Competing Events and the Meaning of Risk

What Does “Risk” Mean When Events Compete?

Suppose an emulated trial compares two treatment strategies for preventing a nonfatal event, but death can occur first. A person who dies cannot subsequently experience the target event during follow-up. The question is therefore not just how often the target event occurs: it is also how the analysis handles the event that makes it impossible to occur.

The central distinction is between the cause-specific hazard of the target event and its cumulative incidence, the probability of experiencing it by a specified time in the presence of competing events. They describe different features of the event process. Neither should be reported as though it were the other.

Definition: Let $g$ denote a treatment strategy. Under that strategy, let $T^g$ be the time to the first event and $J^g$ identify its type. Let $k$ be the target event and let $c$ denote a competing event (or one of several competing-event types). The cumulative incidence of the target event by time $t$ is $$ F_k^g(t)=P(T^g\le t,\ J^g=k). $$ The cause-specific hazard for event type $j$, among people who have experienced no event by time $t$, is $$ \lambda_j^g(t)=\lim_{\Delta\downarrow 0} \frac{P(t\le T^g<t+\Delta,\ J^g=j\mid T^g\ge t)}{\Delta}, $$ when this limit exists. The event-free survival probability is $S^g(t)=P(T^g>t)$.

For the formulas below, assume event times have continuous distributions, there are no tied event types, and cause-specific hazards are nonnegative and locally integrable. The competing events are actual events in the outcome process, not losses to observation. In a trial protocol, the event types and follow-up horizon must be specified, and the same definitions apply separately under each strategy $g$.

From Hazards to Cumulative Incidence

A cause-specific hazard is conditional on still being event-free at that time. The group at risk under one strategy need not be the same kind of group as the group at risk under another: earlier target or competing events can change who remains. Cumulative incidence instead refers to the probability, at baseline, of having the target event by time $t$, with competing events allowed to occur first.

Write $H_j^g(t)=\int_0^t\lambda_j^g(u)\,du$ for the cumulative cause-specific hazard. If the event types $j$ exhaust all first events, event-free survival is $$ S^g(t)=\exp\left\{-\sum_j H_j^g(t)\right\}. $$ The target-event cumulative incidence is $$ F_k^g(t)=\int_0^t S^g(u)\lambda_k^g(u)\,du. $$ Thus the target hazard contributes to incidence only while a person remains free of every event type.

Result: Under the continuous-time conditions above, $F_k^g(t)=\int_0^t S^g(u)\lambda_k^g(u)\,du$. If the event types are exhaustive, then $\sum_j F_j^g(t)=1-S^g(t)$.

Derivation. Given survival without any event to time $u$, the probability of a type-$j$ event in a short interval of length $\Delta$ is $\lambda_j^g(u)\Delta+o(\Delta)$. Therefore the type-$j$ event density at time $u$ is $S^g(u)\lambda_j^g(u)$. Integrating that density from zero to $t$ gives $F_j^g(t)$.

For the survival formula, the total hazard of any first event is $\lambda_{\mathrm{all}}^g(u)=\sum_j\lambda_j^g(u)$. Survival satisfies $dS^g(u)/du=-S^g(u)\lambda_{\mathrm{all}}^g(u)$ with $S^g(0)=1$. Solving this differential equation gives $S^g(t)=\exp\{-\int_0^t\lambda_{\mathrm{all}}^g(u)\,du\}$. Finally,

\sum_j F_j^g(t) =\int_0^t S^g(u)\sum_j\lambda_j^g(u)\,du =-\int_0^t \frac{dS^g(u)}{du}\,du =1-S^g(t).

The last equality uses the fundamental theorem of calculus; local integrability ensures the cumulative hazards and the displayed integrals are defined. $\square$

Worked Example: Equal Target Hazards, Different Target Risks

Consider two strategies with constant cause-specific hazards per year. Under strategy A, $\lambda_k=0.10$ and $\lambda_c=0.20$. Under strategy B, $\lambda_k=0.10$ and $\lambda_c=0.05$. There are no other event types. Find target-event risk by two years under each strategy.

Under a constant total hazard, $S(t)=\exp\{-(\lambda_k+\lambda_c)t\}$. Consequently, $F_k(t)=\lambda_k(\lambda_k+\lambda_c)^{-1}[1-\exp\{-(\lambda_k+\lambda_c)t\}]$.

