Competing Risks

Competing Risks in Economic Models, with Cause-Specific Hazards, Cumulative Incidence and Transition Probabilities

Competing risks arise in a health economic model because a patient who dies of another cause can no longer have the stroke, relapse or fracture that a treatment is meant to prevent, so the value of prevention depends on how many patients live long enough to have the event. The issue matters most in older or frail populations, where death from other causes is common, and in any model state with two or more exits. The two basic quantities are the cause-specific hazard and the cumulative incidence function, one minus a Kaplan-Meier estimate overstates risk, and each regression approach answers a different question. Competing risks also enter cohort state-transition models and discrete event simulation, and in an illustrative stroke-prevention example ignoring other-cause death inflates the strokes avoided by 38%.

Why one event can rule out another

Austin, Lee and Fine define a competing risk as an event whose occurrence precludes the primary event of interest; in a study of cardiovascular death, death from a non-cardiovascular cause is a competing risk. Andersen and colleagues describe competing risks as the rule rather than the exception in epidemiological follow-up studies, and Koller and colleagues, reviewing 50 clinical studies in populations prone to competing events, found competing risks issues in 70% of them.

In decision models the same structure appears whenever a health state has more than one way out. Jones, Epstein and García-Mochón define competing risks in a multistate Markov model as a state from which a person can move to two or more other states. The progression-free state of an oncology model, with exits to progression and to death without progression, is the standard case, and Williams and colleagues treat these two transitions as competing risks in their multi-state cost-effectiveness tutorial.

A tempting idea is a "net" probability for each cause acting alone. Tsiatis showed that observed data on the first event and its cause do not identify these net probabilities without assuming that the latent event times are independent, an assumption the data cannot verify, and Austin and colleagues note that biology often suggests some dependence. Analyses therefore work with quantities that the observed data do identify.

Two quantities, the cause-specific hazard and the cumulative incidence

Each competing event has a cause-specific hazard, the instantaneous rate of that event among people who have not yet had any of the events. The probability of having had a given event by time $t$ is its cumulative incidence, which depends on every cause-specific hazard because competing events shrink the group still at risk. Andersen and colleagues give the relationship for two causes as follows.

$$S(t) = \exp\{-H_1(t) - H_2(t)\}, \qquad F_1(t) = \int_0^t S(u)\, h_1(u)\, du$$

where $h_1(u)$ and $h_2(u)$ are the cause-specific hazards of the event of interest and the competing event at time $u$, $H_1(t)$ and $H_2(t)$ are their cumulative hazards, $S(t)$ is the probability of being free of both events at time $t$, and $F_1(t)$ is the cumulative incidence of the event of interest. When both hazards are constant this becomes the following.

$$F_1(t) = \frac{h_1}{h_1 + h_2}\left[1 - e^{-(h_1 + h_2)t}\right]$$

where $h_1$ and $h_2$ are the constant cause-specific hazards and $t$ is time. Unlike one minus a single-event survival curve, $F_1(t)$ does not approach 1; with constant hazards it levels off at $h_1/(h_1 + h_2)$. Andersen and colleagues call the loss of the one-to-one link between rate and risk the key feature of competing risks, since the hazard rate of an event no longer fixes its risk on its own.

Why one minus Kaplan-Meier overstates risk

A common shortcut treats the competing event as censoring and reports one minus the Kaplan-Meier estimator as the risk of the event of interest. Austin and colleagues state that this biases incidence upward whether or not the competing events are independent, because it estimates the risk in a hypothetical population in which the competing event cannot happen. In their heart failure case study, the five-year incidence of cardiovascular death was 43.0% from one minus Kaplan-Meier and 36.8% from the cumulative incidence function.

The cumulative incidences of all competing events must add up to the probability of having had any event, which gives a quick check. In stem cell transplantation data analysed by Andersen and colleagues, naive Kaplan-Meier estimates of the five-year probabilities of relapse and of death without relapse were 0.515 and 0.569, summing to more than 1 for two mutually exclusive events; the cumulative incidence estimates were 0.316 and 0.475. Nelson-Aalen and Cox methods with competing events censored remain valid for cause-specific hazards; the error lies in reading the result as a probability.

Two regression models, two questions

With covariates or treatment effects, two families of models are available. The cause-specific hazard model is an ordinary Cox model for one event type with competing events censored, and its hazard ratio describes the rate among people still free of all events. The subdistribution hazard model introduced by Fine and Gray keeps people who have had a competing event in the risk set, so its hazard ratio describes the effect on cumulative incidence. Austin and colleagues suggest cause-specific models for questions of aetiology and subdistribution models for predicting absolute risk; Lau, Cole and Gange add resource allocation as a use of the subdistribution measure.

The two can disagree. In Austin and colleagues' heart failure data, cancer had a cause-specific hazard ratio of 0.96 for cardiac death but a subdistribution hazard ratio of 0.82. A decision model that tracks people through health states needs cause-specific hazards, since they are rates among people still in the state and the model turns them into cumulative incidence as it runs. A subdistribution hazard is not such a rate, because its risk set includes people who have already left by the competing exit.

Competing risks in cohort state-transition models

A cohort model needs one transition probability per exit per cycle. Jones and colleagues note that the simple conversion $p = 1 - e^{-rt}$, applied to each rate separately, is always wrong when there are competing risks, because each exit is converted as if it were the only one. For a state whose exits lead to states that are not left again within the cycle, they give the following.

$$p_{k} = \frac{q_{k}}{\sum_{j} q_{j}}\left[1 - e^{-t \sum_{j} q_{j}}\right]$$

where $p_k$ is the probability of leaving by exit $k$ during a cycle of length $t$, $q_k$ is the transition rate for exit $k$, and the sums run over all exits from the state. In their example with two exit rates of 1 and a cycle of 1, the simple formula gives 0.632 for each exit and a staying probability of minus 0.264, while the formula above gives 0.432 for each exit and 0.135 for staying. Where onward transitions within a cycle are possible, they derive the probabilities from the full rate matrix.

