Stochastic Interventions on Competing Events

Why Intervene on the Competing-Event Process Probabilistically?

When a competing event prevents the target event from occurring or being observed, a target-event risk depends on both event processes. One possible causal question asks what the target-event risk would be if the competing event were eliminated. That hypothetical intervention may require unsupported extrapolation when some individuals have essentially no chance of remaining free of the competing event. A stochastic intervention asks a different question: what would target-event risk be if the competing-event mechanism were shifted, while still allowing competing events to occur?

The shift must be defined precisely. Here it changes the conditional odds of a competing event among outcomes that are not the target event. The conditional target-event probability is held fixed at each event-free interval; the remaining probability is assigned to no event. This is a particular intervention definition, not the only possible way to modify a competing-event process.

Definition: Consider an individual who is event-free at the start of interval $k$, with treatment $A=a$ and observed history $H_k$. Let $D_k$ be the interval outcome: target event ($D_k=1$), competing event ($D_k=2$), or no event ($D_k=0$). Write $p_{j,k}(H_k,a)=P(D_k=j\mid H_k,A=a,\text{event-free at }k)$. Define $q_k=p_{2,k}/(p_{0,k}+p_{2,k})$, the probability of a competing event among outcomes that are not the target event. For a shift parameter $\delta>0$, the stochastic intervention changes the odds of this probability by the factor $\delta$: $$ q_{k,\delta}=\frac{\delta q_k}{1-q_k+\delta q_k}. $$ It assigns probabilities $p_{1,k,\delta}=p_{1,k}$, $p_{2,k,\delta}=(1-p_{1,k})q_{k,\delta}$, and $p_{0,k,\delta}=(1-p_{1,k})(1-q_{k,\delta})$. The value $\delta=1$ leaves the competing-event mechanism unchanged; $0<\delta<1$ reduces its conditional odds, and $\delta>1$ increases them.

For finite positive $\delta$, the intervention is stochastic wherever the original conditional odds are finite and positive: it does not set the competing-event probability to zero. The limit $\delta\to0$ approaches elimination in histories where $q_k<1$, but that boundary case is not the focus here. At $\delta=1$, the original probabilities are recovered because $q_{k,1}=q_k$, so $p_{2,k,1}=p_{2,k}$ and $p_{0,k,1}=p_{0,k}$.

Risk and Causal Risk Contrast

Let $R_1^{a,\delta}(\tau)$ denote the probability of experiencing the target event by horizon $\tau$, under treatment level $a$ and the competing-event intervention with parameter $\delta$. The superscript denotes a hypothetical intervention regime, not a subgroup defined by observed events. A risk contrast comparing two competing-event regimes at the same treatment level is $R_1^{a,\delta_1}(\tau)-R_1^{a,\delta_0}(\tau)$. It is an absolute difference in target-event risk by $\tau$, measured in probability units (or percentage points when multiplied by 100).

For a simple case with no time-varying covariates and interval-specific transition probabilities that do not depend on prior history beyond remaining event-free, let $p_{j,k,\delta}$ be the probabilities under the intervention in interval $k$. The probability of remaining event-free through interval $k$ is the product of the no-event probabilities up to that interval. The cumulative target-event risk is therefore

R_1^{a,\delta}(K) = \sum_{k=1}^{K} \left\{\prod_{\ell=1}^{k-1}p_{0,\ell,\delta}\right\}p_{1,k}.

Each summand is the probability of having no event in every preceding interval and then experiencing the target event in interval $k$. With measured covariates or richer histories, the transition probabilities depend on the history generated under the intervention. The risk is then obtained by averaging the corresponding sequential transition probabilities over those histories, as in the competing-risk g-formula.

Why the Shift Formula Works

The intervention is constructed by separating the target event from the two non-target outcomes. Among individuals who do not experience the target event in an interval, the observed competing-event odds are $q_k/(1-q_k)$. Multiplying those odds by $\delta$ gives

\frac{q_{k,\delta}}{1-q_{k,\delta}} = \delta\frac{q_k}{1-q_k}.

Solving this equation gives $q_{k,\delta}=\delta q_k/(1-q_k+\delta q_k)$. The target-event probability remains $p_{1,k}$; the total probability left for competing event or no event is $1-p_{1,k}$. Allocating that remainder using $q_{k,\delta}$ gives the stated $p_{2,k,\delta}$ and $p_{0,k,\delta}$. They are valid probabilities because both are nonnegative and $p_{1,k}+p_{2,k,\delta}+p_{0,k,\delta}=p_{1,k}+(1-p_{1,k})=1$.

This construction also clarifies the interpretation. The shift changes the chance of the competing event relative to no event among those who do not have the target event in that interval. It does not directly multiply the overall competing-event probability by $\delta$, and it does not assert that an individual's competing event can be prevented with certainty.

