Aalen-Johansen Estimation With Right Censoring

Why Right Censoring Changes the Calculation

A subject who is right-censored has no observed event time or event cause after the last time they were known to be event-free. That subject contributes information up to the censoring time: they belong in the risk set before then, but not in risk sets at later event times. They are not an observed event, and censoring does not itself create a jump in estimated event-free survival or cumulative incidence.

The Aalen–Johansen estimator combines two pieces of information at each observed event time: the fraction of the risk set experiencing each cause, and the estimated fraction still event-free just before that time. The second piece matters because an event increment is applied to the event-free fraction remaining from earlier times. Thus, the estimator is not simply a count of observed events divided by the original sample size.

For the calculation, an individual censored at \(t_j\) is included in \(Y_j\): they were observed and event-free immediately before that time. An event at \(t_j\) is also included in \(Y_j\), then counted in \(d_j\) and the appropriate cause count. After time \(t_j\), both event cases and cases censored at that time leave subsequent risk sets. Censoring times with no event do not appear as event times in the product or sum below.

The Estimator With Censoring

Let \(\widehat S(t)\) estimate the probability of remaining free of every event through time \(t\), and let \(\widehat F_k(t)\) estimate the probability of experiencing cause \(k\) by time \(t\) before any competing event. At each distinct observed event time, the updates are

Initialize \(\widehat S(0-)=1\) and \(\widehat F_k(0-)=0\). Between event times, all these estimates remain constant, including at a censoring time that has no event. Equivalently, through time \(t\),

The \(Y_j\) in these expressions is calculated from the observed follow-up, including subjects censored only after \(t_j\), and excluding subjects censored before \(t_j\). Censored subjects therefore affect the estimate indirectly by reducing later risk-set sizes. For these estimates to represent the target event-time distribution, censoring must satisfy an appropriate independent-censoring condition. The calculation itself does not establish that condition; it is an assumption about the data-generating and observation processes.

Why the Event Increment Uses Event-Free Survival

Among the \(Y_j\) subjects at risk just before \(t_j\), the observed fraction with cause \(k\) at that time is \(d_{jk}/Y_j\). This estimates a cause-specific event probability for that time among those still event-free and under observation. But the target population includes subjects who may already have had an event. The estimated fraction of the original population still event-free just before \(t_j\) is \(\widehat S(t_j-)\). Multiplying these two quantities gives the estimated population probability mass for cause \(k\) at \(t_j\).

This also gives a direct check on the calculation. Since \(\sum_k d_{jk}=d_j\), the sum of all cause-specific increments at \(t_j\) is \(\widehat S(t_j-)d_j/Y_j\). The event-free survival decrement at that time is

Starting with \(\widehat S(0-)=1\) and all cumulative incidences equal to zero, this equality holds at each event-time update. Adding the increments over event times therefore gives \(\sum_{k=1}^K\widehat F_k(t)=1-\widehat S(t)\). The identity is useful for detecting arithmetic or coding errors; it does not by itself show that the censoring assumption is appropriate.

Checks, Alternatives, and Failure Modes

Before interpreting an estimate, check that each \(Y_j\) is the number under observation and event-free immediately before the event time, and that \(d_j\leq Y_j\). A subject censored earlier must not contribute to a later risk set. At a time with censoring but no event, the estimates must not jump. At an event time, \(\widehat S(t)\) cannot increase and each \(\widehat F_k(t)\) cannot decrease. Finally, their sum should satisfy \(\widehat S(t)+\sum_k\widehat F_k(t)=1\), apart from rounding.

Common alternatives answer different questions or require different assumptions. A raw proportion of observed cause-k events divided by the starting sample size treats subjects whose outcomes are censored as though their eventual outcomes were known. Treating competing events as censoring in a Kaplan–Meier calculation does not estimate the observed-population probability of cause \(k\) by time \(t\); the Aalen–Johansen calculation instead accounts for event-free survival across all causes. When there is no censoring, the same event-time updates still apply, but risk sets shrink only because of events.

A plausible-looking step function is not proof that the estimator is informative. Sparse risk sets late in follow-up can make estimates sensitive to a few observations. Heavy or systematically different censoring can also undermine the interpretation if the independent-censoring condition is not reasonable. Review follow-up and censoring patterns, verify event coding and risk-set counts, and avoid interpreting a late estimate as well supported merely because it can be calculated. The estimator describes probabilities in context; it does not establish that differences between groups are causal.