Separable Effects for Competing Risks

Why Separate Treatment Components?

A treatment can influence the risk of a target event through more than one process. For example, a hypothetical therapy might affect disease progression and also affect the risk of a competing event that prevents the target event from occurring. The total effect of receiving the therapy combines these influences. A separable effect asks a different question: what counterfactual risk contrast would result from changing one conceptual component of treatment while holding another component fixed?

This is a way to define causal contrasts, not a claim that every treatment can literally be split into two independently assignable parts. The distinction matters: an estimand can be stated mathematically even when its components are difficult or impossible to implement, and defining it does not by itself show that it can be identified from observed data.

Definition: Suppose a binary treatment is represented conceptually by two binary components, $A_Y$ and $A_D$. Let $T^{a_Y,a_D}$ denote the counterfactual time to the first event under component levels $a_Y$ and $a_D$, and let $J^{a_Y,a_D}$ denote its event type. For a target event type $k$, define its counterfactual cumulative incidence by $$ F_k(t;a_Y,a_D)=P\bigl(T^{a_Y,a_D}\le t,\ J^{a_Y,a_D}=k\bigr). $$ A separable effect through $A_Y$, at a fixed level $a_D$, is a contrast between $F_k(t;1,a_D)$ and $F_k(t;0,a_D)$. A separable effect through $A_D$, at a fixed level $a_Y$, contrasts $F_k(t;a_Y,1)$ and $F_k(t;a_Y,0)$. Unless otherwise stated, this tutorial expresses these contrasts as risk differences.

The subscripts $Y$ and $D$ are labels for components, not guarantees about their biological or causal action. In particular, the definition alone does not ensure that $A_Y$ affects only the target event or that $A_D$ affects only the competing event. Those are additional substantive ideas, not consequences of writing down the counterfactual risks.

For a binary treatment that corresponds to both components taking the same value, the usual treatment regimes are represented by $(A_Y,A_D)=(0,0)$ and $(1,1)$. The contrast $F_k(t;1,1)-F_k(t;0,0)$ is then the total effect on the target-event risk at time $t$. Separable contrasts instead compare regimes such as $(1,0)$ and $(0,0)$, or $(0,1)$ and $(0,0)$. These are often called off-diagonal regimes because the components are set to different levels.

Defining and Reading the Contrasts

A common pair of risk-difference estimands uses the untreated level of the other component as the reference:

\begin{aligned} \mathrm{SDE}_k(t) &= F_k(t;1,0)-F_k(t;0,0),\\ \mathrm{SIE}_k(t) &= F_k(t;0,1)-F_k(t;0,0). \end{aligned}

Here, $\mathrm{SDE}_k(t)$ is the separable effect through $A_Y$ when $A_D=0$, and $\mathrm{SIE}_k(t)$ is the separable effect through $A_D$ when $A_Y=0$. These names are conventional; the important information is the pair of component settings in each contrast. A positive difference means a higher counterfactual probability of the target event by time $t$ under the first setting than under the second. Its units are probability points, commonly reported as percentage points.

The reference level is part of the estimand. The effect of changing $A_Y$ while $A_D=1$, for instance, is $F_k(t;1,1)-F_k(t;0,1)$. It need not equal the effect of changing $A_Y$ while $A_D=0$. Nor does the separable effect necessarily describe a biological pathway in isolation: the risk of a target event can depend on both components because the event competes with other causes.

Worked Example: Calculating Two Separable Risk Differences

Consider a fully hypothetical population and a fixed follow-up time of 2 years. Suppose the counterfactual risks of a target event under four component assignments are stipulated to be

$a_Y$	$a_D$	Target-event risk by 2 years
0	0	0.100
1	0	0.160
0	1	0.070
1	1	0.120

With $A_D$ fixed at 0, the separable effect through $A_Y$ is

\mathrm{SDE}_k(2)=0.160-0.100=0.060.

