Why Fix the Competing-Event Process?
A treatment can affect the risk of a target event both by changing the target-event process and by changing whether a competing event happens first. For example, a treatment might change the risk of disease-related death while also changing the risk of death from another cause. The observed cumulative incidence reflects both processes. A controlled direct effect asks a different question: how would target-event risk differ between treatment levels if the competing-event process were governed by the same specified intervention in both cases?
The phrase “controlled direct” does not mean that the contrast is automatically a biological effect that bypasses every pathway involving the competing event. Its meaning depends on the intervention used to control that process. In particular, “set the competing-event risk to 10%” may not fully define an intervention: the timing and circumstances of those events can matter. A regime should specify how the competing-event process is changed over follow-up.
Here, \(g\) might describe a policy that reduces the competing-event hazard during a specified period, or a clinical intervention that delays a particular competing event. It must be sufficiently precise to distinguish interventions that could have different consequences. The outcome is still a first-event outcome: if the competing event occurs first, the target event is not counted as having occurred by \(\tau\).
Interpretation and Assumptions
The risk difference answers a population-level question about two joint intervention scenarios. For example, a positive value means that the risk of the target event by \(\tau\) is higher under \(a\) than under \(a'\), when both are paired with regime \(g\). It does not say that the same individuals would experience the target event under one treatment and avoid it under the other. Both potential outcomes are defined, but only one joint scenario is observed for any individual.
A causal interpretation requires more than calculating two risks. The treatment levels and competing-event regime must be well-defined, including how and when the regime acts. Consistency requires that a person receiving a specified treatment and regime has the corresponding potential outcome. Interference between individuals must be absent or otherwise accounted for. For identification from data, treatment and regime assignment must be randomized, or appropriate exchangeability assumptions must hold given measured covariates and history. There must be positivity: each relevant covariate history must have support for the treatment and regime being compared. Loss to follow-up must also be handled under suitable censoring assumptions.
These conditions can be demanding when \(g\) is hypothetical. For instance, preventing a competing event may be impossible, or there may be multiple ways to prevent it that also affect the target event differently. A numerical value assigned to \(g\) is not enough if it leaves those intervention versions unspecified.
Worked Example: A Controlled Risk Difference Under One Regime
Suppose a hypothetical randomized study evaluates a target event within two years. Under a shared, specified regime \(g\) for the competing-event process, the risk is 120 out of 1,000 under treatment \(a'\) and 180 out of 1,000 under treatment \(a\). Assume the regime and treatment assignments are well-defined and the risks refer to the same target population.
The two risks are \(120/1{,}000=0.120\) and \(180/1{,}000=0.180\). Thus, $$ \widehat{\operatorname{CDE}}_{RD}(2;g)=0.180-0.120=0.060. $$
Under regime \(g\), the estimated target-event risk by two years is 6.0 percentage points higher under \(a\) than under \(a'\). The risk ratio is \(0.180/0.120=1.50\), or 1.50. These are contrasts between joint intervention scenarios, not comparisons restricted to people who happened to avoid a competing event.
Why the Regime Must Be Specified
A controlled direct effect is indexed by \(g\). Changing \(g\) can change each risk and can also change the difference between them. Consider two possible regimes: \(g_1\), which leaves the competing-event process at its standard level, and \(g_2\), which delays competing events during early follow-up. These are distinct causal questions, even when the treatment contrast is unchanged.
Worked Example: Two Regimes, Two Controlled Effects
In a hypothetical population followed for one year, suppose the target-event risks under the four joint scenarios are as follows. The same regime is used for both treatment levels within each row.
| Competing-event regime | Risk under \(a'\) | Risk under \(a\) |
|---|---|---|
| \(g_1\): standard process | 0.10 | 0.14 |
| \(g_2\): delayed process | 0.19 | 0.25 |
For \(g_1\), the risk difference is \(0.14-0.10=0.04\), or 4 percentage points. For \(g_2\), it is \(0.25-0.19=0.06\), or 6 percentage points. The corresponding risk ratios are \(0.14/0.10=1.40\) under \(g_1\) and \(0.25/0.19=1.32\) under \(g_2\), to three significant figures.
The controlled effect is therefore not a single number independent of the competing-event intervention. Here the delayed-process regime raises target-event risk under both treatments, and the risk-difference contrast also changes. This example does not establish that delaying competing events would have these effects in a real population; the numbers are hypothetical and illustrate how the estimand depends on \(g\).
One special choice is a regime that prevents the competing event throughout follow-up. The resulting target-event risk is sometimes called a risk under competing-event elimination or a net risk. It is only one possible controlled contrast, and it should not be treated as equivalent to every controlled direct effect. In particular, a regime that delays competing events for a limited period is not the same intervention as preventing them for all of follow-up.
Identification by Standardization
When \(g\) is a regime assigned to individuals, a basic identification result can be written using baseline covariates \(X\). Suppose treatment and regime assignment are randomized, or that conditional exchangeability holds given \(X\); consistency holds; and treatment-regime positivity holds for each relevant covariate stratum. Then the risk under a joint intervention can be obtained by averaging stratum-specific risks over the covariate distribution in the target population:
For a dynamic regime acting during follow-up, the corresponding identification generally requires assumptions and calculations over the evolving history, rather than treating \(G\) as a single baseline label. The expression above is the simpler case in which assignment to the specified regime is represented as a joint intervention and the stated assumptions are adequate.
