Causal Contrasts With Competing Events · Tutorial 167 of 699

Treatment-Specific Cumulative Incidence as a Causal Outcome

Learn to define and interpret a cause-specific cumulative incidence under a specified treatment strategy, and understand the assumptions needed to identify it from data.

Advanced 10 min read

What You'll Learn

Define a treatment-specific cumulative incidence for a chosen event type and fixed time.
Distinguish the counterfactual outcome from a cause-specific hazard and from risk among survivors.
Derive the cumulative-incidence formula from cause-specific hazards under a treatment strategy.
Identify assumptions linking counterfactual cumulative incidence to observed data.
Calculate the outcome using discrete probabilities, continuous hazards, and covariate standardization.
Recognize common interpretive errors and diagnose threats to identification.

Why Define the Outcome Before Comparing Treatments?

When several event types can occur, “risk by one year” is incomplete unless the event of interest is specified. For example, a treatment may affect both a disease event and death before that event. A clear causal outcome therefore names the treatment strategy, event type, and time horizon. This tutorial defines that outcome; it does not yet define a contrast between treatments.

The earlier material on counterfactual event times and causes supplies the basic potential-outcome framework. For each person, imagine the event time and event type that would occur under each treatment strategy. The treatment-specific cumulative incidence is a population probability calculated from those counterfactual outcomes.

Definition: Let $a$ denote a specified treatment strategy, $T^a$ the counterfactual time to the first event under that strategy, and $J^a$ the type of that event. For event type $k$, the treatment-specific cumulative incidence at fixed time $t$ is $$ F_k^a(t)=P(T^a\leq t,\ J^a=k). $$ It is the probability that, under strategy $a$, a person experiences event type $k$ by time $t$, before any other event type occurs.

The event types are taken to be mutually exclusive first events. If $K$ event types are possible, $J^a$ takes one of the values $1,\ldots,K$ when an event occurs. A person who experiences a competing event first does not subsequently have $J^a=k$ as their first-event type, even if they might otherwise have experienced event $k$ later. The time $t$ must be stated in the units used to measure follow-up, such as years after treatment assignment.

This is a marginal probability in the population under the specified strategy. It is not the probability of event $k$ among people who remain event-free at time $t$, nor is it a hazard among people still at risk. The distinction matters because the cumulative incidence accounts for all ways people can leave the event-free state.

Interpreting the Counterfactual Quantity

A treatment strategy must be sufficiently well defined to give meaning to $T^a$ and $J^a$. A simple example is assignment at baseline to a specified treatment. A strategy may also specify a treatment regime over time, but then its rules—including what happens when a clinical condition occurs—must be clear. Different versions of treatment or poorly specified adherence rules can make “under treatment $a$” ambiguous.

The event of interest and the competing events are part of the outcome definition, not nuisances to be removed from it. For instance, if $k$ is hospitalization and death is a competing first event, then $F_k^a(t)$ counts hospitalization by $t$ only when hospitalization occurs before death. It does not ask what the hospitalization probability would have been if death were prevented.

Worked Example: Reading a Treatment-Specific Probability

Consider a hypothetical population under a clearly specified strategy $a$, followed for one year. Suppose the counterfactual outcomes in that population are: 12% experience the target event first by one year, 8% experience a competing event first, and 80% remain free of either event through one year.

For the target event type $k$, the definition gives $$ F_k^a(1)=P(T^a\leq 1,\ J^a=k)=0.12. $$ Thus the treatment-specific cumulative incidence is 0.12, or 12%, by one year.

The denominator is the whole population under strategy $a$, not just the 92% who avoid the competing event. The 8% with a competing event are not classified as target-event cases by one year. The three outcome probabilities sum to $0.12+0.08+0.80=1.00$, as required in this simplified setting with complete one-year outcome ascertainment.

