Natural Direct and Indirect Effects With Competing Events

Natural Effects When an Event Can Preclude the Outcome

When a competing event can prevent the event of interest from occurring first, an intervention on treatment or a mediator may change both kinds of event. A natural direct effect or natural indirect effect must therefore be defined for the event outcome that is actually of interest, not for a hypothetical outcome in which the competing event has been removed. The central bookkeeping question is: which treatment is set, and which treatment’s mediator value is used, in each counterfactual outcome?

We define natural effects on the risk of a specified event by a fixed horizon. This gives contrasts in probabilities, rather than contrasts in hazards. The effects are estimands: precise causal quantities defined using counterfactuals. Their definitions alone do not establish that they can be identified or estimated from observed data.

Definition: Let $A\in\{0,1\}$ denote treatment, $M$ a mediator measured after treatment and before the event process, and $k$ the event type of interest. Under an intervention setting treatment to $a$ and mediator to $m$, let $T^{a,m}$ be the time to the first event and $J^{a,m}$ its type. For a fixed horizon $\tau$, define the counterfactual cause-specific event outcome $$ Y_k^{a,m}(\tau)=I(T^{a,m}\leq \tau,\ J^{a,m}=k). $$ It equals 1 if event $k$ occurs by $\tau$ as the first event, and 0 otherwise. In particular, it is 0 if a competing event occurs first, or if no event occurs by $\tau$. Let $M^a$ be the mediator value the same person would have under treatment $a$. The nested outcome $Y_k^{a,M^{a'}}(\tau)$ uses treatment $a$ for the event process and that person’s mediator value under treatment $a'$.

The definition treats event type and timing together. A competing event is not ordinary censoring: once it occurs first, the person cannot subsequently have event $k$ as the first event. Thus, changing treatment or the mediator can alter the outcome risk through changes in the target event, the competing event, or both.

Defining the Natural Direct and Indirect Effects

A commonly used decomposition takes the mediator value under the untreated condition as the reference for the direct effect, and treatment as the reference for the indirect effect. For the risk of event $k$ by $\tau$, define the effects on the risk-difference scale as follows.

Definition: The natural direct effect with reference $M^0$ compares treatments while holding each person’s mediator at the value that person would naturally have under treatment 0. The natural indirect effect with treatment reference 1 compares mediator values under treatments 1 and 0 while setting treatment to 1: $$ \operatorname{NDE}_{0,k}(\tau) =E\!\left[Y_k^{1,M^0}(\tau)\right] -E\!\left[Y_k^{0,M^0}(\tau)\right], $$ $$ \operatorname{NIE}_{1,k}(\tau) =E\!\left[Y_k^{1,M^1}(\tau)\right] -E\!\left[Y_k^{1,M^0}(\tau)\right]. $$ The expectations are population mean risks. Both effects are measured in risk points (probability units) by time $\tau$.

In the first term of the direct effect, the event process uses treatment 1 but the mediator is set to $M^0$; in the second term, both treatment and the mediator reference are 0. For the indirect effect, treatment is 1 in both terms, but the mediator is $M^1$ in one and $M^0$ in the other. The mediator value is person-specific, not a single population-wide value.

These are nested counterfactuals: for example, $Y_k^{1,M^0}$ combines the event outcome under treatment 1 with the mediator value that would arise under treatment 0. The notation defines the contrast; it does not imply that these quantities are jointly observed for any person. Identification from observed data requires additional assumptions and methods.

Worked Example: Reading Each Counterfactual in a Natural Effect

Suppose a hypothetical study compares a treatment with usual care and considers the risk of a target event within two years. Let $M$ be a post-treatment biomarker. For one hypothetical person, $M^0=4$ units and $M^1=7$ units. Consider the term $Y_k^{1,M^0}(2)$.

This term sets treatment to 1 for the event process, but uses the person’s mediator value of 4 units, the value they would have had under treatment 0. It equals 1 only if the target event is the first event and occurs within two years under that combination. If a competing event occurs first, the term is 0, even if the target event might otherwise have occurred later.

For this person, $Y_k^{0,M^0}(2)$ instead uses treatment 0 and mediator value 4 units. The contrast between these terms is the person-level component of the direct-effect contrast; the natural direct effect itself is the population average of such differences. The person’s values alone do not determine the population effect.

Decomposition and Reference Choices

The definitions give an exact algebraic decomposition of the total effect on this risk scale. The total effect compares the two treatment-specific natural regimes: treatment 1 with its own mediator value and treatment 0 with its own mediator value.