Thus, in this hypothetical example, changing $A_Y$ from 0 to 1 while setting $A_D=0$ increases the counterfactual 2-year risk of the target event by 6.0 percentage points. With $A_Y$ fixed at 0, the contrast through $A_D$ is

\mathrm{SIE}_k(2)=0.070-0.100=-0.030.

Under that component assignment, changing $A_D$ from 0 to 1 reduces the counterfactual target-event risk by 3.0 percentage points. Neither calculation says that the component changes the target event through a single isolated biological mechanism; each is a contrast in the total probability of the target event under the specified competing-risk regime.

How the Total Effect Relates to the Component Contrasts

There is a useful algebraic decomposition of the total risk difference. One can move from $(0,0)$ to $(1,1)$ by first changing $A_Y$, then changing $A_D$:

\begin{aligned} F_k(t;1,1)-F_k(t;0,0) &=\{F_k(t;1,0)-F_k(t;0,0)\}\\ &\quad+\{F_k(t;1,1)-F_k(t;1,0)\}. \end{aligned}

The identity follows by adding and subtracting $F_k(t;1,0)$; the two occurrences cancel. It decomposes the total risk difference into the effect of changing $A_Y$ with $A_D=0$, followed by the effect of changing $A_D$ with $A_Y=1$. The second term is not the $\mathrm{SIE}_k(t)$ defined above, which changes $A_D$ while $A_Y=0$.

Alternatively, changing $A_D$ first gives

\begin{aligned} F_k(t;1,1)-F_k(t;0,0) &=\{F_k(t;0,1)-F_k(t;0,0)\}\\ &\quad+\{F_k(t;1,1)-F_k(t;0,1)\}. \end{aligned}

These are both exact arithmetic identities, not assumptions about causal mechanisms. If the component effects differ across the level of the other component, the two paths allocate the total contrast differently. In general, the two reference-specific separable effects $\mathrm{SDE}_k(t)$ and $\mathrm{SIE}_k(t)$ do not add to the total effect.

Worked Example: Reference Levels and the Decomposition Path

Use the hypothetical risks in the previous table. The total effect comparing $(1,1)$ with $(0,0)$ is

$$ F_k(2;1,1)-F_k(2;0,0)=0.120-0.100=0.020. $$

Following the path that changes $A_Y$ first gives

$$ (0.160-0.100)+(0.120-0.160)=0.060-0.040=0.020. $$

Following the path that changes $A_D$ first gives

$$ (0.070-0.100)+(0.120-0.070)=-0.030+0.050=0.020. $$

Both paths recover the same total risk difference of 2.0 percentage points. But the effect of changing $A_Y$ is 6.0 percentage points at $A_D=0$ and 5.0 percentage points at $A_D=1$. The effect of changing $A_D$ is −3.0 percentage points at $A_Y=0$ and −4.0 percentage points at $A_Y=1$. The component contrasts therefore depend on the reference setting; choosing a path is not a purely verbal choice.

Conditions for a Meaningful Interpretation

Before interpreting a separable effect causally, the component interventions must have a coherent meaning. At a minimum, the levels $0$ and $1$ of each component must refer to sufficiently well-defined interventions, and the joint assignments used in the contrast must make sense for the target population. If the components are merely labels for two post hoc pathways, without a clear intervention interpretation, the counterfactual risks may not answer an actionable causal question.

There is also a data-support issue. When actual treatment assigns components together, observed individuals typically receive $(0,0)$ or $(1,1)$, not $(1,0)$ or $(0,1)$. The off-diagonal risks in a separable contrast are counterfactual quantities. Observing the total-treatment groups alone does not reveal them. Additional causal assumptions and an identification argument are needed before observed data can be used to estimate those risks.

These requirements are distinct. A proposed component intervention can be conceptually meaningful but unsupported by the observed data; conversely, a statistical model can produce numbers for off-diagonal regimes without making the interventions substantively meaningful. A calculated contrast should not be presented as identified merely because software returns an estimate.