Connection to Cause-Specific Hazards

A useful representation expresses the cumulative incidence through the cause-specific hazards under the same strategy. Let $\lambda_j^a(u)$ be the cause-specific hazard for event type $j$ at time $u$ under strategy $a$: the instantaneous event rate for type $j$ at $u$, conditional on remaining free of every event immediately before $u$. For continuous event times, assume these hazards exist and there are no event-time atoms or tied first events. The event-free survival probability is $$ S^a(u)=P(T^a>u)=\exp\left\{-\int_0^u\sum_{j=1}^K\lambda_j^a(v)\,dv\right\}. $$ Then $$ F_k^a(t)=\int_0^t S^a(u)\lambda_k^a(u)\,du. $$

The formula says that a target event at time $u$ requires two things: the person must have remained free of all event types until $u$, and then the target event must occur at that time. The probability of the first condition is $S^a(u)$; the instantaneous contribution of the second is $\lambda_k^a(u)\,du$. Summing these contributions over times from 0 to $t$ yields the cumulative incidence.

This representation also shows why competing events matter. The hazards of every event type appear in $S^a(u)$. Increasing a competing-event hazard can reduce the number of people still able to experience the target event, even if the target cause-specific hazard itself is unchanged. The cumulative incidence is therefore not obtained by integrating the target hazard alone.

Worked Example: Constant Hazards for Two Event Types

Suppose, hypothetically, that under strategy $a$ there are two event types and constant cause-specific hazards over one year: $\lambda_k^a(u)=0.08$ per year for the target event and $\lambda_2^a(u)=0.12$ per year for the competing event. These hazards are assumed constant and defined among people still free of either event. The total hazard is $0.20$ per year, so $S^a(u)=\exp(-0.20u)$.

Applying the cumulative-incidence formula gives $$ \begin{aligned} F_k^a(1) &=\int_0^1 \exp(-0.20u)(0.08)\,du\\ &=\frac{0.08}{0.20}\left(1-\exp(-0.20)\right)\\ &=0.4\left(1-0.8187308\right)\\ &\approx 0.07251. \end{aligned} $$

The hypothetical target-event probability by one year is approximately 0.07251, or 7.251%. For a check, the competing-event probability is $$ F_2^a(1)=\frac{0.12}{0.20}\left(1-\exp(-0.20)\right)\approx 0.10876, $$ and the event-free probability is $S^a(1)=\exp(-0.20)\approx0.81873$. Their sum is approximately $0.07251+0.10876+0.81873=1.00000$, allowing for rounding.

Justification: In a short interval of width $du$, conditional on being event-free just before $u$, the probability of a type-$j$ event is $\lambda_j^a(u)du+o(du)$. The total event hazard is $\sum_j\lambda_j^a(u)$, so the event-free probability solves $dS^a(u)/du=-S^a(u)\sum_j\lambda_j^a(u)$, with $S^a(0)=1$. Solving gives the exponential expression above. The probability of a type-$k$ event in that interval is $S^a(u)\lambda_k^a(u)du+o(du)$; summing these increments from 0 to $t$ gives the integral formula.

When Can the Counterfactual Probability Be Identified?

The definition is causal because it refers to outcomes under a specified strategy, but a definition alone does not make the probability estimable from observed data. Identification requires assumptions connecting the counterfactual outcomes to the data. For a baseline treatment $A$, a common set of assumptions is:

Consistency: for a person who follows strategy $a$, the observed event time and type equal $T^a$ and $J^a$. The strategy and its relevant treatment versions must be well defined.
Exchangeability: conditional on measured baseline covariates $X$, treatment assignment is independent of the counterfactual event time and type under each strategy. Random assignment supports this condition by design; in observational data, it is an assumption about adequate control of confounding.
Positivity: for each covariate pattern included in the target population, each strategy whose outcome is estimated has positive probability of being received. Otherwise the relevant counterfactual outcome is unsupported by observed data.
Appropriate censoring assumptions: if follow-up is censored before $t$, censoring must be independent of the event outcomes conditional on the variables used to handle it, with adequate probability of remaining observed. Complete ascertainment through $t$ avoids this additional requirement.

Under these assumptions, a baseline-covariate standardization identity is $$ F_k^a(t)=\sum_x P(T\leq t,J=k\mid A=a,X=x)\,P(X=x), $$ for discrete $X$; with continuous covariates, the sum is replaced by an integral over their target-population distribution. The conditional observed cumulative incidences on the right must account for competing events and censoring appropriately. The identity averages stratum-specific risks under $a$ over the covariate distribution of the target population.

