Hypothetical Effects Under Competing-Event Elimination

Why Specify a Competing-Event Intervention?

Suppose a study follows patients for relapse, while death before relapse makes a later relapse impossible. A treatment can affect both events. One question concerns relapse risk under the care that patients actually receive. A different question asks what relapse risk would be if a specified intervention prevented death first. The second is a hypothetical causal question: its answer depends on what preventing death means and what happens to patients who would otherwise die.

The intervention is not merely a mathematical instruction to “remove” the competing event. Keeping a patient alive could involve different procedures, each of which might also affect relapse. A causal contrast under competing-event elimination must therefore state the intervention whose consequences are being imagined. Without that specification, the hypothetical outcome—and hence its risk—is ambiguous.

Definition: Let event 1 be the event of interest and event C be a competing event. Let $a$ denote a treatment level, and let $\mathcal{I}_C$ be a specified intervention that prevents event C through a stated time horizon. Write $T_1^{a,\mathcal{I}_C}$ for the counterfactual time to event 1 under treatment $a$ and intervention $\mathcal{I}_C$. The hypothetical event-1 risk by time $\tau$ is $$ R_1^{a,\mathcal{I}_C}(\tau) = P\left(T_1^{a,\mathcal{I}_C}\leq \tau\right). $$ A hypothetical risk difference comparing treatment levels $a=1$ and $a=0$, under the same intervention, is $$ RD_1^{\mathcal{I}_C}(\tau) = R_1^{1,\mathcal{I}_C}(\tau)-R_1^{0,\mathcal{I}_C}(\tau). $$

The superscript $\mathcal{I}_C$ matters: the risk is defined under that particular intervention, not under an unspecified world in which the competing event “does not happen.” The time horizon $\tau$, event definition, treatment strategies, and prevention strategy are also parts of the estimand. A negative risk difference means lower event-1 risk by $\tau$ under treatment 1 than under treatment 0, when both are paired with the stated competing-event intervention.

What Does “Prevent the Competing Event” Mean?

An intervention must specify what is done, to whom, and when. For example, “prevent death” might mean providing a particular rescue treatment at the point when death would otherwise occur. It might instead mean replacing a fatal disease process with a different, nonfatal process. Those strategies need not have the same effect on the event of interest. If a rescue treatment also changes relapse, the resulting hypothetical relapse risk includes that effect unless the intervention is explicitly defined otherwise.

This is why elimination is not equivalent to selecting people who would survive. It is an intervention-defined counterfactual outcome for a target population. Nor does it mean censoring people when the competing event occurs and treating that censoring as ordinary loss to follow-up. Under the hypothetical intervention, people who would have experienced the competing event instead remain subject to the intervention’s consequences and may later experience event 1.

Worked Example: A Treatment Contrast Under a Stated Prevention Strategy

Consider a fully hypothetical population of 1,000 patients and a two-year horizon. Event 1 is cancer relapse, and event C is death before relapse. Suppose the intervention $\mathcal{I}_C$ is defined as a specified rescue strategy that prevents death through two years while otherwise leaving the treatment assignment and follow-up plan unchanged. Imagine that, in the two counterfactual treatment groups under this same intervention, 240 of 1,000 patients would relapse under $a=0$, while 180 of 1,000 would relapse under $a=1$.

The two hypothetical risks are $$ R_1^{0,\mathcal{I}_C}(2)=\frac{240}{1000}=0.240, \qquad R_1^{1,\mathcal{I}_C}(2)=\frac{180}{1000}=0.180. $$

The two-year risk difference is $$ RD_1^{\mathcal{I}_C}(2)=0.180-0.240=-0.060. $$

Thus, in this hypothetical example, the treatment contrast under the specified rescue strategy is a 6.0 percentage-point lower relapse risk by two years for $a=1$ than for $a=0$. The risk ratio is $0.180/0.240=0.750$, or 25% lower risk relative to the risk under $a=0$. These are effects under $\mathcal{I}_C$, not necessarily effects under the care patients would otherwise receive. The numbers are stipulated counterfactual risks for illustration; they are not calculated from observed competing-risk data.

