Why Competing Events Matter at Selection
An intermediate outcome is useful for arm selection only when its meaning is clear. Suppose a trial team wants to compare arms on time to disease progression at an interim selection look. Death before progression is not ordinary missing follow-up: a participant who has died can no longer experience progression. Death is therefore a competing event. The observed probability of progression depends both on the progression process and on how often the competing event occurs.
This distinction matters in a multi-arm, multi-stage (MAMS) trial. A selection rule based on progression may rank an arm differently depending on whether it uses the progression hazard among participants still event-free or the probability of progression by the selection horizon. A ranking based on either measure may, in turn, disagree with the ranking for a later endpoint. These measures answer different questions; neither alone guarantees a correct later-endpoint selection.
In notation, write the cause-specific hazard as \(\lambda_{ka}(t)\), and the probability of remaining free of every event through time \(t\) as \(S_a(t)=P(T>t\mid a)\). The cause-specific hazard is conditional on still being event-free; the CIF is an absolute probability in the arm’s original population. The hazard is not itself a probability, and a lower cause-specific hazard does not necessarily imply a lower CIF if competing-event hazards differ between arms.
How the CIF Incorporates Competing Events
For continuous event times, the probability of remaining event-free is determined by all cause-specific hazards together. If the hazards are well-defined and the event-time distribution is continuous, then
The second expression follows by considering a short time interval. Among the proportion \(S_a(u)\) still event-free at time \(u\), the probability of an event of cause 1 in an interval of width \(du\) is approximately \(\lambda_{1a}(u)du\). Thus the probability mass for the event of interest in that interval is approximately \(S_a(u)\lambda_{1a}(u)du\). Adding this mass over time gives the integral. Because \(S_a(u)\) depends on every cause-specific hazard, competing events affect the CIF even if the hazard of the event of interest is unchanged.
With observed event counts at distinct event times, the corresponding nonparametric calculation is the Aalen–Johansen estimator. Just before event time \(t_j\), let \(n_j\) be the number still event-free and \(d_{1j}\) the number of cause-1 events. If \(\widehat S(t_j-)\) is the estimated event-free probability just before \(t_j\), the estimated CIF increases by \(\widehat S(t_j-)d_{1j}/n_j\). Events of competing causes reduce the event-free probability available to contribute to later cause-1 events.
These calculations assume that causes are recorded and defined consistently, each participant has at most one first-event cause, and loss to follow-up is handled appropriately. For the ordinary Aalen–Johansen estimate, censoring must be independent of the event process (or conditionally independent given variables included in an appropriate analysis). Randomization supports arm comparisons at baseline, but it does not eliminate the need to assess censoring, event ascertainment, and follow-up.
Worked Example: Constant Hazards Give Different Arm Rankings
Consider a hypothetical randomized trial with two arms and two mutually exclusive first-event causes: progression (cause 1) and death before progression (cause 2). Assume independent participants, complete follow-up to 5 years, and constant cause-specific hazards in each arm. Suppose arm A has progression hazard \(0.10\) per year and death hazard \(0.20\) per year; arm B has corresponding hazards \(0.12\) and \(0.02\) per year.
For constant hazards, \(F_1(t)=\{\lambda_1/(\lambda_1+\lambda_2)\}\{1-\exp[-(\lambda_1+\lambda_2)t]\}\). At 5 years, arm A has
For arm B,
The cause-specific progression hazard is lower in A (0.10 versus 0.12 per year), and the observed 5-year progression CIF is also lower in A (25.9% versus 43.2%). If fewer progressions is the desired intermediate outcome, both measures rank A ahead of B. But death is much more frequent in A. Under the same model, the probability of remaining free of both events at 5 years is \(e^{-0.30(5)}=0.2231\) in A and \(e^{-0.14(5)}=0.4966\) in B. Thus B has higher event-free survival at that horizon. An intermediate-outcome ranking does not automatically establish which arm is better on a later endpoint; competing causes and the later endpoint’s clinical meaning must be considered.