Worked Example: Applying an Odds Shift in One Interval

Suppose, hypothetically, that among individuals event-free at an interval, the probabilities of the target event, competing event, and no event are $0.10$, $0.30$, and $0.60$, respectively. Apply a competing-event odds shift of $\delta=0.5$.

First, among outcomes that are not the target event, the competing-event probability is $q=0.30/(0.30+0.60)=1/3$. Its odds are $(1/3)/(2/3)=0.5$. The intervention halves those odds, giving

q_{0.5} = \frac{0.5(1/3)}{2/3+0.5(1/3)} = \frac{1/6}{5/6} = 0.20.

The remaining probability after the unchanged target-event probability is $1-0.10=0.90$. Thus the modified probabilities are $p_{2,0.5}=0.90(0.20)=0.18$ and $p_{0,0.5}=0.90(0.80)=0.72$, while $p_{1,0.5}=0.10$. They sum to $0.10+0.18+0.72=1.00$. The competing-event probability has fallen from $0.30$ to $0.18$, but it is not zero. In this single interval the target-event risk remains $0.10$, because its conditional probability was defined to remain fixed.

Worked Example: A Two-Interval Risk Contrast

Suppose, hypothetically, that in each of two intervals the probabilities among those still event-free are $p_1=0.10$, $p_2=0.30$, and $p_0=0.60$, regardless of interval. Compare the target-event risk under no shift ($\delta=1$) with the risk under $\delta=0.5$. Assume complete follow-up and that the same transition probabilities apply in both intervals.

Under $\delta=0.5$, the first example gives $p_{1,\delta}=0.10$, $p_{2,\delta}=0.18$, and $p_{0,\delta}=0.72$. The target-event risk by the end of interval 2 is

R_1^{\delta=0.5}(2) = 0.10+(0.72)(0.10) = 0.172.

The first term is the target event in interval 1. The second is no event in interval 1 followed by the target event in interval 2. Under $\delta=1$, the no-event probability remains $0.60$, so

R_1^{\delta=1}(2) = 0.10+(0.60)(0.10) = 0.160.

The risk difference comparing the reduced-competing-event regime with the natural regime is $0.172-0.160=0.012$, or 1.2 percentage points by the end of two intervals. In this hypothetical model, reducing competing events gives more individuals the opportunity to experience the target event later. This calculation is not an estimate of a clinical effect unless the causal assumptions and data analysis justify the intervention interpretation.

Worked Example: Standardizing Across Baseline Risk Groups

Consider a hypothetical population with equal proportions in two baseline groups, $X=0$ and $X=1$. In each of two intervals, assume the conditional probabilities remain constant within each group. For $X=0$, let $(p_1,p_2,p_0)=(0.10,0.20,0.70)$; for $X=1$, let them be $(0.20,0.40,0.40)$. Calculate the standardized target-event risk under $\delta=0.5$ and compare it with the natural-course risk.

For $X=0$, $q=0.20/(0.20+0.70)=2/9$. Therefore $q_{0.5}=(1/9)/(7/9+1/9)=1/8$. The modified probabilities are $p_{2,0.5}=0.90(1/8)=0.1125$ and $p_{0,0.5}=0.90(7/8)=0.7875$. Its two-interval target risk is $0.10+(0.7875)(0.10)=0.17875$.

For $X=1$, $q=0.40/(0.40+0.40)=0.50$, so $q_{0.5}=0.25/(0.50+0.25)=1/3$. The modified probabilities are $p_{2,0.5}=0.80(1/3)=0.2667$ and $p_{0,0.5}=0.80(2/3)=0.5333$. Its two-interval target risk is $0.20+(0.5333)(0.20)=0.3067$, with rounding only at the final step.

Standardizing by the equal group proportions gives a risk of $(0.17875+0.3067)/2=0.2427$, approximately. Under the natural process, the group-specific risks are $0.10+(0.70)(0.10)=0.170$ and $0.20+(0.40)(0.20)=0.280$. Their standardized risk is $(0.170+0.280)/2=0.225$. Thus the standardized contrast is approximately $0.2427-0.225=0.0177$, or 1.77 percentage points by the end of interval 2. This example shows why a population contrast averages risks over the baseline covariate distribution rather than comparing one arbitrary covariate profile.

Assumptions for a Causal Interpretation

The arithmetic definition of a stochastic intervention does not, by itself, make a contrast causal. Identification from observed data requires assumptions about how observed transitions represent counterfactual transitions under the intervention. In a longitudinal setting, the relevant history must contain the measured predictors of both event transitions and subsequent target-event risk, including time-varying covariates when needed.