The justification follows by averaging conditional risks. Under exchangeability, within each covariate stratum the observed risk for individuals assigned \(a\) and \(g\) represents the counterfactual risk under that joint intervention. Consistency links observed outcomes to the assigned intervention. Positivity ensures that the required stratum-specific risks are estimable. Averaging those risks using the target population’s \(P(X=x)\) gives the population risk under the intervention. Subtracting the result for \(a'\) gives the controlled risk difference.
Worked Example: Standardizing Risks Across Covariate Strata
Consider a hypothetical target population with 600 people in a lower-risk covariate stratum and 400 in a higher-risk stratum. Everyone is followed under the same specified competing-event regime \(g\). Suppose the target-event counts by the end of follow-up are:
| Stratum | Population share | Events under \(a'\) | Events under \(a\) |
|---|---|---|---|
| Lower-risk | 0.60 | 60 of 600 | 90 of 600 |
| Higher-risk | 0.40 | 80 of 400 | 120 of 400 |
The stratum-specific risks are 0.10 and 0.20 under \(a'\), and 0.15 and 0.30 under \(a\). Standardizing to the stated population gives $$ \widehat P\{Y^{a',g}=1\}=0.60(0.10)+0.40(0.20)=0.14, $$ and $$ \widehat P\{Y^{a,g}=1\}=0.60(0.15)+0.40(0.30)=0.21. $$ Therefore the standardized controlled risk difference is \(0.21-0.14=0.070\), or 7.0 percentage points.
This calculation is a causal estimate only if the assignment, consistency, positivity, and follow-up assumptions required for the standardization are credible. The arithmetic alone does not establish those assumptions. For example, if the higher-risk stratum had no individuals assigned to \(a\) under \(g\), its risk under that joint intervention could not be estimated from these data without further modeling assumptions.
How This Contrast Differs From Other Comparisons
| Quantity | What is compared? | Key distinction |
|---|---|---|
| Controlled direct effect | Target-event risks under \(a\) and \(a'\), with the same specified regime \(g\) | A joint-intervention contrast indexed by \(g\) |
| Total treatment effect | Risks under \(a\) and \(a'\), each with the competing-event process that would occur under that treatment | Includes treatment-related changes in the competing-event process |
| Risk among observed event-free individuals | Observed groups selected by having had no event up to a time | Conditions on a post-treatment status; it does not set the competing-event process to \(g\) |
| Cause-specific hazard contrast | Instantaneous target-event rates among individuals currently event-free | It is a hazard comparison, not the controlled cumulative-risk contrast defined here |
The key distinction is intervention versus selection. A controlled direct effect compares risks under specified joint interventions. Restricting the analysis to people who remain event-free does not implement an intervention on the competing-event process: those people may differ between treatment groups because treatment affects who remains event-free.
Diagnostics, Limitations, and a Common Mistake
- Check the intervention description. State how \(g\) changes the competing-event process, when it acts, and which versions count as the same regime. If only a cumulative competing-event probability is specified, timing may remain ambiguous.
- Check support. Look for covariate strata or histories with no observations under one of the treatment-regime combinations. Extrapolation into those regions can dominate an estimate.
- Check outcome and follow-up definitions. Confirm that target events, competing events, the time origin, and \(\tau\) are defined consistently across scenarios. Differential censoring can make a risk comparison misleading if it is not addressed.
- Check plausibility and scope. Some regimes may be impractical or impossible to implement. A causal effect under such a regime can still be a hypothetical estimand, but its interpretation should say so.
Common mistake: “Among people who did not experience the competing event, compare treatment groups; that gives the direct effect.” This is wrong because remaining event-free is a post-treatment selection. The two selected groups need not represent comparable populations, and no competing-event intervention has been imposed. The correction is to define a common regime \(g\), specify the target-event risk under each joint intervention, and compare those risks under assumptions that support causal identification.
A second limitation is that a controlled direct effect does not, by itself, explain why treatment affects the target event. It describes the contrast under a chosen intervention on the competing-event process; it does not automatically separate pathways or identify a mechanism. Its value is that the causal question is explicit: the treatment comparison is made under a stated competing-event regime rather than under the different competing-event processes that may arise naturally under each treatment.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- In one sentence, state what is held fixed in a controlled direct effect on target-event risk.
- Under a common regime \(g\), the risks by 18 months are 0.16 under \(a\) and 0.11 under \(a'\). Calculate the risk difference and interpret it.
- Why does comparing only people who remain free of both events fail to define a controlled direct effect?
- Two regimes produce different controlled risk differences. Does this necessarily indicate a calculation error? Explain.
- Name two conditions needed to identify the intervention-specific risks from observational data.
Solutions. (1) The competing-event intervention regime \(g\) is held fixed across the compared treatment levels. (2) The risk difference is \(0.16-0.11=0.05\), or 5 percentage points; under \(g\), risk by 18 months is estimated to be 5 percentage points higher under \(a\). (3) Event-free status is a post-treatment selection, not an intervention, and the selected groups may differ. (4) No: the estimand is indexed by \(g\), so different regimes can produce different causal contrasts. (5) Examples include conditional exchangeability, positivity, consistency, and appropriate censoring assumptions.