Worked Example: What the Observed Treatment Groups Do Not Show

Suppose a hypothetical study assigns only a combined treatment: participants receive either neither component, $(0,0)$, or both components, $(1,1)$. The observed 2-year target-event risks are 0.100 and 0.120, respectively. The treatment-policy risk difference is therefore

$$ 0.120-0.100=0.020, $$

or 2.0 percentage points. This comparison supplies no direct observation of the risks under $(1,0)$ and $(0,1)$, so it does not supply either of the reference-level separable contrasts. For example, $\mathrm{SDE}_k(2)$ requires $F_k(2;1,0)-F_k(2;0,0)$; knowing the second risk is 0.100 does not determine the first. Many values of $F_k(2;1,0)$, consistent with a probability between 0 and 1, would yield different separable effects. The total effect and the separable effects are different estimands, not interchangeable calculations.

Relation to Other Causal Contrasts

A treatment-policy effect compares outcomes under assigning the overall treatment regimes, here typically $(1,1)$ and $(0,0)$. A separable effect instead changes one component while fixing the other. A controlled direct effect fixes a mediator or other variable to a specified level while changing treatment; that is not automatically the same intervention as setting one treatment component. Natural direct and indirect effects use counterfactual mediator values and different assumptions. The fact that all these approaches can be described as “separating” aspects of an effect does not make their estimands equivalent.

The scale also matters. A risk difference is additive and has percentage-point units; a risk ratio compares risks multiplicatively. Component contrasts and their decompositions must be defined on a stated scale. The telescoping identities above apply to differences because differences add. They do not imply an analogous additive decomposition of risk ratios.

Common Mistakes and Diagnostics

Calling a component contrast the total treatment effect. The contrast $F_k(t;1,0)-F_k(t;0,0)$ changes only $A_Y$; the total contrast compares $(1,1)$ with $(0,0)$.
Ignoring the fixed component level. A separable effect must state what is held fixed. If component effects vary by that level, omitting it hides a meaningful feature of the estimand.
Assuming components can be separated because they have names. A conceptual decomposition does not establish feasible interventions, observed support, or identification.
Reading a lower target-event risk as proof of benefit. A competing event can change the opportunity to experience the target event. Interpret the stated cause-specific risk contrast, and consider the competing outcome and clinical objective as well.
Adding the two reference-level effects and expecting the total. The two contrasts may hold the other component at different reference levels from those in a chosen path. Check the component settings before adding terms.

A practical diagnostic is to write the four component assignments in a $2\times2$ table, mark which risks are observed or counterfactual, and label each contrast by the component changed and the component held fixed. If a proposed interpretation cannot be mapped to those settings, the estimand has not been stated precisely.

Key takeaway: A separable effect is a counterfactual risk contrast that changes one treatment component while fixing another. Its definition specifies the event, time horizon, component assignments, and contrast scale; it does not by itself establish that the component interventions are feasible or that their risks are identified.

Check Your Understanding

Answer each question using the definitions and hypothetical risks above.

For the displayed risk table, calculate the effect of changing $A_Y$ from 0 to 1 while holding $A_D=1$.
Calculate the effect of changing $A_D$ from 0 to 1 while holding $A_Y=1$.
Why can the total effect not generally be obtained by adding the two separable effects defined with the other component fixed at 0?
If a study observes only assignments $(0,0)$ and $(1,1)$, what additional risks are needed to calculate $\mathrm{SDE}_k(t)$ and $\mathrm{SIE}_k(t)$?

Solutions. (1) $F_k(2;1,1)-F_k(2;0,1)=0.120-0.070=0.050$, a 5.0 percentage-point increase. (2) $F_k(2;1,1)-F_k(2;1,0)=0.120-0.160=-0.040$, a 4.0 percentage-point decrease. (3) Those two effects each hold the other component at 0, so they do not follow the same path from $(0,0)$ to $(1,1)$; component effects can vary with the fixed component. (4) In addition to the $(0,0)$ risk, the contrasts require $F_k(t;1,0)$ for $\mathrm{SDE}_k(t)$ and $F_k(t;0,1)$ for $\mathrm{SIE}_k(t)$. The observed diagonal groups alone do not provide those off-diagonal counterfactual risks.