Why Decompose a Treatment?
A treatment can be a package of actions with different intended roles. For example, a regimen might include an anti-tumor action intended to reduce cancer progression and a toxicity-management action intended to reduce non-cancer death. When cancer progression and non-cancer death are competing events, the total effect of the package does not by itself say how changing either action would alter the risk of progression.
A separable-effects analysis represents the treatment by two components and asks about interventions that set them separately. The key scientific task is not merely to give the components labels. It is to define interventions on them that are meaningful, including combinations that may not occur in ordinary treatment. The earlier material on separable effects motivates the resulting contrasts; here the emphasis is on what the component intervention means.
Write \(T^{a_Y,a_D}\) for the event time under the joint component intervention and \(J^{a_Y,a_D}\) for the event type, with \(J=Y\) denoting the event of interest and \(J=D\) the competing event. The cumulative incidence of the event of interest by time \(t\) under that intervention is
This risk is the probability of experiencing \(Y\) by \(t\) before a competing event, under the specified joint intervention. It is not a risk in a hypothetical world where \(D\) has been removed. Changing \(A_D\) can change the opportunity to experience \(Y\), because a person who experiences \(D\) can no longer experience \(Y\) first. Therefore, a contrast in \(F_Y\) across \(A_D\) values is a contrast in the observed competing-risk outcome under different component interventions; it need not represent a direct biological effect of that component on \(Y\).
What Makes a Component Intervention Meaningful?
A component is a proposed part of the treatment that can be described as an intervention, not simply a statistical label attached after seeing the outcomes. The intervention should specify what is changed and what is held fixed well enough that the combination has a scientific interpretation. In a clinical setting, that can mean specifying distinct treatment modules, such as an anti-tumor agent and a separately assignable toxicity-prevention strategy.
The two components must also be distinguishable as treatment versions. If changing the supposed \(A_Y\) component necessarily changes the supposed \(A_D\) component, then an intervention setting one while holding the other fixed may not be coherent. Similarly, if a combination such as \((1,0)\) has no defined implementation, its potential risk is not given a clear scientific interpretation merely by writing the notation.
The subscripts express intended roles, not facts established by the labels. Calling a component \(A_Y\) does not prove that it affects only the event-of-interest process; calling another \(A_D\) does not prove that it affects only the competing-event process. The assumptions that would support those restrictions are a separate matter. Nor does a plausible decomposition alone identify the off-diagonal risks from observed data. A well-defined intervention and an identifiable estimand are distinct requirements.
Worked Example: Defining the Two Components
Consider a hypothetical two-year study of a cancer regimen. The event of interest is cancer progression, and non-cancer death is the competing event. The usual control regimen is coded \(A=0\), and the usual active regimen is coded \(A=1\). Investigators propose to decompose the active regimen into (i) an anti-tumor module and (ii) a toxicity-prevention module.
Let \(A_Y=1\) mean assignment to the anti-tumor module and \(A_D=1\) mean assignment to the toxicity-prevention module; value 0 means the corresponding control version. Then \((0,0)\) is the control package and \((1,1)\) is the active package. The off-diagonal regime \((1,0)\) means receiving the anti-tumor module without the toxicity-prevention module. The regime \((0,1)\) means receiving toxicity prevention without the anti-tumor module.
These definitions are scientifically interpretable only if each module can be specified and the combinations can actually be implemented. If toxicity prevention is inseparable from the drug formulation, the proposed \((1,0)\) intervention may not be a well-defined regimen. The component notation would then describe an aspiration, not a usable causal intervention. Even if all four combinations are implementable, their interpretation as components with distinct causal roles still requires additional justification.
Component Contrasts and Their Interpretation
At a fixed follow-up time \(t\), a contrast that changes \(A_Y\) while holding \(A_D\) fixed is a component effect of the \(Y\)-component at that value of the other component. On the risk-difference scale, it is
Likewise, changing \(A_D\) while holding \(A_Y\) fixed gives the contrast
Each contrast compares population risks under two joint interventions. For example, a negative first contrast means that assigning the \(A_Y=1\) component rather than \(A_Y=0\), with \(A_D\) fixed at \(a_D\), lowers the cumulative incidence of \(Y\) by time \(t\) in the target population. It is not automatically an effect on a cause-specific hazard, a biological pathway effect, or an effect in a subgroup selected by surviving event-free.
Worked Example: Interpreting Component Risks
Suppose, purely hypothetically, that the two-year population risks of cancer progression under four well-defined component interventions are as follows. Treat the numbers as stipulated causal risks for illustration, not as estimates from observed data.
| \(A_Y\) | \(A_D\) | Two-year risk of progression |
|---|---|---|
| 0 | 0 | 0.200 |
| 1 | 0 | 0.120 |
| 0 | 1 | 0.250 |
| 1 | 1 | 0.160 |
At \(A_D=0\), the risk-difference contrast for changing \(A_Y\) is
Thus, under these stipulated risks, assigning the anti-tumor component while keeping toxicity prevention off lowers two-year progression risk by 8.0 percentage points. At \(A_D=1\), the corresponding contrast is \(0.160-0.250=-0.090\), or a 9.0 percentage-point reduction. These are context-specific contrasts: they differ because the value of the other component differs.