Result: For the specified event $k$, horizon $\tau$, and reference choice above, $$ \begin{aligned} &E\!\left[Y_k^{1,M^1}(\tau)\right] -E\!\left[Y_k^{0,M^0}(\tau)\right]\\ &\quad = \operatorname{NIE}_{1,k}(\tau) +\operatorname{NDE}_{0,k}(\tau). \end{aligned} $$

To verify the identity, add and subtract the same counterfactual mean risk, $E[Y_k^{1,M^0}(\tau)]$. This is an algebraic step, not an assumption about how the effects can be estimated.

\begin{aligned} &E[Y_k^{1,M^1}(\tau)]-E[Y_k^{0,M^0}(\tau)]\\ &=\{E[Y_k^{1,M^1}(\tau)]-E[Y_k^{1,M^0}(\tau)]\}\\ &\quad+\{E[Y_k^{1,M^0}(\tau)]-E[Y_k^{0,M^0}(\tau)]\}\\ &=\operatorname{NIE}_{1,k}(\tau)+\operatorname{NDE}_{0,k}(\tau). \end{aligned}

There is another exact decomposition using $M^1$ as the reference for the direct effect and treatment 0 as the reference for the indirect effect:

\begin{aligned} \operatorname{NDE}_{1,k}(\tau) &=E[Y_k^{1,M^1}(\tau)]-E[Y_k^{0,M^1}(\tau)],\\ \operatorname{NIE}_{0,k}(\tau) &=E[Y_k^{0,M^1}(\tau)]-E[Y_k^{0,M^0}(\tau)]. \end{aligned}

These alternative natural effects also sum to the same total effect. They need not equal the effects using $M^0$ and treatment 1 as references. When effects of treatment and mediator interact, the allocation of the total contrast between direct and indirect parts can depend on the reference choice. Therefore, report the definitions and reference choices, not just the labels “direct” and “indirect.”

Worked Example: Calculating Both Natural Decompositions

Suppose hypothetical population mean risks by two years for the target event are $E[Y_k^{0,M^0}]=0.07$, $E[Y_k^{1,M^0}]=0.10$, $E[Y_k^{0,M^1}]=0.11$, and $E[Y_k^{1,M^1}]=0.16$.

Using $M^0$ as the direct-effect reference, the natural direct effect is $0.10-0.07=0.03$, or 3 percentage points. The natural indirect effect with treatment reference 1 is $0.16-0.10=0.06$, or 6 percentage points. Their sum, $0.03+0.06=0.09$, equals the total effect $0.16-0.07=0.09$, or 9 percentage points.

Using $M^1$ as the direct-effect reference, the direct effect is $0.16-0.11=0.05$, or 5 percentage points. The indirect effect with treatment reference 0 is $0.11-0.07=0.04$, or 4 percentage points. These also sum to 9 percentage points. The total is unchanged, but the decomposition differs: in this example, the reference choice allocates different portions of the total risk difference to the direct and indirect effects.

Each subtraction compares population probabilities of the target event occurring first by two years. It is not a comparison of risks among people who remained event-free, nor does it remove the competing event.

What the Competing Event Contributes

Because $Y_k^{a,m}(\tau)$ is defined through the first event, it can change when a treatment or mediator intervention changes the competing event, even if the target-event process itself is unchanged. This is part of the cause-specific risk effect, not a defect in the definition. A natural effect on this outcome includes pathways through the mediator that alter the competing event and thereby alter whether event $k$ occurs first by $\tau$.

Worked Example: A Mediator Changes Competing-Event Risk

Consider hypothetical mean risks by one year under the four treatment-mediator settings. Suppose the target-event risks are 0.08 under $(0,M^0)$, 0.12 under $(1,M^0)$, and 0.18 under $(1,M^1)$. Suppose the corresponding competing-event risks are 0.10, 0.20, and 0.25. These risks refer to mutually exclusive first events by the horizon.

The natural direct effect with reference $M^0$ is $0.12-0.08=0.04$, a 4-percentage-point increase in target-event risk. The natural indirect effect with treatment reference 1 is $0.18-0.12=0.06$, a 6-percentage-point increase. Their sum is 10 percentage points, matching the total target-event risk difference $0.18-0.08=0.10$.

Under the mediator contrast at treatment 1, the competing-event risk also changes, from 0.20 to 0.25. The indirect effect of 6 percentage points is therefore an effect on the target-event cumulative incidence with the competing event present; it is not an effect isolated from changes in competing-event risk. For each setting, the target-event risk plus competing-event risk must be at most 1, with the remainder being the probability of neither event by one year. Here those sums are 0.18, 0.32, and 0.43, respectively, so all three pairs are compatible with probabilities.

