From Event Increments to Cumulative Incidence
A cause-specific event increment describes how much estimated event probability is added at one event time. To estimate the probability of experiencing a particular cause by a horizon, those increments must be accumulated over time. The key is that an increment at a later time applies only to the fraction of the cohort still estimated to be event-free just before that time.
This weighting matters in competing-risks data: an event from another cause removes a person from the possibility of later experiencing the cause of interest as the first event. The Aalen–Johansen estimator combines the estimated event-free survival across all causes with the cause-specific event increments.
Here \(\widehat S(t_j-)\) is the estimated probability of remaining free of every event just before \(t_j\). The factor \(d_{jk}/Y_j\) is the estimated cause-\(k\) event increment among those at risk at that time. Their product is the estimated amount of probability added to cause \(k\) at that time. Censoring does not itself create an event increment, but it affects subsequent risk-set sizes.
Conditions for the Estimate
The formula is a nonparametric estimator for a cohort whose observed event process is subject to right censoring. Its interpretation as an estimate of the population cumulative incidence depends on how subjects enter observation, how censoring occurs, and whether outcomes and causes are recorded correctly. In the simple setting considered here, the main conditions are:
- Correct risk sets: \(Y_j\) includes subjects who are under observation and have experienced no event immediately before \(t_j\). A person is removed after a first event or after censoring.
- Independent censoring: censoring must not selectively hide event experience in a way that leaves the observed risk sets unrepresentative of the target cohort. In a simple marginal analysis this is often stated as censoring independent of event time and cause. More complex conditional censoring may require additional methods.
- Accurate event classification: each first event is assigned to the appropriate cause, and censoring is not mistakenly coded as an event.
- Appropriate time origin and target: the time scale and population represented by the cohort match the question being answered.
The calculation does not require competing causes to be statistically independent. A person can experience only one first event in this setup; the estimator accounts for each competing event by including it in the all-cause event decrement used to update \(\widehat S\). The earlier material on independent censoring explains why this assumption is distinct from the fact that competing events remove subjects from the risk set.
How the Pieces Accumulate
At each event time, first calculate the all-cause event fraction \(d_j/Y_j\). This reduces estimated event-free survival from just before to just after that time. Separately, calculate the fraction \(d_{jk}/Y_j\) attributable to cause \(k\). Weighting this cause-specific fraction by survival just before the time gives the contribution to \(\widehat F_k\).
For each event time, identify \(Y_j\) and the number of events of each cause. Use all causes when calculating \(d_j\).
Multiply the current survival estimate by \(1-d_j/Y_j\) after the event time. Keep the value from before the event time for that time’s cumulative-incidence contribution.
For cause \(k\), add \(\widehat S(t_j-)d_{jk}/Y_j\) to the contributions from earlier times.
A useful arithmetic check follows from the same updates. At an event time, the all-cause event-free survival decrement is \(\widehat S(t_j-)d_j/Y_j\). Summing the cause-specific contributions at that time gives
Thus the event-free survival lost at an event time is exactly the probability allocated across the causes at that time. Starting from survival 1 and cumulative incidence 0, this implies that the sum of estimated cumulative incidences through a time plus estimated event-free survival after that time equals 1, when all event causes are included.
Worked Example: Two Causes at Three Event Times
Suppose a hypothetical cohort has distinct event times at 2, 5, and 8 months. The risk sets and observed first events are as follows. The risk-set sizes reflect any censoring before each event time.
| Time (months) | At risk \(Y_j\) | Cause A events \(d_{jA}\) | Cause B events \(d_{jB}\) | All-cause events \(d_j\) |
|---|---|---|---|---|
| 2 | 10 | 1 | 0 | 1 |
| 5 | 8 | 0 | 1 | 1 |
| 8 | 5 | 1 | 0 | 1 |
Estimate the probability of cause A by 8 months. Immediately before month 2, everyone is event-free, so \(\widehat S(2-)=1\). The cause-A contribution at month 2 is \(1(1/10)=0.1000\). After the event, estimated event-free survival is \(1(1-1/10)=0.9000\).
At month 5, the event is from cause B. The cause-A contribution at that time is zero, while the survival update is \(0.9000(1-1/8)=0.7875\). At month 8, the cause-A increment is \(1/5\), weighted by survival just before month 8:
The estimated cause-A cumulative incidence by 8 months is 0.2575, or 25.75%. Cause B contributes \(0.9000(1/8)=0.1125\), so the two estimated cumulative incidences sum to \(0.3700\). Event-free survival after month 8 is \(0.7875(1-1/5)=0.6300\); \(0.2575+0.1125+0.6300=1.0000\), as expected.
Worked Example: Censoring Changes Later Risk Sets
In a second hypothetical cohort, there are 12 subjects at risk at month 1, when one experiences cause A. Two other subjects are censored before month 4. At month 4, one of the 9 remaining subjects experiences cause B. At month 6, one of the 8 remaining subjects experiences cause A. Assume no ties between event and censoring times.
The month-1 cause-A contribution is \(1(1/12)=0.08333\), and survival after month 1 is \(1-1/12=11/12=0.9167\). The two censorings do not reduce this survival estimate by an event fraction; they reduce the later risk set from 11 to 9. At month 4, the cause-B event changes event-free survival to
At month 6, the cause-A contribution is \(0.8148(1/8)=0.1019\). Therefore,
The estimate is 18.52% by month 6. This calculation uses censoring through the risk-set size at month 4, rather than treating the censored subjects as having experienced an event or as remaining at risk indefinitely.
Worked Example: Checking All-Cause Probability Allocation
Consider a hypothetical cohort with distinct event times at months 2, 4, and 7. Suppose the risk sets are 20, 15, and 10, respectively; the events are one cause-A event at month 2, one cause-B event at month 4, and one cause-A event at month 7.
The event-free survival immediately before month 2 is 1, so cause A contributes \(1/20=0.05000\). After month 2, survival is \(19/20=0.9500\). Cause B then contributes \(0.9500/15=0.06333\) at month 4, and survival after that event is \(0.9500(14/15)=0.8867\). At month 7, cause A contributes \(0.8867/10=0.08867\). Hence \(\widehat F_A(7)=0.05000+0.08867=0.1387\), and \(\widehat F_B(7)=0.06333\).
Survival after month 7 is \(0.8867(9/10)=0.7980\). The check is \(0.1387+0.06333+0.7980\approx1.000\), with the small discrepancy attributable to rounding. A failure of this check beyond rounding error often signals that an event was omitted from the all-cause decrement or that a cause-specific contribution was added with the wrong risk set or survival value.
Interpretation, Comparisons, and Failure Modes
At a chosen horizon \(t\), \(\widehat F_k(t)\) estimates the probability that a member of the target cohort experiences cause \(k\) as the first event by \(t\), in the presence of the other competing causes. It is a probability, so it is between 0 and 1 and cannot decrease as time increases. It is not an instantaneous rate, and a value such as 0.20 does not mean that the cause-specific hazard is 0.20 per unit time.
The Aalen–Johansen estimator is the nonparametric approach for estimating cumulative incidence from observed competing-risks data with right censoring under suitable censoring assumptions. It differs from a cause-specific hazard analysis, which describes an instantaneous event rate among those currently event-free; a hazard is not itself the probability of the event by a horizon. It also differs from one minus a Kaplan–Meier estimate that treats competing events as censoring. That latter calculation removes competing events from the risk set as if they could still later experience the event of interest, so it generally does not estimate the observed cause-specific cumulative incidence. The earlier material on cumulative incidence as an event probability provides the probability interpretation underlying this distinction.
Several checks help identify mistakes or limitations:
- Wrong survival weight: use \(\widehat S(t_j-)\), not survival after the event at \(t_j\). The event increment is applied to those event-free just before the time.
- Wrong all-cause update: update survival using events from every cause. Updating it using only cause \(k\) overstates the fraction still eligible to experience cause \(k\) later.
- Wrong risk set: a subject censored before an event time is no longer in that later risk set. Incorrect risk sets distort both the event fraction and subsequent estimates.
- Overinterpretation near the end of follow-up: few subjects at risk can make the estimate unstable, and the estimate cannot correct for informative censoring or unobserved outcome patterns.
The estimator is descriptive of event probabilities for the population and time scale represented by the analysis. It does not by itself establish causal effects of an exposure or treatment. Its finite-sample precision also requires separate uncertainty estimation; the cumulative-incidence formula alone is a point estimate, not a confidence interval.
Check Your Understanding
Answer each question, then check the solutions below.
- At a distinct event time, \(Y_j=25\), \(d_j=2\), and \(d_{jk}=1\). If \(\widehat S(t_j-)=0.80\), what is the cause-\(k\) contribution at that time, and what is survival immediately after it?
- Why does a cause-B event affect a later cause-A contribution even though it adds nothing directly to \(\widehat F_A\) at its own time?
- Suppose the estimated cause-A incidence is 0.12 and cause-B incidence is 0.08 through a horizon. What should event-free survival be at that horizon, up to rounding, if these are the only causes?
- A researcher omits cause-B events when updating event-free survival but still uses the observed cause-A risk sets and increments. What is the likely direction of the error in later cause-A contributions, and why?
Solutions. 1. The cause-\(k\) contribution is \(0.80(1/25)=0.032\). Survival after the time is \(0.80(1-2/25)=0.736\). 2. The cause-B event reduces the estimated fraction still event-free, so it lowers the survival weight on later cause-A increments. 3. Event-free survival is \(1-0.12-0.08=0.80\), subject to rounding. 4. Later cause-A contributions will tend to be too large because omitting cause-B events makes estimated event-free survival too high, giving later cause-A increments excessive weights.