Avoiding Kaplan-Meier and Censoring Misinterpretations

Why This Reporting Distinction Matters

A report of “20% risk of the target event by one year” makes a claim about the probability of experiencing that event by one year in the population being studied. That quantity must account for what can happen first. If a participant experiences a competing event, such as death before disease recurrence, they can no longer experience the target event as the first event. Treating that competing event as though it were ordinary loss to follow-up can therefore change the quantity being estimated.

The key reporting distinction is between the cumulative incidence of a cause, which is the observed-world probability of that cause occurring first by a specified time, and the complement of a Kaplan–Meier estimate calculated after treating other causes as censored. The latter is not generally the observed absolute risk of the target cause.

Definition: Let \(T\) be the time to the first event and \(J\) the cause of that event. For cause \(k\), the cumulative incidence function is \(F_k(t)=P(T\le t,\ J=k)\). It is the probability that the event of interest is the first event and occurs by time \(t\), in the presence of the competing causes that occur in the population.

This distinction is about the estimand, not just the name of a calculation. A cause-specific hazard describes the instantaneous event rate for cause \(k\) among people who have not yet experienced any event. A cumulative incidence function combines that rate with the probability of still being event-free. The introductory material on competing risks establishes the role of these quantities; here the focus is on avoiding a particular interpretation error when reporting them.

What the Two Calculations Use

At an event time \(t_j\), let \(Y_j\) be the number at risk just before that time, \(d_{kj}\) the number of cause-\(k\) events, and \(d_j\) the number of events from all causes. Under the usual convention for tied event times, every event at \(t_j\) is counted against the risk set just before that time. With complete cause classification and independent right censoring of observation, the Aalen–Johansen estimator of the cause-\(k\) cumulative incidence is

\widehat{F}_k(t)=\sum_{t_j\le t}\widehat{S}(t_j-)\frac{d_{kj}}{Y_j}, \qquad \widehat{S}(t_j)=\widehat{S}(t_j-)\left(1-\frac{d_j}{Y_j}\right).

Here \(\widehat{S}(t_j-)\) is the estimated probability of having experienced no event of any cause just before \(t_j\). The cumulative incidence adds the target-cause event fraction at each time, weighted by the chance of still being event-free immediately before that time.

If the other causes are instead recorded as censored in a Kaplan–Meier calculation for cause \(k\), the product-limit estimate uses the target events in its decrement:

\widehat{S}_{k,\mathrm{KM}}(t)= \prod_{t_j\le t}\left(1-\frac{d_{kj}}{Y_j}\right), \qquad 1-\widehat{S}_{k,\mathrm{KM}}(t) = 1-\prod_{t_j\le t}\left(1-\frac{d_{kj}}{Y_j}\right).

The complement can be described as a net-risk or net-failure quantity derived from the cause-specific hazard. It is not automatically the observed-world probability \(F_k(t)\). A causal interpretation as risk under an intervention that eliminates competing events would require additional assumptions about that intervention and how it affects the target-event process; the calculation alone does not establish such an interpretation.

Why the Estimates Can Differ

The difference follows directly from the risk weighting. The cumulative incidence weights each target-cause event fraction by survival free of all causes. The Kaplan–Meier complement weights target-cause increments through a product that removes only target events. A person who has a competing event is no longer at risk for the target event in the observed process, but the target-only Kaplan–Meier calculation treats that person as censored rather than as having left the event-free state.

\widehat{F}_k(t)= \sum_{t_j\le t}\widehat{S}(t_j-)\frac{d_{kj}}{Y_j}, \qquad 1-\widehat{S}_{k,\mathrm{KM}}(t)= \sum_{t_j\le t}\widehat{S}_{k,\mathrm{KM}}(t_j-)\frac{d_{kj}}{Y_j}.

At each event time, the all-cause event-free estimate decreases by \(d_j/Y_j\), whereas the target-only Kaplan–Meier estimate decreases by \(d_{kj}/Y_j\). Since \(d_j\ge d_{kj}\), the all-cause event-free estimate cannot exceed the target-only estimate when both start at one and use the same risk sets. Thus, with these conventions, the target-only Kaplan–Meier complement is at least as large as the cumulative incidence. It can be strictly larger when competing events occur before or alongside target events.