For the competing-event component, with \(A_Y=1\) fixed, the contrast is \(0.160-0.120=0.040\). Under the stipulated risks, adding the toxicity-prevention component increases the cumulative incidence of progression by 4.0 percentage points in this comparison. This does not show that toxicity prevention biologically promotes cancer. A change in competing-event occurrence can leave more people able to experience progression before the time horizon.
How Component Contrasts Relate to the Total Effect
The total treatment-package risk difference compares the diagonal regimes: the full active package \((1,1)\) versus the control package \((0,0)\). One can insert an off-diagonal regime to express that total contrast as a sum. For the route through \((1,0)\), the identity is
This follows by adding and subtracting the same quantity:
The equality is algebraic; it does not establish that the terms are identified or that the decomposition is unique. One could instead route through \((0,1)\), yielding a component contrast for \(A_D\) at \(A_Y=0\) followed by a contrast for \(A_Y\) at \(A_D=1\). The two routes always add to the same total contrast, but the component terms can differ when effects depend on the level of the other component.
Worked Example: A Total Effect Has More Than One Component Decomposition
Use the hypothetical two-year risks in the previous table. The total risk difference comparing the active package with control is
Taking the route through \((1,0)\), the two terms are \(-0.080\) and \(0.040\), so their sum is \(-0.040\), a 4.0 percentage-point reduction. Taking the route through \((0,1)\), the first term is \(0.250-0.200=0.050\), and the second is \(0.160-0.250=-0.090\). Their sum is also \(0.050-0.090=-0.040\).
The different component terms are not contradictory. The first route changes \(A_Y\) with \(A_D=0\), then changes \(A_D\) with \(A_Y=1\). The second changes \(A_D\) with \(A_Y=0\), then changes \(A_Y\) with \(A_D=1\). In this example, the \(A_Y\) contrasts are \(-0.080\) and \(-0.090\), so the component contrast depends on the setting of the other component. The total effect alone cannot choose a unique allocation between components.
Assumptions, Diagnostics, and Alternatives
Before interpreting a component contrast, check the scientific definition of the intervention. Can each component be assigned or otherwise specified separately? Are all combinations being compared coherent treatment versions? Does the follow-up horizon and event definition match the clinical question? These questions concern the meaning of the estimand. They should be answered before choosing an estimation method.
Then distinguish the target contrast from what the data can support. In a study that assigns only the whole treatment, the observed treatment groups provide information about the diagonal regimes. They do not directly reveal the risks under the off-diagonal interventions. Additional assumptions or design information are needed to learn those risks. In particular, a numerical decomposition should not be presented as an empirical component effect merely because the algebra of the total effect permits an intermediate term.
Useful warning signs include components defined only by a post hoc story, off-diagonal combinations that cannot be implemented, treatment versions that vary substantially within a component label, and claims that an intended component affects only one event process without supporting evidence. Sensitivity to the chosen route through off-diagonal regimes is also scientifically informative: it shows that component contrasts depend on the setting of the other component.
| Approach | Question answered | How it differs from component decomposition |
|---|---|---|
| Treatment-policy contrast | What is the effect of assigning the whole treatment package? | Compares the diagonal regimes and does not allocate the contrast to components. |
| Cause-specific hazard contrast | How do cause-specific event rates compare among those currently event-free? | Is a rate contrast, not by itself a component intervention or a cumulative-risk effect. |
| Hypothetical competing-event elimination | What would the event-of-interest risk be in a regime that prevents the competing event? | Changes the event process itself; it is not the same as setting a treatment component. |
| Separable component contrast | What risk would occur under a joint intervention setting each treatment component? | Requires meaningful component interventions, including the combinations used in the contrast. |
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- In a proposed decomposition, what does the off-diagonal regime \((A_Y,A_D)=(1,0)\) mean?
- A component contrast in the cumulative incidence of \(Y\) is positive when \(A_D\) changes from 0 to 1. Does that alone establish that the \(A_D\) component biologically causes \(Y\)?
- Why can two different sequences of component changes give different component terms but the same total risk difference?
- A study randomizes only the complete treatment package. Does randomization alone reveal risks under both off-diagonal component assignments?
Solutions. (1) It means assigning the version with \(A_Y=1\) and \(A_D=0\); its interpretation depends on whether that combination is a well-defined intervention. (2) No. The cumulative incidence of \(Y\) can change because the competing-event process changes who remains able to experience \(Y\), and the contrast does not alone establish a biological pathway. (3) The two sequences use different reference values for the component changed at each step. Adding and subtracting the relevant intermediate risk gives the same total difference, even if the individual terms differ. (4) No. Randomization of the package supports comparisons of the diagonal treatment regimes, subject to the study’s other requirements; it does not by itself provide outcomes under interventions that were not assigned.