Worked Example: Standardizing Across Two Covariate Strata

Suppose an observational dataset is being used to estimate the one-year cumulative incidence of a target event under strategy $a=1$. Let $X$ denote a baseline risk group. In a hypothetical target population, 60% have $X=0$ and 40% have $X=1$. Among those receiving strategy 1, assume the observed one-year target-event cumulative incidences are 12% in $X=0$ and 24% in $X=1$. Assume the competing events are correctly counted as competing first events, follow-up is complete to one year, there is treatment positivity in both strata, and conditional exchangeability holds given $X$.

Standardize the stratum-specific probabilities to the target population: $$ \begin{aligned} F_k^{1}(1) &=P(X=0)P(T\leq1,J=k\mid A=1,X=0)\\ &\quad+P(X=1)P(T\leq1,J=k\mid A=1,X=1)\\ &=(0.60)(0.12)+(0.40)(0.24)\\ &=0.072+0.096\\ &=0.168. \end{aligned} $$

The standardized treatment-specific cumulative incidence is 0.168, or 16.8% by one year in the target population under strategy 1. The calculation is justified only under the stated assumptions; if important confounders are omitted, the conditional observed probabilities need not represent the corresponding counterfactual probabilities.

Diagnostics, Alternatives, and Limits

Several checks help reveal when an analysis or interpretation is going wrong. First, verify that event types are mutually exclusive first events and that the event coding matches the scientific question. Second, inspect whether each treatment strategy is represented across the covariate strata being standardized. Third, examine follow-up and censoring: sparse observation near the chosen horizon or highly treatment-dependent loss to follow-up can make the estimated probability unstable or biased. Finally, check that the estimated probabilities are between 0 and 1 and that the cumulative incidence does not decrease with time.

The cumulative incidence is one possible outcome definition, not a universal summary of treatment effect. A cause-specific hazard describes an instantaneous rate among those still event-free; an all-event probability combines event types; and a hypothetical event probability that removes competing events answers a different question. The treatment-specific cumulative incidence retains competing events as events that prevent the target first event from occurring. Which quantity is relevant depends on the scientific question and the intervention being considered.

A common alternative calculation treats competing events as if they were ordinary censoring and uses one minus a Kaplan–Meier estimate for the target cause. That quantity generally does not equal the observed-world cumulative incidence: it behaves as though people with competing events could remain at risk for the target event. The correct cumulative incidence incorporates the competing-event hazards in the event-free survival probability, as the derivation shows.

Common pitfall: “The target-event risk among those who avoid the competing event is the treatment-specific cumulative incidence.” This is wrong because it conditions on a post-treatment event-free status and changes the denominator. The cumulative incidence is $P(T^a\leq t,J^a=k)$ in the full population under strategy $a$, with competing first events counted as outcomes that preclude the target first event by $t$.

The principal limitation is that counterfactual outcomes are not jointly observed for the same person under multiple strategies. Assumptions and study design are therefore needed to identify each treatment-specific probability. Even when identified, a single fixed-time probability does not describe the full timing pattern, and a different time horizon or event definition yields a different estimand. Report the strategy, event type, population, and time horizon together so the quantity is interpretable.

Check Your Understanding

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

In a study where death is a competing first event, what does $F_k^a(2)$ mean if $k$ denotes hospitalization?
Under constant hazards $\lambda_k^a=0.10$ and $\lambda_2^a=0.10$ per year, calculate the target-event cumulative incidence by one year.
Why can the target cause-specific hazard not, by itself, determine the target cumulative incidence?
A covariate stratum has no people receiving strategy $a$. Which identification condition fails, and what is the consequence?
Why is treating every competing event as ordinary censoring generally not a calculation of the treatment-specific cumulative incidence?

Solutions. (1) It is the probability under strategy $a$ of hospitalization as the first event by two years, in the full population under that strategy. (2) The total hazard is $0.20$ per year, so $F_k^a(1)=\frac{0.10}{0.20}(1-\exp(-0.20))=0.5(0.1812692)\approx0.09063$, or 9.063%. (3) The probability of remaining event-free depends on the hazards of all event types, so competing-event hazards also affect the integral $F_k^a(t)=\int_0^t S^a(u)\lambda_k^a(u)\,du$. (4) Positivity fails; the stratum provides no observed support for estimating its outcome under $a$ without additional modeling or extrapolation. (5) Such censoring acts as if people with competing events could still experience the target event, whereas the cumulative incidence counts only target events that occur first.

Back to Tutorials