This inequality is a useful check, not a reason to substitute one quantity for the other. The Kaplan–Meier complement answers a different question, and both estimates depend on the observed event-time and censoring information.

Worked Example: Shared Event Time and a Necessary Assumption

Suppose a hypothetical cohort has 100 people at risk at the beginning of a single grouped interval. By its end, 20 experience the target event and 30 experience a competing event. Assume explicitly that all 50 events occur at the same event time, or that the data are analyzed as one grouped interval with simultaneous events. There is no loss to follow-up before that time.

For the cumulative incidence, the target-cause increment is \(20/100\). There were no earlier events, so the event-free estimate before this time is one:

Cumulative incidence: \(1\times(20/100)=0.20\), or 20%.

Under the stated tied-time convention, the target-only Kaplan–Meier decrement at that time also uses the 100 people at risk just before the time:

Kaplan–Meier complement: \(1-(1-20/100)=0.20\), or 20%.

The equality here does not show that the two estimands are generally interchangeable. It follows from the explicit single tied-time assumption: there is no earlier competing-event time to alter a later target-event risk set. If the 30 competing events occurred earlier, the target-only Kaplan–Meier calculation would instead use a smaller risk set for the target events, as the next example shows.

Worked Example: Competing Events Occur Before Target Events

Consider another hypothetical cohort of 100 people. At day 10, 30 people experience a competing event. At day 20, 20 of the 70 remaining people experience the target event. There is no censoring before day 20.

For the target cumulative incidence, the competing events reduce the event-free proportion to \(70/100=0.70\) by day 10. The target event fraction at day 20 is \(20/70\). Therefore:

Cumulative incidence at day 20: \(0.70\times(20/70)=0.20\), or 20%.

For the target-only Kaplan–Meier calculation, the 30 competing events are treated as censored at day 10. The target events occur with 70 still in the risk set:

Kaplan–Meier complement at day 20: \(1-(1-20/70)=20/70=0.2857\), or about 28.6%.

The estimates differ because the Kaplan–Meier complement does not weight the day-20 target-event fraction by the 0.70 probability of remaining free of every event. In this example, the counts alone would not have been enough to calculate the target-only Kaplan–Meier complement: the order of the competing and target events matters.

Competing Events Are Not Ordinary Loss to Follow-Up

Ordinary right censoring means that follow-up ends before an event is observed, so the person’s later event status is unknown. A competing event is observed and is itself an event in the study’s endpoint structure. For example, death before recurrence is not an unobserved recurrence time; it is a known first event that prevents recurrence from being observed as the first event.

For estimation of a cause-specific hazard, people who experience another cause leave the risk set at that time. Software may represent those records as censored for the purpose of fitting a cause-specific hazard model. This computational convention does not turn a competing event into ordinary loss to follow-up, nor does it make the complement of a target-only Kaplan–Meier curve an observed cumulative incidence.

By contrast, independent loss-to-follow-up censoring is an assumption about how observation ends. For standard nonparametric estimation, the event process must remain observable among those still followed, or the censoring must be independent conditional on the variables used in the analysis. If that assumption fails, estimates can be biased. Competing events are handled as events in the cumulative-incidence calculation; losses to follow-up are censored observations, subject to the censoring assumption.

Worked Example: Different Denominators, Different Absolute Risks

Suppose that in a hypothetical cohort of 100 people, 20 experience a competing event at day 5. At day 10, 8 of the 80 people who remain event-free experience the target event. The other people are administratively censored after day 10, so their censoring occurs after the target-event time considered here.

The observed target cumulative incidence at day 10 is the probability of remaining event-free to day 5, multiplied by the target-event fraction at day 10:

Cumulative incidence: \((80/100)\times(8/80)=0.08\), or 8%.

