Common Mistakes in Competing-Risk Causal Analysis

Why Causal Competing-Risk Analyses Are Easy to Misread

A competing event changes more than the event label: once it occurs, the event of interest can no longer occur first. That feature creates several tempting but incorrect shortcuts. Censoring competing events may silently change the question; comparing people who remain event-free may compare selected groups; and a hazard ratio may be mistaken for an absolute change in risk. Each error can produce a number that is mathematically computable but does not answer the causal question being asked.

The starting point is to specify the outcome and the intervention. Let \(A\in\{0,1\}\) denote treatment, \(T^a\) the time to the first event under treatment level \(a\), and \(J^a\) its cause. Cause \(k\) is the event of interest; other causes can prevent it. The treatment-policy cumulative incidence is the probability that cause \(k\) has occurred by time \(t\) under treatment \(a\), with competing events allowed to occur naturally.

For a continuous-time model with absolutely continuous cause-specific hazards \(\lambda_j^a(u)\), the event-free probability is \(S^a(u)=\exp\{-\int_0^u\sum_j\lambda_j^a(v)\,dv\}\). The cumulative incidence is obtained by accumulating the cause-\(k\) event hazard among people who have not yet had any event:

This formula assumes absolutely continuous cause-specific hazards, so event probability accumulates through a hazard density over time. If event-time jumps are allowed, the corresponding measure-based expression is \(F_k^a(t)=\int_0^t S^a(u-)\,d\Lambda_k^a(u)\), where \(\Lambda_k^a\) is the cause-specific cumulative hazard measure and \(S^a(u-)\) is the probability of remaining event-free just before \(u\). The left limit matters at jumps: people experiencing an event at \(u\) must be in the risk set immediately before that event.

Mistake 1: Censoring Competing Events Without Defining the Intervention

A common analysis treats each competing event as if it were ordinary right-censoring and then uses a Kaplan–Meier-type calculation for the event of interest. That calculation does not generally estimate the treatment-policy cumulative incidence. Instead, it removes competing events from the calculation of event-free survival, producing a net-risk quantity associated with a hypothetical world in which competing causes have been eliminated while the cause-specific hazard for the event of interest is held fixed. In the continuous model, this quantity is \(1-\exp\{-\int_0^t\lambda_k^a(u)\,du\}\).

That hypothetical quantity is not automatically the risk under a real intervention. Such an interpretation needs a well-defined intervention that eliminates or prevents the competing event, together with assumptions that support the intervention’s effect on the event of interest. For example, it may be implausible that preventing a competing event leaves the target-event process unchanged. A hypothetical calculation also needs appropriate control of confounding and censoring, positivity for the intervention being considered, and consistency between the intervention and the observed data. Merely relabeling a competing event as “censored” establishes none of these conditions.

A practical diagnostic is to ask what the reported estimand says happens to competing events. If the scientific question concerns risk under usual treatment assignment, competing events remain part of the outcome process and the target is a cumulative incidence. If the question is explicitly hypothetical, state the intervention and explain why its effect on the target event can be identified. A mismatch between the question and the analysis is a warning even when the calculation itself is correct.

Mistake 2: Treating Survivors as Though They Were Comparable

A cause-specific hazard at time \(t\) is the instantaneous rate of the event of interest among people who are still event-free just before \(t\). Under treatment \(a\), that risk set consists of people satisfying \(T^a\geq t\). If treatment changes the frequency of competing events, the people who remain event-free can differ between treatment regimes. A comparison of their hazards therefore compares treatment-specific, selected risk sets, not necessarily the same underlying people.

Randomization makes treatment groups comparable at baseline, but it does not make the groups still event-free at a later time comparable. Remaining event-free is affected by treatment and by prognosis. Conditioning on it can induce an association between treatment and prognostic characteristics, even when they were independent at baseline. A hazard ratio can still be a useful description of the ratio of the two regime-specific cause-specific hazards; it is not, without further assumptions and a carefully stated target, an absolute causal effect for a common population.

When a question concerns an effect among people who would survive under either treatment, it requires a principal-stratum estimand involving jointly defined potential survival outcomes. Membership in that stratum is not observed for any individual, so it is not identified by simply restricting each arm to its observed survivors. Other estimands, such as treatment-policy risks from baseline, avoid that particular survivor restriction and should be reported when they match the scientific question.

Mistake 3: Reading a Hazard Ratio as an Absolute Effect

A hazard ratio is a relative comparison of instantaneous rates among people still event-free at each time. It is not a risk difference, a risk ratio, or a fixed percentage change in the probability of experiencing the event by a specified time. Even if proportional cause-specific hazards hold, absolute risks depend on the baseline hazards, follow-up time, and hazards of all competing causes.

Checks That Prevent These Errors

These checks do not replace identification analysis. For observational data, causal interpretations require assumptions such as consistency, conditional exchangeability given measured covariates, and positivity for the treatment regimes being compared. Censoring also requires suitable assumptions or methods. The earlier material on causal contrasts, hypothetical competing-event elimination, and positivity provides tools for defining and estimating such quantities; the essential safeguard here is not to claim an estimand that the procedure and assumptions do not support.

A final pitfall is to assume that a statistically precise estimate must answer the intended causal question. Precision concerns sampling uncertainty under an analysis; it cannot repair a mismatch between the estimand and the calculation, selection into a risk set, or unsupported assumptions about eliminating competing events. Interpret every estimate first by asking: what population, intervention, event process, and time horizon does this number describe?