Why the distinction matters
Suppose a study follows patients for a particular event, such as hospital readmission, while some patients die first. The question “What proportion experience readmission by one year?” concerns the event probability in the world where death can also occur. A calculation that treats death as censoring answers a different question: it removes those deaths from the risk set as though they were losses to follow-up. The resulting Kaplan–Meier complement is not, in general, the observed-world probability of readmission.
The distinction is not merely technical. If competing events are common, a Kaplan–Meier complement can give a noticeably larger number than the cumulative incidence estimate. The Aalen–Johansen estimator accounts for the fact that people who have already experienced a competing event can no longer experience the event of interest.
The two calculations
Consider event times \(t_1,\ldots,t_m\) observed by a chosen horizon \(t\). Let \(Y_j\) be the number at risk immediately before \(t_j\), \(d_{1j}\) the number of cause-1 events at \(t_j\), and \(d_j\) the number of events of all causes at \(t_j\). Let \(\widehat S(t_j-)\) denote the estimated probability of remaining free of every event just before \(t_j\). The Aalen–Johansen estimate is
Each term has a direct interpretation: among those still free of events just before \(t_j\), the observed fraction with a cause-1 event is \(d_{1j}/Y_j\). The term multiplies that fraction by the estimated fraction still event-free at that point. Summing the increments gives the estimated probability of cause 1 occurring by \(t\).
For the competing-cause Kaplan–Meier calculation, competing events are instead treated as censored at their event times. The risk set immediately before each event time is still \(Y_j\), but only cause-1 events reduce the Kaplan–Meier survival estimate:
The last equality follows by telescoping the successive decreases in the Kaplan–Meier product. It shows the key difference: the Kaplan–Meier complement weights cause-1 increments by \(\widehat S_{KM,1}(t_j-)\), which accounts for having avoided cause 1, but not for having avoided competing events. The Aalen–Johansen estimate uses \(\widehat S(t_j-)\), which accounts for avoiding every event.
Why the Kaplan–Meier complement is larger
Assume the same observed risk sets are used in both calculations, event counts are nonnegative, and there are no complications from tied event times in the calculation. The all-cause event-free estimate includes competing events in its product; the cause-1 Kaplan–Meier estimate does not. Since \(d_j\ge d_{1j}\), each all-cause survival factor is no larger than the corresponding cause-1 factor:
Products of these nonnegative factors preserve the ordering. Therefore, just before every event time, \(\widehat S(t_j-)\le\widehat S_{KM,1}(t_j-)\). The cause-1 event fraction \(d_{1j}/Y_j\) is nonnegative, so each Aalen–Johansen increment is no larger than the corresponding increment in the Kaplan–Meier complement. Summing gives
The inequality is strict when a competing event has reduced the event-free estimate before a later cause-1 increment that is positive. If there are no competing events before any cause-1 events contributing by \(t\), the estimates agree. This is a comparison of the two estimates on the same data, not a claim that the Kaplan–Meier complement is always wrong for every possible target. It is not the observed-world cause-1 probability when competing events are possible.
Worked Example: Two Competing Events Reduce the Estimated Cause-1 Risk
Consider hypothetical follow-up of 10 people through time \(t=4\). There is one cause-2 event at time 1, one cause-1 event at time 2, another cause-2 event at time 3, and another cause-1 event at time 4. No one is censored before the horizon. The distinct event times and risk sets are:
| Time | At risk \(Y_j\) | Cause-1 events \(d_{1j}\) | Cause-2 events |
|---|---|---|---|
| 1 | 10 | 0 | 1 |
| 2 | 9 | 1 | 0 |
| 3 | 8 | 0 | 1 |
| 4 | 7 | 1 | 0 |
For the Aalen–Johansen estimate, the all-cause event-free probability just before time 2 is \(9/10\). The cause-1 increment at time 2 is therefore \((9/10)(1/9)=0.1\). Just before time 4, the all-cause event-free probability is \((9/10)(8/9)(7/8)=7/10\). The cause-1 increment at time 4 is \((7/10)(1/7)=0.1\). Thus
For the cause-1 Kaplan–Meier estimate, the cause-2 events are treated as censored. The cause-1 survival factors are \(1-1/9=8/9\) at time 2 and \(1-1/7=6/7\) at time 4. Hence
The Kaplan–Meier complement is about 0.0381, or 3.81 percentage points, higher. The Aalen–Johansen estimate is the estimated probability of cause 1 by time 4 in the setting where cause 2 can occur. The Kaplan–Meier complement has not counted the competing events as failures, so its cause-1 increments are given relatively larger weights.