The Intervention Version Changes the Question

A causal estimand needs a sufficiently well-defined intervention: different versions of an intervention cannot simply be assumed to have identical consequences. This issue is especially important when the intervention acts on the competing event itself. Preventing a death by a treatment that has no direct effect on the event of interest is a different hypothetical strategy from preventing death by a treatment that also changes the event-of-interest process.

Worked Example: Two Ways of Preventing the Same Competing Event

Suppose, purely hypothetically, that under treatment $a=0$ and over a fixed horizon of one year, the event-1 risks in the target population would be 0.20 under prevention strategy $\mathcal{I}_C^{A}$ and 0.12 under prevention strategy $\mathcal{I}_C^{B}$. Strategy A prevents the competing event using a rescue procedure that can prolong exposure to a cause of event 1. Strategy B prevents it using a different procedure that also reduces exposure to that cause. Assume both strategies prevent event C for the full year, but they are not otherwise equivalent.

For strategy A, the hypothetical number of event-1 cases per 100 people is $100(0.20)=20$. For strategy B, it is $100(0.12)=12$. The risk difference between the two prevention strategies is $$ R_1^{0,\mathcal{I}_C^{A}}(1)-R_1^{0,\mathcal{I}_C^{B}}(1) =0.20-0.12=0.08. $$

The eight-percentage-point difference shows why “eliminate event C” alone does not define a unique hypothetical risk. It does not establish that either strategy has this effect in real patients; the risks here are hypothetical. To compare treatments under competing-event prevention, the same well-defined strategy must be specified in both treatment arms. Otherwise, the contrast combines a treatment difference with a difference in how the competing event was prevented.

A Discrete-Time Justification

A simple two-period calculation shows how an intervention preventing a competing event can change the event-1 risk. It also makes the key assumption visible. Consider a hypothetical cohort at time zero. In period 1, assume event 1 occurs with probability 0.20 and event C with probability 0.30; the remaining probability, 0.50, corresponds to neither event. In period 2, among those who remain event-free at its start, assume the probability of event 1 is 0.25.

Under the observed competing-event process, the period-1 event-1 risk is 0.20. Only the 0.50 who experienced neither event remain at risk for period 2. Their contribution to the event-1 risk by the end of period 2 is $0.50(0.25)=0.125$. Therefore, $$ P(\text{event 1 by period 2 under the stated process}) =0.20+(0.50)(0.25) =0.325. $$

Now consider an idealized intervention that prevents event C in period 1 and has no other effect on event 1 or on the period-2 event-1 probability. Event 1 still occurs in period 1 with probability 0.20. The remaining 0.80 are now available to experience event 1 in period 2, so the hypothetical risk is $$ P(\text{event 1 by period 2 under }\mathcal{I}_C) =0.20+(0.80)(0.25) =0.400. $$

The difference, $0.400-0.325=0.075$, arises because some people who would have experienced event C can now remain at risk of event 1. This calculation is valid for the stipulated intervention only because it assumes that prevention does not change the period-1 event-1 probability or the period-2 event-1 probability among those event-free at its start. If the prevention strategy changes either probability, the calculation must use the probabilities under that strategy instead. The example is a justification of the intervention-specific risk, not a general recipe for recovering it from observed data.

Key distinction: The event-1 risk under a competing-event-prevention intervention is a counterfactual risk under a specified strategy. It is not, by definition, the observed event-1 cumulative incidence, a cause-specific hazard ratio, or a coefficient from a competing-risk regression model.

Assumptions, Interpretation, and Diagnostics

The definition of an estimand does not by itself show that its value can be estimated from a particular data set. At minimum, interpretation requires a coherent intervention and counterfactual outcome: the prevention strategy must be described well enough to say what would happen under it. If treatment is also contrasted, the two treatment levels must be compared under the same prevention strategy and time horizon.