The target-only Kaplan–Meier calculation treats the 20 competing events at day 5 as censored. Its target-event factor at day 10 is \(1-8/80=0.90\):

Kaplan–Meier complement: \(1-0.90=0.10\), or 10%.

Thus, the observed absolute risk of the target event by day 10 is 8%, not the 10% Kaplan–Meier complement. The administrative censoring after day 10 does not alter either calculation at day 10. If people had instead been lost before day 10, estimating the risks would additionally require an appropriate assumption or adjustment for that loss to follow-up.

Reporting Checks and Common Pitfalls

When reviewing a results section, check that the reported number matches the stated estimand and method. If the aim is the observed probability of a particular cause by a horizon, the report should identify a cumulative incidence estimate, not label a target-only Kaplan–Meier complement as that probability.

Check what was counted as an event. State the target cause and the competing causes, and distinguish both from loss to follow-up.
Check whether event timing is available. Cause-specific event counts alone do not generally determine the Kaplan–Meier complement. The event-time order and risk sets matter, unless a specific tied-time or grouped-interval convention is stated.
Check the denominator at each event time. Cumulative incidence uses the all-cause event-free probability before each time; the target-only Kaplan–Meier product does not remove competing events as events.
Check the wording of the conclusion. “The target cumulative incidence by day 20 was 20%” describes observed absolute risk. “One minus the target-only Kaplan–Meier estimate was 28.6%” reports a different calculation and should not be presented as the same risk.
Check censoring assumptions. A competing event is known, while loss to follow-up is incomplete observation. Independent censoring of follow-up does not mean competing events are ordinary censoring for interpretation.

A common mistake is to reason, “The other cause was censored in the software, so one minus Kaplan–Meier is the observed risk of the target cause.” The correction is to separate the model or software handling from the estimand: the competing event may leave a cause-specific hazard risk set, but the observed absolute risk must account for the chance of experiencing any competing event first. Also, do not infer a Kaplan–Meier complement from a summary table of cause counts unless its event-time structure is specified.

The main alternative approaches answer related but distinct questions. Cumulative incidence estimation describes observed cause-specific absolute risk. Cause-specific hazard analysis describes an event rate among those still event-free. The target-only Kaplan–Meier complement transforms a target cause-specific hazard into a net-risk quantity. The choice should follow the scientific question, and the result should be named accordingly. No one of these numbers, by itself, establishes what would happen under an intervention that removes a competing cause.

Key takeaway: For observed absolute risk in the presence of competing events, report the cause-specific cumulative incidence. Do not describe one minus a Kaplan–Meier estimate that censors competing causes as that observed risk. If reporting the Kaplan–Meier complement, name its net-risk interpretation and make the event-time and censoring assumptions clear.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

In one sentence, distinguish a competing event from loss to follow-up.
A cohort has 100 people at risk; 10 target events and 20 competing events all occur at one explicitly tied event time. What is the target cumulative-incidence increment? Under the tied-time convention, what is the target-only Kaplan–Meier complement at that time?
Why can counts of 20 target events and 30 competing events fail to determine the target-only Kaplan–Meier complement?
At day 5, 25 of 100 people experience a competing event. At day 10, 15 of the remaining 75 experience the target event. With no earlier censoring, calculate the target cumulative incidence and the target-only Kaplan–Meier complement at day 10.
Is it correct to call a competing event “ordinary censoring” because it is coded as censored when fitting a cause-specific hazard model? Explain.

Solutions. (1) A competing event is an observed first event that prevents the target event from occurring first; loss to follow-up is the end of observation before the event status is known. (2) Both are \(10/100=0.10\), or 10%, under the explicit tied-time convention. (3) The Kaplan–Meier factors depend on the risk set at each target-event time, which depends on when competing events occurred relative to target events. (4) The cumulative incidence is \((75/100)(15/75)=0.15\), or 15%; the Kaplan–Meier complement is \(15/75=0.20\), or 20%. (5) No. That coding is a risk-set convention for estimating a cause-specific hazard; it does not change the competing event into incomplete observation or make a Kaplan–Meier complement the observed cumulative incidence.