Worked examples: when the estimates differ or agree
Worked Example: A Competing Event Before the Event of Interest
In a hypothetical group of five people, one person experiences a competing event at time 1, and one experiences cause 1 at time 2. The other three remain event-free through time 2. There is no censoring before the horizon \(t=2\). At time 2, \(Y=4\), and the estimated event-free probability just before that time is \(4/5\).
The Aalen–Johansen estimate is the cause-1 increment weighted by the fraction who remained free of all events:
When the competing event is treated as censored, the cause-1 Kaplan–Meier estimate starts with four people at risk at time 2. Its complement is \(1-(1-1/4)=0.2500\). Thus, on these data, the two estimates are 20% and 25%, respectively. The competing event is a real outcome that prevents a later cause-1 event for that person; it is not simply missing follow-up for the observed-world probability.
Worked Example: No Competing Events Before the Horizon
Suppose eight people are followed in a hypothetical example. One has a cause-1 event at time 1, another has a cause-1 event at time 2, and the remaining six are administratively censored after time 2. There are no competing events before \(t=2\). The risk sets are \(Y_1=8\) and \(Y_2=7\).
The Aalen–Johansen estimate is
The cause-1 Kaplan–Meier complement is
The estimates agree because no competing event reduces the all-cause event-free estimate before a cause-1 increment. Agreement in this special case does not make the two methods interchangeable when competing events occur.
Assumptions, interpretation, and diagnostics
These calculations assume that event types and event times are recorded correctly and that the risk set at each time contains the people still under observation and free of every event immediately beforehand. For estimates intended to represent a population, right censoring must be independent of the event process under the analysis design, or independent conditional on variables appropriately accounted for. A competing event is not independent censoring for the observed-world cause-1 probability: it determines that cause 1 cannot subsequently occur. Treating it as censored changes the estimand.
The Kaplan–Meier complement can be described as a net risk associated with the cause-1 hazard: it reflects cause-1 failure while the competing causes are removed from the calculation. Interpreting that quantity as the risk under an intervention that eliminates competing events requires additional assumptions. It should not automatically be read as a causal effect or as the probability that a patient would experience cause 1 if other causes were prevented.
Before interpreting a reported number, check the event coding, the risk-set convention, and which causes were treated as censoring. A useful diagnostic is to compare the cause-1 Kaplan–Meier complement with the Aalen–Johansen estimate at the same horizon. A substantial gap is plausible when competing events occur often or early. If the Kaplan–Meier complement is below the Aalen–Johansen estimate using the same event data and risk sets, check for a coding, implementation, or comparison error: the product-and-increment argument above gives the expected ordering.
| Quantity | How competing events are handled | Interpretation |
|---|---|---|
| Aalen–Johansen \(\widehat F_1(t)\) | Included in the all-cause event-free probability and prevent later cause-1 events | Estimated observed-world probability of cause 1 by time \(t\) |
| \(1-\widehat S_{KM,1}(t)\) | Treated as censored in the cause-1 Kaplan–Meier calculation | Net risk associated with cause 1; not generally the observed-world cause-1 probability |
Neither estimate repairs informative loss to follow-up, incorrect cause classification, or a poorly chosen horizon. The Aalen–Johansen estimator answers the observed-world probability question under its censoring and data-quality assumptions; it does not by itself explain why risks differ between groups or establish a causal effect.
Common mistake
Check Your Understanding
Work through each question, then compare with the solutions.
- In one sentence, explain why a competing event is not ordinary censoring when the target is the observed-world probability of cause 1 by time \(t\).
- Suppose a competing event occurs before any cause-1 event, and a cause-1 event occurs later. Which estimate is larger on the same data: the Aalen–Johansen estimate or the cause-1 Kaplan–Meier complement? Why?
- For eight people, suppose there is one competing event at time 1 and one cause-1 event at time 2; the remaining six are event-free through time 2. Calculate both estimates at time 2.
- Give a condition under which the two estimates agree through the horizon.
Solutions. (1) A competing event is an observed outcome that prevents a later event of interest, whereas censoring means later outcomes are unobserved. (2) The Kaplan–Meier complement is larger: the competing event reduces the all-cause event-free weight used by Aalen–Johansen but is treated as censoring in the cause-1 Kaplan–Meier calculation. (3) At time 2, \(Y=7\), and the all-cause event-free probability is \(7/8\). Thus \(\widehat F_1(2)=(7/8)(1/7)=0.1250\), while \(1-\widehat S_{KM,1}(2)=1-(1-1/7)=1/7\approx0.1429\). (4) They agree if there are no competing events before any cause-1 increments contributing by the horizon; in particular, they agree when no competing event occurs by that horizon.