Competing-Risk Outcomes in a Time-to-Event Trial

Why Competing Events Change the Question

In a time-to-event trial, a participant may experience the event of clinical interest, a different event that prevents it from occurring, or neither event before follow-up ends. For example, a trial might study time to clinical improvement while death before improvement is a competing event. Death is not simply missing follow-up: after death, that participant cannot later improve. The probability of improvement by a specified horizon therefore depends on both the chance of improvement and the chance of the competing event.

The target of this tutorial is the cumulative incidence of a specified event type: the probability that this event has occurred by a given time, in the presence of the other event types. The analysis will use a discrete-time Bayesian model. Its central operation is to estimate the probability of each event type among participants still event-free in each interval, then combine those probabilities over time.

Definition: A competing-risks outcome has mutually exclusive event types. For event type $k$, its cumulative incidence at time $t$, written $F_k(t)$, is the probability that a participant experiences event type $k$ by $t$, before experiencing any other event type. A competing event removes a participant from the risk of the event of interest; it is an observed outcome, not ordinary censoring.

A Discrete-Time Bayesian Model

Divide follow-up into intervals $j=1,\ldots,J$. In each treatment arm, a participant who is event-free at the start of interval $j$ has exactly one of three interval outcomes: event of interest, competing event, or no event during the interval. Let $q_{1j}$, $q_{2j}$, and $q_{0j}$ denote the corresponding conditional probabilities. They sum to one. These are interval probabilities conditional on being event-free at the interval’s start; they are not probabilities conditional on having survived the entire study.

For a simple model, assume the event probabilities are constant within each interval and that, conditional on treatment arm and the specified model, the interval outcome probabilities apply to all participants at risk in that interval. Assign each interval an independent Dirichlet prior, $(q_{1j},q_{2j},q_{0j})\sim\text{Dirichlet}(a_{1j},a_{2j},a_{0j})$. If $y_{1j}$, $y_{2j}$, and $y_{0j}$ are the observed counts in those three categories among participants with outcome status observed through the interval, the posterior is Dirichlet with parameters $a_{kj}+y_{kj}$.

The simple multinomial likelihood requires careful follow-up accounting. A participant known to be event-free through the interval contributes to the no-event count. A participant who experiences either event contributes to that event’s count and leaves subsequent risk sets. A participant censored before the interval outcome is known is not counted as event-free. The examples assume censoring only at interval boundaries, so participants’ interval outcomes are ascertainable. With censoring within an interval, the likelihood must incorporate the partial follow-up; treating that participant as a full-interval no-event observation would be incorrect. Independent censoring, conditional on the variables included in the analysis, is needed for the observed follow-up to support inference about the trial population.

Key result: Let $S_j$ be the probability of remaining free of all event types through the end of interval $j$, with $S_0=1$. Then $$ S_j=S_{j-1}q_{0j}, \qquad F_{1j}=F_{1,j-1}+S_{j-1}q_{1j}, \qquad F_{2j}=F_{2,j-1}+S_{j-1}q_{2j}, $$ where $F_{10}=F_{20}=0$. The event of interest can occur in interval $j$ only among those still event-free at its start.

The cumulative-incidence expression follows by partitioning the event of interest according to the interval in which it first occurs: $$ F_{1J}=\sum_{j=1}^{J}\left(\prod_{\ell=1}^{j-1}q_{0\ell}\right)q_{1j}. $$ For example, the contribution from interval 2 is $q_{01}q_{12}$: the participant must have no event in interval 1, then experience event type 1 in interval 2. This reasoning includes competing events in the probability of leaving the event-free group. Replacing them with ordinary censoring would instead remove their contribution to the event-free survival calculation and would answer a different, generally hypothetical question.

Worked Examples

Worked Example: Posterior Event Probabilities in One Interval

Suppose a hypothetical trial follows 100 participants in one arm through a fixed horizon. By that horizon, 20 have the event of interest, 15 have a competing event, and 65 remain event-free. Use the prior $\text{Dirichlet}(1,1,1)$, in the order event of interest, competing event, and no event.