For A, the total hazard is $0.30$, so $F_k^A(2)=(0.10/0.30)(1-e^{-0.60})=(1/3)(0.45119)=0.15040$. For B, the total hazard is $0.15$, so $F_k^B(2)=(0.10/0.15)(1-e^{-0.30})=(2/3)(0.25918)=0.17279$. The cause-specific target hazards are equal, but strategy A has lower target-event risk by two years: $0.15040-0.17279=-0.02239$. Its higher competing-event hazard removes more people from the event-free population before they can experience the target event.

Interpreting Contrasts Between Strategies

For two strategies $g_1$ and $g_0$, a cause-specific hazard ratio at time $t$, when the denominator is positive, is $\lambda_k^{g_1}(t)/\lambda_k^{g_0}(t)$. It compares instantaneous target-event rates among the people still event-free under each strategy. It is not, by itself, a contrast in the probability of the target event in the original trial population.

A cumulative-incidence contrast at a prespecified horizon $\tau$, such as $F_k^{g_1}(\tau)-F_k^{g_0}(\tau)$, directly compares target-event risks under the two strategies. A risk ratio can also be used when the reference risk is nonzero. These risks incorporate both the target-event hazard and the competing-event process under each strategy.

When an observational emulation is intended to estimate these strategy-specific risks, the causal interpretation requires assumptions appropriate to the design: well-defined strategies and consistency, exchangeability (randomization in a randomized trial or adequate control of confounding in an observational study), positivity, and adequate outcome measurement. If loss to follow-up occurs, its handling requires additional assumptions; loss to follow-up is not the same thing as a competing event. Competing events themselves should be recorded as outcomes in the event process, not treated as missing follow-up.

Worked Example: A Strategy Can Lower Target Risk but Raise Any-Event Risk

Compare two strategies over three years. Under $g_0$, the constant target and competing hazards are $0.08$ and $0.12$ per year. Under $g_1$, they are $0.06$ and $0.18$ per year. Calculate the target risk and the risk of any first event under each strategy.

For $g_0$, the total hazard is $0.20$ per year. Thus $1-S^{g_0}(3)=1-e^{-0.60}=0.45119$. The target fraction of the total hazard is $0.08/0.20=0.40$, so $F_k^{g_0}(3)=0.40(0.45119)=0.18048$. The competing-event incidence is $0.12/0.20$ times $0.45119$, or $0.27071$. These add to $0.45119$.

For $g_1$, the total hazard is $0.24$ per year. Hence $1-S^{g_1}(3)=1-e^{-0.72}=0.51325$. The target fraction is $0.06/0.24=0.25$, giving $F_k^{g_1}(3)=0.25(0.51325)=0.12831$. The competing-event incidence is $0.18/0.24$ times $0.51325$, or $0.38494$; the two incidences sum to $0.51325$.

The target risk difference is $0.12831-0.18048=-0.05216$, while the risk difference for any first event is $0.51325-0.45119=0.06206$. These are not contradictory: the target risk falls while the competing-event risk rises. A composite endpoint of “target event or competing event” answers a different question from the target-event risk.

Why Treating a Competing Event as Censoring Changes the Question

If a competing event is simply censored in a Kaplan–Meier calculation, the calculation acts as though those people could continue to contribute event-free time for the target event. Under suitable conditions, that calculation corresponds to a “net” quantity built from the target cause-specific hazard alone, $1-\exp\{-H_k^g(t)\}$. It is not the actual-world cumulative incidence $F_k^g(t)$ when competing events occur.

One can define a hypothetical question in which competing events are eliminated, but that is a different estimand and needs a clearly specified intervention and additional causal assumptions. Merely censoring competing events does not establish that hypothetical interpretation. For the actual risk of a target event in the presence of competing events, retain those events in the survival term $S^g$ and use the cumulative-incidence formula.

Result: In a model with target hazard $\lambda_k^g(t)$ and total competing-event hazard $\lambda_c^g(t)$, removing the competing hazards while holding the target hazard fixed changes the target incidence from $$ F_k^g(t)=\int_0^t e^{-H_k^g(u)-H_c^g(u)}\lambda_k^g(u)\,du $$ to the net quantity $$ F_{k,\mathrm{net}}^g(t)=\int_0^t e^{-H_k^g(u)}\lambda_k^g(u)\,du. $$ The two are equal at horizon $t$ if and only if $H_c^g(u)=0$ for $\lambda_k^g(u)\,du$-almost every $u\in[0,t]$.