Evidence often arrives as probabilities over a period rather than as rates. Gidwani and Russell show that converting such probabilities one exit at a time fails with three possible transitions. In their heart transplant example it projected 23 people still waiting at three years against an observed 9. The ISPOR-SMDM report on state-transition modelling recommends that, when the cause of death matters or causes differ in cost, competing causes be modelled in an unbiased way, for example by estimating the probability of death first and then a conditional distribution of causes. Background mortality competes with disease-specific death in the same way; combining the two without double counting is covered on the all-cause mortality page.

Competing risks in discrete event simulation

Discrete event simulation samples event times directly, so competing risks are handled by how the next event is chosen. The ISPOR-SMDM report on discrete event simulation describes sampling a separate time for each possible event and taking the earliest, or sampling one time to the next event and then its type. The first is easier to fit to observed data; the second estimates the timing of competing events jointly, which the report prefers where feasible. Degeling and colleagues compared four implementations and recommend sampling the time to event first and the event type second, although in their colorectal cancer case study the choice did not change health economic outcomes substantially.

Worked example of a stroke-prevention drug in an older cohort

All numbers are illustrative. A cohort of older adults has a constant stroke hazard of 0.04 a year and a constant hazard of death from other causes of 0.08 a year. A new drug multiplies the stroke hazard by 0.75, giving 0.03, and does not change other-cause death. Each first stroke costs GBP 20,000, the horizon is 10 years and only first strokes are counted.

1. The naive estimate. Treating other-cause death as censoring gives a 10-year stroke risk of 1 − exp(−0.4) = 0.32968 without the drug and 1 − exp(−0.3) = 0.25918 with it, so 0.07050 strokes are avoided per person.

2. The cumulative incidence. Without the drug the total hazard is 0.12, so the stroke risk is (0.04/0.12) × (1 − exp(−1.2)) = 0.33333 × 0.69881 = 0.23294. The other-cause death risk is 0.66667 × 0.69881 = 0.46587, and 0.30119 remain event free; the three sum to 1. With the drug the total hazard is 0.11 and the stroke risk is (0.03/0.11) × (1 − exp(−1.1)) = 0.27273 × 0.66713 = 0.18194. Strokes avoided are 0.23294 − 0.18194 = 0.05100 per person.

3. The cost offsets. The naive estimate implies savings of 0.07050 × 20,000, or GBP 1,410 per person, against 0.05100 × 20,000, or GBP 1,020, from the cumulative incidence. The naive figure overstates strokes avoided and savings by 38% (0.07050 / 0.05100 = 1.38).

4. The annual transition probabilities. Without the drug the probability of any event in a year is 1 − exp(−0.12) = 0.11308, split one third to stroke (0.03769) and two thirds to other-cause death (0.07539), leaving 0.88692 event free. Ten annual cycles reproduce the 10-year stroke risk of 0.23294 exactly. Converting each rate separately gives 0.03921 and 0.07688, and the same run gives 0.23943, about 2.8% too high.

5. Interpretation. The large error comes from the input, not from the cycle conversion, because a stroke risk from one minus Kaplan-Meier assumes patients cannot die of other causes. The hazard ratio of 0.75 also does not carry over to risk, since the ratio of cumulative incidences is 0.18194 / 0.23294 = 0.78. People spared a stroke remain at risk of other-cause death, whose 10-year risk rises from 0.46587 to 0.48519 with the drug, so life years and QALYs gained are smaller than a single-risk analysis suggests.

Competing risks and neighbouring ideas

Censoring and competing events differ in what happens next. 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. The cumulative incidence page covers risk estimation more generally, and survival analysis places competing risks among other survival methods.

Misreadings that distort model inputs

Most errors with competing risks overstate the risk of the event of interest. They therefore overstate the benefit of preventing it.

One minus Kaplan-Meier with competing events censored overstates risk, more so as the competing hazard grows; checking that event-specific risks add up to the all-event risk exposes it. Applying a cause-specific hazard ratio directly to a cumulative incidence assumes a link that Andersen and colleagues show is lost. Converting each exit rate with the single-event formula overstates exits, a smaller error with short cycles and low rates, and can give negative staying probabilities when rates are high. Estimating net risks, as if other causes could be removed, rests on an independence assumption the data cannot test.

Sources

• Andersen PK, Geskus RB, de Witte T, Putter H. Competing risks in epidemiology: possibilities and pitfalls. International Journal of Epidemiology. 2012;41(3):861-870. https://doi.org/10.1093/ije/dyr213
• Austin PC, Lee DS, Fine JP. Introduction to the analysis of survival data in the presence of competing risks. Circulation. 2016;133(6):601-609. https://doi.org/10.1161/CIRCULATIONAHA.115.017719
• Degeling K, Koffijberg H, Franken MD, Koopman M, IJzerman MJ. Comparing strategies for modeling competing risks in discrete-event simulations: a simulation study and illustration in colorectal cancer. Medical Decision Making. 2019;39(1):57-73. https://doi.org/10.1177/0272989X18814770
• Gidwani R, Russell LB. Estimating transition probabilities from published evidence: a tutorial for decision modelers. PharmacoEconomics. 2020;38(11):1153-1164. https://doi.org/10.1007/s40273-020-00937-z
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