A generalized theory of separable effects in competing event settings

In competing event settings, a counterfactual contrast of cause-specific cumulative incidences quantifies the total causal effect of a treatment on the event of interest. However, effects of treatment on the competing event may indirectly contribute to this total effect, complicating its interpretation. We previously proposed the separable effects to define direct and indirect effects of the treatment on the event of interest. This definition was given in a simple setting, where the treatment was decomposed into two components acting along two separate causal pathways. Here we generalize the notion of separable effects, allowing for interpretation, identification and estimation in a wide variety of settings. We propose and discuss a definition of separable effects that is applicable to general time-varying structures, where the separable effects can still be meaningfully interpreted as effects of modified treatments, even when they cannot be regarded as direct and indirect effects. For these settings we derive weaker conditions for identification of separable effects in studies where decomposed, or otherwise modified, treatments are not yet available; in particular, these conditions allow for time-varying common causes of the event of interest, the competing events and loss to follow-up. We also propose semi-parametric weighted estimators that are straightforward to implement. We stress that unlike previous definitions of direct and indirect effects, the separable effects can be subject to empirical scrutiny in future studies. Supplementary Information The online version supplementary material available at 10.1007/s10985-021-09530-8.

1 Treatment decomposition of A into A Y , A D and A Z Hitherto we have described settings in which the treatment is decomposed into 2 components, A D and A Y . Consider now a hypothetical treatment decomposition into 3 components A D , A Y and A Z , as illustrated in Figure 1, which is similar to Robins and Richardson's decomposition in a mediation setting [1,Figure 6(d)]. Analogous to the 2 way decomposition, we define a generalized decomposition assumption: 3 way generalized decomposition assumption: The treatment A can be decomposed into three binary components A Y ∈ {0, 1}, A D ∈ {0, 1} and A Z ∈ {0, 1} such that, in the observed data, the following determinism holds be the indicator of the event of interest by interval k + 1 had, possibly contrary to fact, he/she been We assume that an intervention that assigns A = a results in the same outcome as an intervention that assigns for Q k+1 ∈ {Y k+1 , D k+1 , Z k+1 }. Analogous to the 2 way decomposition, the 3 way decomposition may be practically interesting in settings where we can conceive interventions on all 3 components of A. Furthermore, in settings where Z k partition fails, it may be possible to define a 3 way decomposition that allows identifiability of separable effects. For example, Figure 1 can represent an alternative decomposition of the setting described in Figure 5a, where Z k partition fails.
To define identifiability conditions that apply to settings with 3 way decompositions, we continue to use superscripts to denote counterfactuals and for notational simplicity we consider settings without censoring, such that e.g.
Here we will only consider settings that satisfy the following assumptions: the only causal paths from A Y to D k+1 and Z k+1 , k ∈ {0, . . . , K} are through Y j , the only causal paths from A D to Y k+1 and Z k+1 , k = 0, . . . , K are through D j+1 , the only causal paths from A Z to Y k+1 and D k+1 , k = 0, . . . , K are through Z j+1 , For k = 0, . . . , K, consider the separable effects for a D , a Z ∈ {0, 1}, for a Y , a Z ∈ {0, 1}, and for a Y , a D ∈ {0, 1}. Similar to the two component decomposition, the total effect can be expressed as a sum of the separable direct and indirect effects, in particular,

Interpretation of the 3 component decomposition
Under (3)-(5), the 3 way decomposition of A into A D , A Y and A Z allows us to interpret the separable effects as direct and indirect effects; (6) is the effect not emanating from A D or A Z , i.e. a separable direct effect, (7) is the separable indirect effect on the event of interest only emanating from A D , and (8) is the separable indirect effect on the event of interest only emanating from A Z . In our running example, where Z k = L k encodes the (systolic and diastolic) blood pressure, it is not obvious that the 3 part decomposition is of interest; to interpret effects defined by the 3 part decomposition, we would need to conceptualize a treatment decomposition of blood pressure therapy into 3 components: the A D component could now be defined as the component that exerts effects on mortality not through blood pressure reduction or kidney injury; that is, the substantive meaning of an intervention on A D fundamentally changes. The A Z component would affect the outcome of interest only through blood pressure reduction; the effect exerted by A Z is analogous to an indirect mediation effect described by Didelez [2] under an agnostic causal model, but in our setting we also allow for competing risks. We note that under this 3 way decomposition, the A Y component is identical to the A Y component in the 2 way decomposition, that is, the component of blood pressure therapy only exerting direct effects on kidney injury not through blood pressure reduction.
In other settings, however, the 3 part decomposition may be feasible. For example, Robins and Richardson [1, Figure 6(d)] consider a similar decomposition in a conceptual example on the effect of cigarettes on lung cancer; they consider the effect of cigarettes smoking through nicotine, tar and other pathways.

Identification of the 3 component decomposition
The identifiability conditions are straightforward extensions of the conditions in Section 6. Now we must identify First the exchangeability, consistency and positivity conditions are identical to the condition in Section 6. The dismissible component conditions read Under these assumptions we can identify Pr(Y a Y ,a D ,a Z k+1 = 1) for k = 0, . . . , K from which follows from a similar derivation from that in Appendix B. The identifiability conditions under the 3 component decomposition require stronger restrictions on the unmeasured variables, compared to the settings in Section 6; unmeasured common causes of any pair in (Y k+1 , D j+1 , L m+1 ), k, j, m ∈ {0, . . . , K} can violate the dismissible component conditions. In particular, an unmeasured common cause U L,Y of L k and Y k will violate the dismissible component condition, as shown in grey in Figure 2. Fig. 1: Treatment A is decomposed into 3 components.