Consistency: for a person who actually receives a specified treatment and the specified prevention strategy, the observed outcome agrees with the corresponding counterfactual outcome. Vague or multiple versions of “prevention” make this correspondence unclear.
No interference: one person’s treatment or prevention strategy does not alter another person’s outcome, unless the estimand and analysis explicitly account for such effects.
Identification from data: estimating counterfactual risks requires additional conditions connecting observed data to outcomes under the intervention. For example, treatment assignment and the processes governing intervention and event occurrence may need to be adequately controlled or randomized, along with suitable positivity and measurement conditions. The definition alone supplies none of these conditions.

A practical diagnostic is to ask what would happen to a patient at the moment when event C would otherwise occur. If the answer is unspecified, the hypothetical intervention is underspecified. Then ask whether the proposed intervention could change event 1 directly, change later care, or change the population remaining at risk. If it could, those consequences belong to that intervention’s hypothetical risk; they cannot be ignored by calling the intervention “elimination.”

A second diagnostic is to check that the interpretation matches the quantity calculated. A treatment-specific cumulative incidence describes event-1 occurrence under the event processes that actually operate in that treatment condition. A cause-specific hazard concerns the instantaneous event-1 rate among those currently event-free. A Fine-Gray coefficient describes a model parameter for a subdistribution hazard under its model assumptions. None automatically equals the risk contrast under a specified intervention that prevents event C. Translating any of them into that hypothetical causal effect requires further assumptions and justification.

Common Pitfalls and Limitations

Leaving the intervention unspecified. “What if no one died?” is not enough. State what prevents death, for whom, and for how long, including relevant consequences for event 1.
Treating competing events as ordinary censoring. Censoring at event C does not itself enact an intervention that keeps the person alive and at risk. It cannot, by itself, establish the hypothetical risk.
Assuming observed treatment randomization answers the hypothetical question. Randomizing treatment can support comparisons of treatment strategies under the trial’s actual event processes. It does not automatically identify outcomes under an additional intervention that prevents event C.
Changing the prevention strategy across treatment arms. If the prevention versions differ, the resulting contrast does not isolate treatment under a common intervention.
Interpreting the hypothetical effect as a real-world policy effect. A strategy that can prevent the competing event may be unavailable, harmful, or infeasible. A well-defined hypothetical estimand can still clarify a scientific question, but its practical relevance depends on whether the intervention is meaningful and its consequences credible.

The principal limitation is that the counterfactual course after prevention may not be directly observed: people who experience the competing event cannot reveal what their event-1 outcome would have been had it been prevented. Consequently, identifying the effect generally requires substantive assumptions, an appropriate intervention model, or data that provide relevant information about the intervention. A numerical estimate without a clear intervention and an identification argument should not be presented as the causal effect defined here.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

Write a formula for the risk difference by time $\tau$ comparing treatments 1 and 0 under the same intervention $\mathcal{I}_C$ that prevents the competing event.
Under a specified intervention, hypothetical risks are 0.31 for treatment 1 and 0.37 for treatment 0. Calculate and interpret the risk difference.
Why can two strategies that both prevent the competing event define different hypothetical event-1 risks?
In a two-period example, event 1 has period-1 probability 0.10 and event C has probability 0.40. Among those event-free at the start of period 2, the event-1 probability is 0.20. Under an intervention that prevents event C in period 1 and leaves the other probabilities unchanged, calculate the period-2 event-1 risk.
Does random assignment to treatment alone establish the treatment contrast under a hypothetical competing-event-prevention intervention? Explain.

Solutions. (1) $RD_1^{\mathcal{I}_C}(\tau)=P(T_1^{1,\mathcal{I}_C}\leq\tau)-P(T_1^{0,\mathcal{I}_C}\leq\tau)$. (2) $0.31-0.37=-0.06$, a 6-percentage-point lower risk by $\tau$ under treatment 1, under the stated intervention. (3) The prevention strategies may have different effects on the event-1 process or later care, so they imply different counterfactual outcomes. (4) With event C prevented, $0.10+(1-0.10)(0.20)=0.10+0.18=0.28$, or 28%. The calculation assumes that prevention leaves the stated event-1 probabilities unchanged. (5) No. Randomization supports a treatment comparison under the trial’s assigned treatment and actual care, but identifying outcomes under an additional prevention intervention requires further assumptions or data.