Estimating Cumulative Incidence From Event Counts
When event times are recorded, the CIF is built up from the event-free risk set over time. Competing events are counted as events, not censored observations, in this calculation. The following small example has no loss to follow-up before the last listed event time, so the risk sets and calculations can be read directly.
Worked Example: Aalen–Johansen CIF From Two Event Times
In a hypothetical trial, 100 participants in each arm are followed through two event times. The event of interest is progression; death before progression is competing. Assume independent participants, complete ascertainment through the second time, and no other event types.
| Arm | At first time: progression | At first time: death | At second time: progression | At second time: death |
|---|---|---|---|---|
| A | 10 | 20 | 7 | 7 |
| B | 15 | 2 | 10 | 2 |
For A, the first-time progression increment is \(1.00(10/100)=0.10\). After 30 events of either cause, the event-free estimate is \(70/100=0.70\). At the second time, the progression increment is \(0.70(7/70)=0.07\). Therefore the estimated progression CIF through the second time is \(0.10+0.07=0.170\), or 17.0%.
For B, the first-time increment is \(1.00(15/100)=0.15\), leaving an event-free estimate of \(83/100=0.83\). At the second time, the increment is \(0.83(10/83)=0.10\). The estimated CIF is \(0.15+0.10=0.250\), or 25.0%. At this horizon, A has the lower estimated progression probability. This is a descriptive arm comparison; with samples this small, uncertainty would be substantial and should be reported rather than ignored.
Choosing the Intermediate-Outcome Quantity
The choice of quantity should follow the selection question. A cause-specific hazard describes the instantaneous event rate among those who remain event-free. The CIF describes the probability of observing the event of interest by a specified horizon in the presence of the competing events that actually occur. A fixed-horizon selection question such as “Which arm has the lower probability of progression by the interim horizon?” calls for a CIF, not just a hazard ratio.
| Quantity | Question answered | Key caution for arm selection |
|---|---|---|
| Cause-specific hazard | Among those still event-free, what is the instantaneous rate of this event? | Risk sets may differ in composition across arms; the hazard is not an absolute event probability. |
| Cumulative-incidence function | What proportion experiences this cause by a specified time, before a competing cause? | It reflects hazards for all competing causes, not just the cause of interest. |
| Composite endpoint | What is the probability or rate of experiencing any included event? | Combining causes changes the clinical question and may hide differences between causes. |
| One minus Kaplan–Meier with competing events censored | What is the net-risk quantity obtained when competing events are treated as censoring? | It is not the observed probability of the event in the presence of competing risks. |
Worked Example: Why Censoring Competing Events Can Mislead
Use the hypothetical constant-hazard model from the first example, where the progression hazards are 0.10 per year in A and 0.12 per year in B. Suppose someone censors deaths and calculates one minus the Kaplan–Meier estimate for progression. In this constant-hazard setting, that calculation gives the net-risk values \(1-e^{-\lambda_1t}\). At 5 years they are
These values exceed the observed progression CIFs of 0.2590 and 0.4315 from the first example. The Kaplan–Meier calculation treats deaths as if they were censoring rather than events that remove participants from the possibility of later progression. Its result is therefore not the observed probability of progression by 5 years in either arm. Interpreting 39.4% and 45.1% as actual progression probabilities would be incorrect. A net-risk interpretation also requires assumptions beyond simply computing the curve, so it should not be substituted for the CIF without a clearly stated target question.
Using the Results in MAMS Selection
At an interim selection look, specify the event of interest, the competing causes, the time horizon, and the direction of benefit before comparing arms. For a time-to-progression intermediate outcome, for example, state whether the selection summary is a cause-specific hazard contrast or the progression CIF at the selection horizon. If death before progression is clinically important, report its CIF as well; a low progression CIF can arise partly because many participants experience death first.
Then compare the intermediate ranking with the rationale for selecting arms for the later endpoint. An arm can have a favorable intermediate measure yet an unfavorable later outcome if the intermediate measure is an imperfect surrogate, if competing events affect the measure, or if the intermediate and later endpoints capture different clinical effects. The earlier material on surrogacy and on restricted mean survival time at an interim landmark provides related ways to think about selection, but neither makes the competing-risk distinction disappear. A numerical association between an intermediate measure and a later endpoint is not, by itself, proof that the intermediate ranking predicts the later ranking.
A cause-specific hazard model may be useful for describing cause-specific rates, while a CIF directly summarizes absolute event probabilities. A composite endpoint can avoid treating one component as “competing” when the clinical question concerns any event, but it answers a different question and can conceal a harmful component. The method is not selected by convenience: it is selected by the estimand the trial needs.
Specify the intermediate event of interest, each event that precludes it, the time origin, and the selection horizon.
Use a cause-specific hazard for an instantaneous rate among event-free participants, or a CIF for the probability of the event by a stated time in the presence of competing events.
Review cause coding, censoring patterns, risk-set counts, and whether follow-up is comparable across arms.
Report competing-event experience and uncertainty, then explain why the chosen intermediate measure is relevant to the later endpoint.
Diagnostics, Limitations, and Common Pitfalls
Check whether cause definitions are clinically coherent and applied consistently across arms. Inspect event counts by cause and time, numbers at risk, and the amount and timing of censoring. A CIF based on few participants remaining at risk late in follow-up may be unstable. If censoring differs markedly by arm or is related to prognosis, a simple nonparametric comparison may be biased; address this with an appropriate analysis and sensitivity assessment rather than relying on the displayed curve alone.
Cause-specific hazards can also be difficult to interpret as treatment effects. The event-free participants at a given time are selected by their prior event histories, and the composition of these risk sets may differ across arms. A hazard contrast is not generally a risk ratio or a causal effect among a common fixed group of participants. Conversely, CIF differences reflect both the event-of-interest process and competing-event processes. These features are not defects; they are consequences of asking different questions.
- Pitfall: censoring competing events and calling one minus Kaplan–Meier the observed risk. This treats participants who can no longer experience the event of interest as if they remained eligible for it. Use a CIF to describe the observed probability in the presence of competing events.
- Pitfall: treating the cause-specific hazard as an absolute probability. A hazard is an instantaneous rate among those still event-free. State the horizon and use a CIF when the question concerns the probability of an event by that time.
- Pitfall: choosing an arm from the intermediate ranking alone. A favorable intermediate measure does not guarantee a favorable later endpoint. State the selection rationale and show how competing events may influence the ranking.
No competing-risk summary resolves uncertainty about whether the intermediate outcome is a useful guide to the later endpoint. Small interim samples, immature follow-up, uncertain cause classification, and changes in treatment effects over time can all weaken a ranking. Prespecify the estimand and selection use, and report uncertainty alongside point estimates.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- In one sentence, distinguish the cause-specific hazard for progression from the progression CIF at 2 years.
- At an event time, 80 people are event-free just before the time; 8 have a progression event and 12 have a competing event. If the estimated event-free probability just before the time is 0.60, what is the progression CIF increment?
- Why is it generally incorrect to censor deaths and interpret one minus Kaplan–Meier as the observed probability of progression?
- Two arms have equal progression cause-specific hazards, but one arm has a higher death hazard before progression. Explain how their progression CIFs can differ.
Solutions. (1) The cause-specific hazard is the instantaneous progression rate among participants still event-free at 2 years; the CIF is the probability of progression by 2 years before a competing event. (2) The increment is \(0.60(8/80)=0.060\), or 6.0 percentage points. (3) Death prevents later progression, so treating it as censoring does not estimate the observed event probability in the presence of death; the CIF accounts for removal from the event-free risk set. (4) The higher death hazard reduces the number remaining at risk for progression, so the arm with more deaths can have a lower progression CIF even when progression hazards are equal. The CIF depends on both hazards.