The posterior is $\text{Dirichlet}(21,16,66)$, because each prior parameter is increased by its corresponding count. The posterior mean probability of the event of interest by the horizon is therefore $$ E(q_1\mid\text{data})=\frac{21}{21+16+66}=\frac{21}{103}=0.2039. $$ The posterior mean for the competing event is $16/103=0.1553$, and for remaining event-free it is $66/103=0.6408$. These three means sum to one. Under this one-interval model, $F_1=q_1$, so the posterior mean cumulative incidence is 20.39% by the stated horizon.

This is an absolute event probability that retains the observed competing outcomes. It is not a claim that the competing-event participants would have had the same event probability as those remaining event-free.

Worked Example: Cumulative Incidence Across Two Intervals

In a separate hypothetical arm, 100 participants are event-free at the start of interval 1. The interval outcomes are 10 events of interest, 5 competing events, and 85 with no event. Of the 85 who remain event-free, 85 have outcome status observed through interval 2: 8 experience the event of interest, 7 experience the competing event, and 70 remain event-free. Use independent $\text{Dirichlet}(1,1,1)$ priors in each interval.

The interval 1 posterior parameters are $(11,6,86)$, with total 103. In interval 2 they are $(9,8,71)$, with total 88. The posterior mean probabilities are thus $E(q_{11})=11/103$, $E(q_{01})=86/103$, and $E(q_{12})=9/88$. The mean cumulative incidence by the end of interval 2 is $$ E(F_{12}\mid\text{data})=\frac{11}{103}+\frac{86}{103}\frac{9}{88} =0.1068+0.0854=0.1922. $$ The multiplication in the second term is essential: the interval 2 event probability applies only to participants who were event-free through interval 1.

This posterior expectation can be calculated by substituting posterior means because the posterior distributions for the two intervals are independent. More generally, posterior uncertainty should be carried through the recursion using posterior draws, rather than using a single set of interval estimates.

Worked Example: Comparing Arms With Posterior Probability

Consider a hypothetical one-interval trial where the event of interest is clinical improvement and the competing event is death before improvement. Use independent $\text{Dirichlet}(1,1,1)$ priors in each arm. In the treatment arm, 2 of 4 participants improve, 1 dies before improvement, and 1 has neither event. In the control arm, 0 of 2 improve, 1 dies before improvement, and 1 has neither event.

The posterior in the treatment arm is $\text{Dirichlet}(3,2,2)$, so the marginal posterior for its improvement probability $p_T$ is $\text{Beta}(3,4)$. The control posterior is $\text{Dirichlet}(1,2,2)$, giving $p_C\sim\text{Beta}(1,4)$. These arm posteriors are independent. Their posterior mean risk difference is $$ E(p_T-p_C\mid\text{data})=\frac{3}{7}-\frac{1}{5}=0.2286. $$ Thus, the posterior mean improvement probability is 22.86 percentage points higher in the treatment arm.

The posterior probability that treatment has the higher improvement probability can also be calculated exactly. For $p_C\sim\text{Beta}(1,4)$, $P(p_C<p_T\mid p_T)=1-(1-p_T)^4$. Since $p_T\sim\text{Beta}(3,4)$, $$ P(p_T>p_C\mid\text{data})=1-E[(1-p_T)^4] =1-\frac{4\cdot5\cdot6\cdot7}{7\cdot8\cdot9\cdot10} =\frac{5}{6}=0.8333. $$ Under this model and prior, the posterior probability that the treatment-arm improvement probability exceeds the control-arm probability is 83.33%. The small sample leaves substantial uncertainty; this probability alone does not establish that a decision threshold has been met.