Proof. Subtract the actual cumulative incidence from the net quantity:

F_{k,\mathrm{net}}^g(t)-F_k^g(t) =\int_0^t e^{-H_k^g(u)} \left(1-e^{-H_c^g(u)}\right)\lambda_k^g(u)\,du.

Every factor in the integrand is nonnegative, and $e^{-H_k^g(u)}>0$ when the cumulative hazard is finite. The integral is zero exactly when $1-e^{-H_c^g(u)}=0$ for $\lambda_k^g(u)\,du$-almost every $u$, which is equivalent to $H_c^g(u)=0$ for $\lambda_k^g(u)\,du$-almost every $u$. This proves the condition. $\square$

In particular, it is not enough that the target and competing hazards are positive at disjoint times. A competing event can occur before a later period of target-event hazard and reduce the number who survive to that period.

Worked Example: Disjoint Hazard Periods Still Change Target Risk

Suppose the competing-event hazard is $\log 2$ per year from time 0 to time 1 and zero afterward. The target-event hazard is zero through time 1 and $0.20$ per year from time 1 to time 2. There are no other event types. The hazards are never positive at the same time.

By time 1, competing-event survival is $e^{-\log 2}=0.5$. Between times 1 and 2, those who remain event-free have target-event survival $e^{-0.20}$. Therefore the actual target cumulative incidence by time 2 is $0.5(1-e^{-0.20})=0.5(0.18127)=0.09063$. If competing events were eliminated while the target hazard stayed fixed, the net quantity would be $1-e^{-0.20}=0.18127$.

The cumulative competing hazard is already $\log 2$ when the target hazard begins. Thus it is positive on a set with positive target-hazard measure, and the equality condition does not hold. Disjoint timing of positive hazards does not imply equal risks.

Diagnostics, Comparisons, and Limitations

Before interpreting a contrast, check that the protocol distinguishes the target event, each competing event, and loss to follow-up; that event times and event types are measured consistently; and that the horizon is the same across strategies. Inspect strategy-specific numbers and timing of both target and competing events. A large treatment effect on a competing event can materially alter target-event risk even if the target cause-specific hazards are similar.

The main alternatives answer distinct questions:

Cumulative incidence gives the actual probability of the specified target event by a horizon, accounting for competing events.
Cause-specific hazards describe instantaneous event rates among those still event-free. They can help characterize event processes but do not alone give the target risk contrast.
A composite endpoint counts any of a set of events as the outcome. It can be useful when that combined outcome is clinically meaningful, but it is not interchangeable with target-event incidence.
A net-risk calculation that removes competing hazards describes a hypothetical or hazard-based quantity, not the observed-world target risk without further interpretation and assumptions.

A cause-specific hazard contrast is also not automatically a direct biological effect among a common, fixed population: the event-free populations being compared can differ because the strategies affect earlier events. Likewise, a cumulative-incidence contrast combines effects through the target and competing event processes; it does not isolate a pathway-specific effect. Estimates may also be unstable when few participants remain event-free late in follow-up, and risks beyond the observed support require extrapolation.

Common pitfall: “We censored deaths, so we estimated the probability of the nonfatal event as if everyone could remain at risk.” This is not justified by censoring alone. The correction is to decide which question is intended: use cumulative incidence for actual target-event risk with death as a competing event, or explicitly define a hypothetical estimand that eliminates death and state the extra assumptions it requires.

Check Your Understanding

Answer each question, then check the solutions.

In one sentence, distinguish the target-event cause-specific hazard from the target-event cumulative incidence.
If two strategies have the same target-event hazard but different competing-event hazards, must their target-event risks be equal? Explain.
When can the net quantity and actual target cumulative incidence be equal at a fixed horizon?
Why is a composite endpoint not interchangeable with the cumulative incidence of one component?

Solutions. (1) The cause-specific hazard is the instantaneous target-event rate among those still event-free, whereas cumulative incidence is the baseline probability of having the target event by the horizon in the presence of competing events. (2) No; different competing hazards change event-free survival and therefore the integral of the target hazard over the population still at risk. (3) Exactly when the cumulative competing hazard is zero for almost every time with respect to the target-hazard measure on that horizon. (4) The composite counts any included event, while the component-specific cumulative incidence counts only the specified target event; the example with changing target and competing hazards shows these risks can move in different directions.