Why Estimate Several Causes Separately?
When the first event can occur for several mutually exclusive reasons, an analysis often needs to describe each cause’s probability over time, not merely the probability of any event. For example, a clinical analysis may distinguish a particular complication from other complications and death. These cause-specific probabilities must be calculated together: an event from one cause removes a person from being able to experience a different first-event cause.
The cumulative incidence function (CIF) for cause \(j\) at time \(t\) is the probability that the first event has occurred by \(t\) and is of cause \(j\). The Aalen–Johansen estimator accumulates each cause’s event increment while accounting for the proportion still event-free just before that event time.
The relevant risk set contains individuals who have not yet had any first event immediately before \(t_k\). The event counts for causes at the same time are treated together: their sum is the total event count \(d_k\). Cause categories must be mutually exclusive and collectively cover the first-event types included in the analysis. If the cause labels overlap, or a relevant cause is omitted, the cause-specific estimates will not describe a complete partition of first events.
For the calculations below, all individuals are followed through the stated event times and there is no censoring. This makes the risk-set counts easy to verify. In analyses with censoring, the risk sets and conditions needed for estimation require additional care. The formula still illustrates the central multi-cause calculation: weight each cause’s event fraction by the estimated probability of remaining event-free up to that time.
How the Cause Estimates Add Up
At time \(t_k\), the estimated event-free probability just before the event time is \(\widehat S(t_{k-1})\). The fraction \(d_{jk}/Y_k\) estimates the conditional probability of a cause-\(j\) event at that time among those still at risk. Their product is the contribution to the cause-\(j\) CIF at \(t_k\). The same weighting is applied at every event time, separately for each cause.
The sum across causes has a useful exact identity. At a given event time, summing the cause-specific increments gives the all-cause increment because \(\sum_j d_{jk}=d_k\). Also, the decrease in event-free survival at that time is \(\widehat S(t_{k-1})d_k/Y_k\). Summing these decreases over time telescopes to one minus the final event-free estimate.
Thus the sum of the estimated CIFs is the estimated probability of having experienced any first event by \(t\). This is an internal consistency check, not an additional model assumption. Small discrepancies in hand calculations usually indicate an omitted cause, a wrong risk set, or an arithmetic error.
Worked Example: Three Causes at Three Event Times
Consider a hypothetical cohort of 10 people with three possible first-event causes, A, B, and C. Everyone is observed through the three event times, with no censoring. The risk set falls from 10 to 7 after the first time and from 7 to 5 after the second.
| Time | At risk \(Y_k\) | Cause A \(d_{Ak}\) | Cause B \(d_{Bk}\) | Cause C \(d_{Ck}\) | All events \(d_k\) |
|---|---|---|---|---|---|
| \(t_1\) | 10 | 1 | 2 | 0 | 3 |
| \(t_2\) | 7 | 1 | 0 | 1 | 2 |
| \(t_3\) | 5 | 0 | 1 | 1 | 2 |
At \(t_1\), \(\widehat S(t_0)=1\), so the estimated event-free probability after the time is \(1(1-3/10)=0.70\). The cause-specific increments are \(1(1/10)=0.10\) for A, \(1(2/10)=0.20\) for B, and 0 for C.
At \(t_2\), the event-free probability just before the time is 0.70. It becomes \(0.70(1-2/7)=0.50\). The new CIF increments are \(0.70(1/7)=0.10\) for A, 0 for B, and \(0.70(1/7)=0.10\) for C.
At \(t_3\), the event-free probability just before the time is 0.50. It becomes \(0.50(1-2/5)=0.30\). The increments are 0 for A, \(0.50(1/5)=0.10\) for B, and \(0.50(1/5)=0.10\) for C. Therefore, at \(t_3\),
The sum is \(0.20+0.30+0.20=0.70\), which equals \(1-\widehat S(t_3)=1-0.30=0.70\). In this hypothetical cohort, the estimated probability of any first event by \(t_3\) is 0.70, partitioned into 0.20 for A, 0.30 for B, and 0.20 for C.
A tempting but incorrect calculation is to add the raw event fractions across times without weighting them by prior event-free survival. That gives A: \(1/10+1/7\approx0.2429\); B: \(2/10+1/5=0.4000\); and C: \(1/7+1/5\approx0.3429\). Their sum is approximately \(0.9857\), not the observed any-event proportion \(7/10=0.70\). The raw fractions count later event-time risk sets as if the full cohort were still event-free; the Aalen–Johansen weights correct for the events that have already occurred.
More Than Three Causes and Grouped Causes
Nothing in the calculation is restricted to three causes. With \(K\) mutually exclusive causes, the same risk set and all-cause event-free estimate are used for every cause. At each time, the cause-specific event counts must sum to the total count. A zero count for one cause at a time contributes zero to that cause’s CIF but does not alter the shared event-free update.
Worked Example: Four Causes With Different Event Patterns
Suppose a hypothetical cohort has 20 people at risk at \(t_1\). Four events occur there: 2 from cause 1, 1 from cause 2, 1 from cause 3, and none from cause 4. At \(t_2\), 16 remain at risk and the counts are 1, 2, 1, and 2. At \(t_3\), 10 remain at risk and the counts are 0, 1, 2, and 1. Follow-up is complete with no censoring.