Posterior Comparisons and Interpretation

For multiple intervals, draw each arm’s interval probabilities from their joint posterior and apply the cumulative-incidence recursion to every draw. This produces a posterior distribution for $F_1(t)$ in each arm and for contrasts such as the risk difference $F_{1,T}(t)-F_{1,C}(t)$. Report a posterior mean or median, a credible interval, and—when relevant—the posterior probability that the contrast exceeds a prespecified clinically meaningful threshold. State the horizon and the event definition: a contrast at 6 months need not match a contrast at 12 months.

The event of interest may be beneficial or harmful, so the direction of a favorable contrast depends on the endpoint. A positive risk difference favors treatment for improvement, but may be unfavorable for a harmful event. The absolute cumulative incidences also matter: the same relative contrast can have different practical importance at different baseline risks. Treatment comparisons require appropriate randomized-arm interpretation and do not by themselves resolve issues such as adherence, missing outcomes, or the target population.

Alternatives, Diagnostics, and Limitations

Approach	What it estimates or describes	Key caution
Bayesian cumulative incidence	Posterior distribution for the probability of each event type by a horizon	Depends on the interval model, prior, and censoring assumptions
Cause-specific hazard analysis	Rate of a particular event among participants still free of all events	A cause-specific hazard is not itself the cumulative probability of that event
Kaplan–Meier with competing events censored	A net-risk quantity under a hypothetical removal of competing events	One minus this curve generally overstates the observed-world event probability when competing events occur
Subdistribution-hazard approach	Models a hazard linked to cumulative incidence while retaining a modified risk set	Its parameterization differs from the interval event-probability model used here

The most important diagnostic is whether the data and risk sets match the model. Confirm that each participant contributes at most one first event, that competing events are assigned their own category, and that no-event counts include only participants observed through the interval. Examine event counts by arm and interval; very sparse intervals can make posterior estimates strongly influenced by the prior. Assess sensitivity to reasonable prior choices, particularly when events are rare, and use posterior predictive checks to compare replicated and observed event counts.

The model also has limits. Constant probabilities within broad intervals can obscure event timing, while very narrow intervals may produce sparse data. Independent interval priors do not borrow information across time or arms. Informative censoring, misclassified event types, and unmodeled differences in prognosis can undermine the analysis. More detailed time-to-event likelihoods or covariate models may be appropriate when these features are important; they still need to represent competing events as distinct outcomes when the target is cumulative incidence.

A Common Mistake

A common error is to censor participants when they experience the competing event, fit a Kaplan–Meier curve for the event of interest, and interpret one minus that curve as the observed-world probability of the event. The calculation effectively treats those competing events as if the participants could remain at risk for the event of interest under a hypothetical scenario. It does not estimate the probability that the event of interest actually occurs in the presence of the competing event. For that probability, include the competing event as an event type and compute cumulative incidence.

Key takeaway: Estimate each event type’s conditional probability among participants still event-free, and accumulate the event of interest only over the probability of remaining event-free to each interval. A competing event is part of this calculation, not a censoring code.

Check Your Understanding

Answer each question, then compare with the solutions below.

In a two-event model, what does $q_{1j}$ condition on, and why can it not be interpreted directly as $F_{1j}$?
Suppose $q_{01}=0.8$, $q_{11}=0.1$, and $q_{12}=0.2$. What is the event-of-interest cumulative incidence through interval 2?
Why should a participant censored before an interval’s outcome is known not be counted in that interval’s no-event category?
In the worked arm-comparison example, what is the posterior probability that control has a higher improvement probability than treatment?

Solutions. (1) $q_{1j}$ is conditional on being event-free at the start of interval $j$. $F_{1j}$ accumulates event-of-interest probability over all intervals, weighted by the probability of remaining event-free until each interval. (2) The first-interval contribution is $q_{11}$; the second is $q_{01}q_{12}$. Thus $F_{12}=0.1+(0.8)(0.2)=0.26$, or 26%. (3) The participant’s interval outcome is unknown, so classifying them as event-free would assert an outcome not observed and distort the likelihood. (4) The posterior probability is $1-5/6=1/6=0.1667$, because the posterior distributions are continuous and ties have probability zero.