Consistency and a well-defined intervention. The observed outcome process agrees with the potential outcome process under the regime actually followed, and the intervention specifies how the competing-event mechanism is modified at each history. Versions of the intervention with materially different consequences should not be treated as interchangeable.
Sequential exchangeability. Conditional on the measured history among individuals still event-free, there are no unmeasured common causes of the event transitions and future potential outcomes needed to identify the regime-specific risk. This is the event-process analogue of the no-unmeasured-confounding condition used in causal g-formula analyses.
Positivity and support. The observed data must support the transition probabilities needed to evaluate the intervention across histories that can arise under it. In particular, the conditional probabilities used to define and shift the competing-versus-no-event odds must be estimable, and relevant histories cannot be wholly absent. A stochastic shift can reduce reliance on extreme counterfactual histories compared with deterministic elimination, but it does not guarantee adequate support.
Appropriate handling of censoring. Loss to follow-up must be absent or handled under suitable conditional independent-censoring assumptions, with enough support for continued observation. Censoring is not the competing event and must not be coded as one.

Under these conditions, the observed conditional transition laws can be combined sequentially under the intervention and averaged over the covariate histories it generates. The resulting g-formula identifies $R_1^{a,\delta}(\tau)$; contrasting two such risks identifies the corresponding causal risk difference. If exchangeability or support fails, the computed quantity may still describe a model-based or observed-data calculation, but its causal interpretation is not established.

Diagnostics, Alternatives, and Limitations

Useful checks begin with the transition model and the intervention definition. Verify that each set of modified probabilities is nonnegative and sums to one, that $\delta=1$ reproduces the observed transition probabilities, and that $p_{1,k,\delta}=p_{1,k}$ as specified. Inspect the distribution of histories and estimated transition probabilities; sparse histories or probabilities near zero and one can make sequential estimates unstable. Compare estimated risks across time and groups, and investigate whether a few histories dominate the standardized result. These are warnings about model fit or support, not proofs that assumptions hold.

The main alternatives answer different questions. A natural-course risk retains the observed competing-event process. A hypothetical elimination contrast asks about a regime in which competing events are prevented, a stronger intervention that may face more demanding support requirements. A stochastic shift instead describes a specified probabilistic change, with the shift parameter giving a direct sensitivity or policy scale. Separable-effect analyses rely on a different causal structure and assumptions; they should not be interpreted as equivalent to this odds-shift intervention.

The intervention is also limited by its chosen odds scale and by the decision to hold the target-event probability fixed at each history. A different redistribution rule—for example, changing the target-event transition probability as well—would define a different intervention and generally a different causal contrast. The odds multiplier is not a risk ratio: $\delta=0.5$ halves the odds of a competing event among non-target outcomes, not necessarily its overall probability. Further, an intervention that appears statistically well-defined need not correspond to a feasible clinical policy. The causal question should be stated in terms of the intervention actually represented by the formula.

Common Mistake

A common error is to multiply the overall competing-event probability $p_2$ by $\delta$ and then leave both the target-event and no-event probabilities unchanged. The resulting probabilities generally fail to sum to one. For example, changing $p_2=0.30$ to $0.15$ while retaining $p_1=0.10$ and $p_0=0.60$ gives a total of $0.85$, not 1. The correction is to define the odds shift on $q=p_2/(p_0+p_2)$, preserve $p_1$ as specified, and redistribute the remaining probability between competing event and no event. Another mistake is to interpret a lower competing-event rate as automatically proving a causal increase in target-event risk. That conclusion depends on the assumptions, intervention definition, and calculated risk contrast.

Key takeaway: A stochastic competing-event intervention changes the conditional competing-versus-no-event odds by a specified factor while leaving the target-event probability fixed at each history. Its causal effect is expressed as a contrast between cumulative target-event risks under two explicitly defined intervention regimes.

Check Your Understanding

Answer the questions, then compare your reasoning with the brief solutions.

If $q=0.40$ and $\delta=2$, what is $q_\delta$?
In one interval, why can a competing-event intervention leave target-event risk unchanged while changing competing-event risk?
For two intervals with constant probabilities $p_1=0.20$ and $p_{0,\delta}=0.50$, what is the cumulative target-event risk?
Does $\delta=0.5$ mean the overall competing-event probability is halved? Explain.
Name one assumption needed to interpret a standardized risk contrast causally.

Solutions. (1) $q_2=2(0.40)/(0.60+2(0.40))=0.80/1.40\approx0.5714$. (2) By definition, the intervention holds the conditional target-event probability fixed and redistributes only the remaining probability between competing event and no event. (3) The risk is $0.20+(0.50)(0.20)=0.30$, or 30% by the end of interval 2. (4) No. It halves the odds of a competing event relative to no event among non-target outcomes; the resulting probability change depends on the original probability. (5) Examples include consistency, sequential exchangeability conditional on measured history, adequate positivity, and suitable handling of censoring.