After \(t_1\), \(\widehat S=1-4/20=0.80\). The four CIFs are \(0.10,\ 0.05,\ 0.05,\ 0\). At \(t_2\), the event-free probability before the time is 0.80, and the total event fraction is \(6/16=0.375\), so \(\widehat S=0.80(1-6/16)=0.50\). The CIF increments at \(t_2\) are \(0.80(1/16)=0.05,\ 0.80(2/16)=0.10,\ 0.05,\) and \(0.10\). The cumulative estimates are now \(0.15,\ 0.15,\ 0.10,\ 0.10\).
At \(t_3\), the increments are \(0,\ 0.50(1/10)=0.05,\ 0.50(2/10)=0.10,\) and \(0.50(1/10)=0.05\). The final CIFs are therefore \(0.15,\ 0.20,\ 0.20,\) and \(0.15\). Their sum is \(0.70\), equal to \(1-\widehat S(t_3)=1-0.30=0.70\). The four cause probabilities partition the estimated probability of any event by \(t_3\).
Causes can also be combined after estimating them. If a composite category consists of causes 2 and 3, its CIF is \(\widehat F_2(t)+\widehat F_3(t)\), because those events are mutually exclusive. The remaining causes retain their own estimates, and the total across the composite and remaining categories is unchanged. Combining causes changes the detail of the reported partition, not the estimated probability of any event.
Worked Example: Combining Two Causes
In a hypothetical cohort of 12 people, four first events occur at \(t_1\): 1 of cause A, 1 of cause B, and 2 of cause C. At \(t_2\), 8 people remain at risk; 2 have cause A, 1 has cause B, and 1 has cause C. There is no censoring.
At \(t_1\), \(\widehat S=1-4/12=2/3\), and the CIFs for A, B, and C are \(1/12,\ 1/12,\ 2/12\). At \(t_2\), the weighted increments are \((2/3)(2/8)=1/6\) for A, \((2/3)(1/8)=1/12\) for B, and \(1/12\) for C. The final estimates are therefore \(\widehat F_A=1/12+1/6=0.25\), \(\widehat F_B=1/12+1/12\approx0.1667\), and \(\widehat F_C=2/12+1/12=0.25\). Their sum is \(2/3\), equal to \(1-\widehat S(t_2)=1-1/3=2/3\). If B and C are reported as one composite cause, its estimate is \(0.1667+0.25\approx0.4167\).
Checks, Alternatives, and Limitations
A useful calculation check is to verify each event-time total and the final identity. At each time, confirm that \(\sum_j d_{jk}=d_k\), that the risk set is measured just before the time, and that the event-free estimate is updated using all causes together. Then check that all CIFs are nondecreasing and nonnegative, and that their sum equals \(1-\widehat S(t)\), up to rounding.
The main alternative errors are not merely arithmetic. Adding unweighted cause fractions ignores the changing event-free population. Estimating a cause-specific risk as one minus a Kaplan–Meier curve that treats other causes as censored also does not estimate that cause’s observed-world CIF; it removes competing events from the probability calculation. The earlier material on the cumulative incidence function explains this distinction. For multiple causes, the Aalen–Johansen calculation keeps competing events in the shared all-cause event-free update and allocates each event increment to its observed cause.
The estimator’s interpretation depends on sound event and cause coding, correctly formed risk sets, and appropriate handling of incomplete follow-up. In the examples, complete follow-up and no censoring make the arithmetic transparent. With right censoring, estimation requires suitable censoring conditions; the risk-set calculation and its justification need to account for that observation process. The estimates here are point estimates: they do not by themselves quantify sampling uncertainty, and they do not establish a causal effect. Estimates near the end of follow-up can also be based on few people at risk. These limitations should inform reporting alongside the cause-specific probabilities.
Check Your Understanding
Answer each question, then compare your work with the solutions below.
- At one event time, \(Y_k=25\), with 2 events of cause A and 3 of cause B. What is the total event-free multiplier, and what are the CIF increments if the event-free estimate just before the time is 0.80?
- Three cause-specific CIF estimates at a horizon are 0.12, 0.08, and 0.25. What is the corresponding estimated probability of any event and the event-free estimate?
- Why is it incorrect to add \(d_{jk}/Y_k\) across event times without weighting each term by the event-free estimate before that time?
- If two causes are combined into one category, how is the composite CIF calculated, and does the total across causes change?
Solutions. (1) The total event count is \(d_k=2+3=5\), so the multiplier is \(1-5/25=0.80\). The increments are \(0.80(2/25)=0.064\) for A and \(0.80(3/25)=0.096\) for B. (2) The any-event estimate is \(0.12+0.08+0.25=0.45\); the event-free estimate is \(1-0.45=0.55\). (3) Later risk sets contain only those who have not yet had an event, so each later cause fraction must be weighted by the estimated probability still event-free. Otherwise, the full starting cohort is implicitly counted repeatedly. (4) Add the CIFs of the mutually exclusive causes in the composite category. The total across the composite and all remaining causes is unchanged.