/competing-risks (read 12 October 2026). The concept page is published under CC BY 4.0. The page's trust record stated that this version had not yet been verified.
References cited on the concept page and used in this demonstrator: