Dismissible Component Conditions

Why the Components Must Be Distinguished

A treatment can sometimes be conceptualized as having two components: $A_Y$, intended to act on the event of interest, and $A_D$, intended to act on the competing event. The separable-effects framework asks what would happen under interventions that set these components separately. Its central challenge is that the components may affect both the chance of experiencing an event and the chance of remaining event-free long enough to experience the other event.

Dismissible component conditions (DCC) are assumptions about transition-specific counterfactual hazards. They specify which component may influence which event’s hazard among people still at risk. They do not say that a component has no effect on the overall probability of an event, and they do not, on their own, identify a separable effect from observed data.

Definition: Let $T^{a_Y,a_D}$ be the counterfactual time to the first event under components set to $A_Y=a_Y$ and $A_D=a_D$, and let $J^{a_Y,a_D}$ denote its type: $J=1$ for the event of interest and $J=2$ for the competing event. Given pretreatment covariates $L=l$, the cause-specific counterfactual hazard for event type $k$ is $$ \lambda_k^{a_Y,a_D}(t\mid l) = \lim_{\Delta t\downarrow 0} \frac{ P\!\left(t\leq T^{a_Y,a_D}<t+\Delta t,\ J^{a_Y,a_D}=k \mid T^{a_Y,a_D}\geq t,\ L=l\right) }{\Delta t}, \qquad k\in\{1,2\}. $$ It is the instantaneous event rate at time $t$ among those who would still be event-free at $t$ under that same component pair. The hazard has units of inverse time.

This definition uses a continuous-time representation with well-defined hazards. The corresponding idea in discrete time compares the conditional probability of each event during an interval, among those event-free at the start of that interval. Both formulations condition on remaining at risk under the component pair being considered. The DCC are stated within levels of $L$; they are not marginal equalities that average over possibly different covariate distributions.

The Two Dismissible Component Conditions

For binary components, the conditions can be stated by comparing the four possible component pairs. The first condition holds the component intended to act on the event of interest fixed, and changes the competing-event component. The second holds the competing-event component fixed, and changes the component intended to act on the event of interest.

Definition: The dismissible component conditions are $$ \lambda_1^{a_Y,a_D}(t\mid l) = \lambda_1^{a_Y,a_D'}(t\mid l) \quad\text{for all }a_Y,a_D,a_D',t,l, $$ and $$ \lambda_2^{a_Y,a_D}(t\mid l) = \lambda_2^{a_Y',a_D}(t\mid l) \quad\text{for all }a_Y,a_Y',a_D,t,l. $$ Thus, conditional on $L$, the event-of-interest hazard does not depend on $A_D$, and the competing-event hazard does not depend on $A_Y$.

The first condition concerns the counterfactual transition to event type 1; the second concerns the counterfactual transition to event type 2. “Dismissible” does not mean that a component can never have any influence on the other event’s cumulative incidence. It means that, at each time and within the relevant counterfactual risk set, changing the other component does not change that particular cause-specific hazard.

The assumptions require a meaningful intervention on each component and well-defined counterfactual outcomes under each component pair. They are substantive restrictions on how the treatment components act. They are distinct from exchangeability, positivity, and censoring assumptions, which may also be needed for an analysis using observed data. The conditions are not guaranteed by randomization of the original bundled treatment: randomization alone does not ensure that its components have the specified transition-specific effects.

Worked Example: Checking Both Conditions in a Constant-Hazard Model

Suppose a hypothetical model has constant cause-specific hazards, in events per person-year, as shown below. Assume these rates apply within each level of $L$, and that all four component combinations are well-defined.

$(a_Y,a_D)$	Event-of-interest hazard $\lambda_1$	Competing-event hazard $\lambda_2$
(0, 0)	0.04	0.12
(0, 1)	0.04	0.06
(1, 0)	0.09	0.12
(1, 1)	0.09	0.06

For the first condition, compare the event-of-interest hazards at fixed $a_Y$. When $a_Y=0$, changing $a_D$ gives $0.04=0.04$; when $a_Y=1$, it gives $0.09=0.09$. Therefore, $\lambda_1^{a_Y,a_D}$ is invariant to $a_D$ in this model.

For the second condition, compare the competing-event hazards at fixed $a_D$. When $a_D=0$, changing $a_Y$ gives $0.12=0.12$; when $a_D=1$, it gives $0.06=0.06$. Thus, $\lambda_2^{a_Y,a_D}$ is invariant to $a_Y$. Both DCC hold for the displayed hazards. The event-of-interest component changes $\lambda_1$, while the competing-event component changes $\lambda_2$, in keeping with the restrictions.

What the Conditions Do Not Imply

Each hazard is conditional on being event-free under the component pair used to define it. If a component changes the other event’s hazard, the composition of the event-free risk set can change over time. Consequently, equality of one type’s cause-specific hazard across component settings does not guarantee equality of that event’s cumulative incidence.

The distinction can be derived directly in a constant-hazard model. Suppose the two hazards under a component pair are $\lambda_1$ and $\lambda_2$, with no other way to leave the event-free state. The total event-free survival probability at time $u$ is $S(u)=\exp[-(\lambda_1+\lambda_2)u]$. The probability of event type 1 by time $t$ is the integral of its instantaneous event rate among those still event-free:

F_1(t) = \int_0^t S(u)\lambda_1\,du = \int_0^t \lambda_1 e^{-(\lambda_1+\lambda_2)u}\,du = \frac{\lambda_1}{\lambda_1+\lambda_2} \left\{1-e^{-(\lambda_1+\lambda_2)t}\right\}.

The expression depends on $\lambda_2$, even when $\lambda_1$ is unchanged. The derivation assumes constant hazards and mutually exclusive first events; it illustrates the implication of the conditions rather than establishing them.

Worked Example: A Dismissible Change Can Still Alter Cumulative Incidence

Use the hazards from the previous hypothetical model and consider $a_Y=1$ through year 2. The event-of-interest hazard is $0.09$ per year for either value of $a_D$, so the first DCC holds. The competing-event hazard is $0.12$ per year when $a_D=0$, and $0.06$ per year when $a_D=1$.

With $a_D=0$, the total hazard is $0.09+0.12=0.21$ per year. The event-of-interest cumulative incidence by year 2 is $$ F_1(2\mid 1,0) = \frac{0.09}{0.21}\left(1-e^{-0.21(2)}\right) = 0.4286(1-e^{-0.42}) \approx 0.1470. $$

With $a_D=1$, the total hazard is $0.09+0.06=0.15$ per year. Therefore, $$ F_1(2\mid 1,1) = \frac{0.09}{0.15}\left(1-e^{-0.15(2)}\right) = 0.6000(1-e^{-0.30}) \approx 0.1555. $$

The event-of-interest cumulative incidence is about 0.0085 higher under $a_D=1$, despite the unchanged event-of-interest hazard. In this model, lowering the competing-event hazard leaves more people event-free and able to experience event type 1. The difference is a probability difference by year 2, not a change in the event-of-interest cause-specific hazard.

Discrete-Time Check and Interpretation

In discrete time, let $q_k^{a_Y,a_D}(m\mid l)$ be the probability of event type $k$ during interval $m$, conditional on being event-free at its start under that component pair and on $L=l$. The competing event and event of interest are mutually exclusive first events within an interval, so $q_1+q_2\leq 1$. The discrete-time DCC compare $q_1$ across $a_D$ at fixed $a_Y$, and compare $q_2$ across $a_Y$ at fixed $a_D$.

Worked Example: Checking Discrete-Time Transition Probabilities

Suppose that in a hypothetical one-month interval, the conditional event probabilities are as follows, and the same values apply within each level of $L$.

$(a_Y,a_D)$	Event of interest $q_1$	Competing event $q_2$
(0, 0)	0.02	0.10
(0, 1)	0.02	0.04
(1, 0)	0.08	0.10
(1, 1)	0.08	0.04

At fixed $a_Y=0$, the event-of-interest probability is $0.02$ under either value of $a_D$; at fixed $a_Y=1$, it is $0.08$ under either value. At fixed $a_D=0$, the competing-event probability is $0.10$ under either value of $a_Y$; at fixed $a_D=1$, it is $0.04$ under either value. Thus both discrete-time conditions hold. The largest sum of the two probabilities is $0.08+0.10=0.18$, which is below 1, so the entries are valid mutually exclusive interval-event probabilities.

Diagnostics, Alternatives, and Limitations

The DCC are causal assumptions, not routine empirical tests. A dataset in which each person receives only one bundled treatment generally does not reveal all four counterfactual component combinations for that person. Even a lack of an observed association between $A_D$ and event type 1 does not demonstrate the first condition: observed risk sets can differ, and treatment assignment, covariates, and selection into event-free survival may affect the comparison. A factorial design or strong mechanistic knowledge can inform plausibility, but neither automatically proves the counterfactual equalities.

A useful assessment asks whether it is scientifically credible that the component assigned to the competing event has no effect on the event-of-interest transition hazard, and whether the reverse restriction is credible for the competing-event hazard. Mechanisms that cross pathways, treatment interactions, or time-dependent effects may make one or both restrictions implausible. The conditions must hold over the relevant times and covariate levels, not merely at one time point or in an overall average. In applied work, it is also important to specify the time scale, event types, component versions, and conditioning covariates clearly.

Approach or claim	What it says	How it differs from DCC
Dismissible component conditions	Each component is excluded from the other event type’s counterfactual cause-specific hazard, conditional on $L$ and being at risk.	Transition-specific restrictions; do not assert unchanged cumulative incidence.
No effect of $A_D$ on event type 1 at all	Changing $A_D$ has no effect on the event-of-interest outcome under the stated meaning of “effect.”	Stronger or differently defined claim; DCC still permits a change in event-type-1 cumulative incidence through event-free survival.
Observed cause-specific hazard comparison	Compares hazards in observed groups, possibly conditional on measured variables.	Does not by itself establish equality between counterfactual hazards under component interventions.

A further limitation is that DCC alone do not identify separable effects. Identification also requires assumptions connecting observed data to the relevant counterfactual quantities, along with suitable treatment and follow-up conditions. The conditions specify a particular causal structure for transitions; they are not a substitute for those additional requirements.

Common Mistake

Wrong reasoning: “The event-of-interest hazard is unchanged when $A_D$ changes, so the event-of-interest risk must also be unchanged.” This treats a cause-specific hazard as if it were a cumulative probability and ignores that $A_D$ can change who remains event-free.

Correction: The DCC compare the event-of-interest hazard among those at risk under each component pair. A competing component can alter the competing-event hazard, change event-free survival, and thereby change the event-of-interest cumulative incidence, even while its cause-specific hazard is invariant. Also, conditioning on being event-free does not mean the comparisons involve the same individuals under different counterfactual regimes.

Key takeaway: The first dismissible component condition links $A_Y$ to the event-of-interest hazard and excludes $A_D$ from that hazard; the second links $A_D$ to the competing-event hazard and excludes $A_Y$ from it. Both are counterfactual, risk-set-specific assumptions, not claims of equal cumulative risks or evidence that the assumptions hold.

Check Your Understanding

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

In the first DCC, which component is held fixed, which is varied, and which transition hazard is compared?
Suppose $\lambda_2^{0,1}(t\mid l)=0.07$ and $\lambda_2^{1,1}(t\mid l)=0.07$ at one time and covariate level. What part of the second DCC does this check, and what remains unchecked?
Why can event-type-1 cumulative incidence change when $\lambda_1$ is invariant to $A_D$?
Does observing similar event-type-1 hazards in two treatment groups establish the first DCC? Explain.

Solutions. (1) Hold $a_Y$ fixed, vary $a_D$, and compare the counterfactual event-type-1 cause-specific hazards among those event-free under each component pair. (2) It checks the second condition for $a_D=1$, at that time and $l$, by comparing across $a_Y$; the condition for other times, covariate levels, and $a_D$ remains unchecked. (3) Changing $A_D$ can change the competing-event hazard and hence event-free survival, altering the number who remain able to experience event type 1. (4) No. An observed comparison is not generally a comparison of the required counterfactual hazards; confounding and differences in risk-set composition can prevent that inference.