Assumptions, Interpretation, and Common Pitfalls

The counterfactual definitions require a meaningful treatment intervention, a meaningful mediator intervention, a mediator measured before the event process, a well-defined event type, and a specified horizon. The event types must be classified as mutually exclusive first events, including a consistent rule for any tied event times. These requirements make the counterfactual outcomes well-defined; they do not by themselves identify their mean risks from observed data.

The natural-effect contrasts combine mediator values and treatment settings that may not occur together in observed data. Estimating them from data requires assumptions beyond the algebraic decomposition, including assumptions that justify the relevant cross-world counterfactual comparisons. The effects can also be difficult to interpret if the mediator has multiple versions or if interventions that set it to a given value are not meaningful. Defining a natural effect is not evidence that these requirements hold.

Keep the target event and horizon explicit. A direct effect on the risk of event $k$ by $\tau$ is not automatically a direct effect on the competing event, on the probability of any event, or on a hazard. The earlier material on treatment-specific cumulative incidence and controlled direct effects distinguishes risk contrasts and fixed-mediator interventions; here the key distinction is that a natural effect uses each person’s counterfactual mediator value under a specified treatment.

Contrast	Mediator value used	What it compares
Total effect on target-event risk	$M^1$ versus $M^0$, paired with their treatments	$E[Y_k^{1,M^1}]$ versus $E[Y_k^{0,M^0}]$
Natural direct effect with $M^0$ reference	$M^0$ in both terms	Treatment 1 versus 0 for the event outcome
Natural indirect effect with treatment 1 reference	$M^1$ versus $M^0$	Mediator shift under treatment 1 for the event outcome
Controlled direct effect	A fixed value $m$ in both terms	Treatments compared after setting the mediator to the same specified value

A frequent mistake is to describe the natural indirect effect as the effect of changing the mediator while “holding the competing event fixed.” That is not what the contrast does: both event types are generated under each treatment-mediator setting, and the target outcome records whether the target event occurs first. Another mistake is to substitute one common mediator value for $M^0$ or $M^1$; that changes the estimand to a controlled direct effect or another intervention contrast.

Key takeaway: For natural effects with competing events, define the outcome as the target event occurring first by a specified horizon. Then state explicitly which treatment generates the event process and whether the mediator in each term is $M^0$ or $M^1$. Both target and competing event pathways remain part of the risk contrast.

Check Your Understanding

Use the definitions above. The solutions follow the questions.

In $Y_k^{0,M^1}(\tau)$, which treatment is used for the event process, and which treatment determines the mediator value?
Suppose $E[Y_k^{1,M^0}]=0.14$ and $E[Y_k^{0,M^0}]=0.09$. Calculate the natural direct effect with $M^0$ reference and interpret it.
If $E[Y_k^{1,M^1}]=0.20$ and $E[Y_k^{1,M^0}]=0.14$, calculate the natural indirect effect with treatment reference 1.
Why is a competing event before $\tau$ coded as $Y_k^{a,m}(\tau)=0$ rather than treated as ordinary censoring?
Must the natural direct effect using $M^0$ equal the natural direct effect using $M^1$? Explain.

Solutions. (1) Treatment 0 is used for the event process, while $M^1$, the mediator value under treatment 1, is used. (2) The effect is $0.14-0.09=0.05$, a 5-percentage-point increase in target-event risk by the specified horizon under the stated natural direct-effect contrast. (3) The effect is $0.20-0.14=0.06$, a 6-percentage-point increase in target-event risk under the mediator contrast at treatment 1. (4) The outcome is whether the target event occurs first by $\tau$; a prior competing event prevents that outcome, so the indicator is 0. (5) No. The treatment contrasts use different mediator reference values, and the resulting direct effects can differ, although each valid direct-indirect pair sums to the same total risk difference.

Contrast	Mediator value used	What it compares
Total effect on target-event risk	\(M^1\) versus \(M^0\), paired with their treatments	\(E[Y_k^{1,M^1}]\) versus \(E[Y_k^{0,M^0}]\)
Natural direct effect with \(M^0\) reference	\(M^0\) in both terms	Treatment 1 versus 0 for the event outcome
Natural indirect effect with treatment 1 reference	\(M^1\) versus \(M^0\)	Mediator shift under treatment 1 for the event outcome
Controlled direct effect	A fixed value \(m\) in both terms	Treatments compared after setting the mediator to the same specified value