Why Estimate Cumulative Incidence Within Groups?
A cumulative incidence estimate describes the probability of experiencing a particular event by a given time in the presence of competing events. When this probability is needed separately for groups—such as treatment groups or baseline risk categories—the event counts and risk sets must also be kept separate. An event in one group cannot be used to update another group’s estimate.
The group-specific approach estimates each group’s cumulative incidence function directly using the Aalen-Johansen estimator. It does not, by itself, adjust for differences between groups or establish that group membership caused a difference in risk. Its purpose is to estimate and describe the event probabilities within the groups as observed.
Here, \(\widehat S_g(t_j-)\) is the estimated probability of remaining free of every event just before \(t_j\). The increment in cause \(k\)’s cumulative incidence at \(t_j\) is that event-free probability multiplied by the observed cause-\(k\) fraction of the group’s risk set at that time. The risk set includes people still under observation and event-free immediately before the time; people censored earlier are no longer included.
Assumptions and the Group-Specific Calculation
For the nonparametric estimate to represent event probabilities in group \(g\), the event cause must be observed and classified, and censoring must be independent of the future event process within that group. In practical terms, after conditioning on group membership, those censored at a given time should not have systematically different subsequent event experience from those remaining under observation. This assumption cannot generally be verified from the observed data alone. If censoring depends on important characteristics within a group, a simple group-specific estimate may be biased unless the groups or analysis are refined to address that dependence.
Individuals must be assigned to the relevant groups according to a clearly defined rule, typically using baseline information. Groups should be mutually exclusive for a straightforward comparison, and the event types should be competing causes: an individual can experience at most one first failure cause. Within each group, use the appropriate risk set, including any delayed-entry rules if people enter observation at different times. At a time with failures from more than one cause, all such failures contribute to \(d_g(t_j)\), while only cause \(k\) contributes to \(d_{gk}(t_j)\). A censoring time reduces later risk sets but does not count as a failure.
A short justification explains why the event-free estimate appears in the cumulative incidence formula. At \(t_j\), the estimated fraction of people in group \(g\)’s risk set who fail from cause \(k\) is \(d_{gk}(t_j)/Y_g(t_j)\). The chance of being event-free up to just before that time is estimated by \(\widehat S_g(t_j-)\). Multiplying these quantities estimates the unconditional probability contribution from cause \(k\) at that time. Adding those contributions over times up to \(t\) gives the cumulative incidence. Meanwhile, the event-free probability updates by removing failures from every cause:
Worked Example: Calculating Both Causes in Two Groups
Suppose a hypothetical cohort has two baseline groups. In Group A, the event-time risk sets and counts are: at time 1, \(Y_A=10\), with 2 cause-1 and 1 cause-2 failures; at time 2, \(Y_A=7\), with 1 cause-1 failure; and at time 3, \(Y_A=6\), with 2 cause-2 failures. Assume no censoring occurs before these event times. Calculate the estimates through time 3.
For Group A, the estimated event-free probability starts at 1. After time 1 it is \(1-(2+1)/10=0.7\). After time 2 it is \(0.7(1-1/7)=0.6\). The risk sets are consistent with these counts: 10 people at risk at time 1, 7 after the three failures, and 6 after the time-2 failure.
The cause-1 estimate is \(2/10\) at time 1, plus \(0.7(1/7)\) at time 2; there are no cause-1 failures at time 3. Thus \(\widehat F_{A1}(3)=0.2+0.1=0.3\). For cause 2, the increments are \(1/10\) at time 1 and \(0.6(2/6)=0.2\) at time 3, giving \(\widehat F_{A2}(3)=0.1+0.2=0.3\). The total event probability estimated through time 3 is 0.6, and the event-free estimate is 0.4.
For comparison, suppose Group B starts with 8 people. At time 1, \(Y_B=8\) and 2 experience cause 2; at time 2, \(Y_B=6\) and 2 experience cause 1; at time 3, \(Y_B=4\), with 1 failure from each cause. The event-free estimates after the three times are \(6/8=0.75\), \(0.75(4/6)=0.5\), and \(0.5(2/4)=0.25\). Therefore,
The estimates are group-specific: by time 3, the estimated cause-1 incidence is 0.300 in Group A and 0.375 in Group B. These are proportions (or probabilities) over the stated follow-up horizon, not rates per unit time. They describe the observed groups and are not, on their own, evidence that group membership caused a difference.
Censoring and Risk Sets Within Each Group
Censoring affects the group-specific calculation through later risk sets, even though it contributes no direct increment to a cause-specific cumulative incidence. At a failure time, the denominator must be the number at risk in that same group immediately before the time. Replacing this denominator with the group’s original sample size, or with a risk set pooled across groups, changes the estimator.
Worked Example: Updating a Risk Set After Censoring
Consider a hypothetical group of 6 individuals. At time 1, 1 individual experiences cause 1 and 1 is censored. At time 2, 1 of the 4 remaining individuals experiences cause 2 and 1 is censored. At time 3, 1 of the 2 remaining individuals experiences cause 1. The counts imply \(Y(1)=6\), \(Y(2)=4\), and \(Y(3)=2\).
At time 1, the event-free estimate becomes \(1-1/6=5/6\), and the cause-1 incidence increases by \(1/6\). The censoring at time 1 does not count as an event; it helps explain why only 4 individuals are at risk at time 2. The time-2 event-free update is \((5/6)(1-1/4)=5/8\), and cause 2’s incidence increases by \((5/6)(1/4)=5/24\). At time 3, cause 1’s incidence increases by \((5/8)(1/2)=5/16\).
Thus, through time 3, \(\widehat F_1=1/6+5/16=23/48\approx0.4792\), and \(\widehat F_2=5/24\approx0.2083\). The remaining event-free estimate is \((5/8)(1-1/2)=5/16=0.3125\). The three probabilities sum to 1: \(23/48+10/48+15/48=1\). This check is useful for the arithmetic, but it does not establish that the censoring assumption is plausible.
Interpreting, Comparing, and Checking Group Estimates
A group-specific estimate at time \(t\) answers: among individuals represented by group \(g\), what is the estimated probability of having experienced cause \(k\) by time \(t\), before any competing cause occurs? The time horizon and cause must be named when reporting it. For example, an estimate of 0.30 at 3 years means an estimated 30% probability of cause 1 by 3 years in that group, accounting for competing causes—not a 30% annual rate.
The main alternative is to pool the groups and estimate one cumulative incidence function. That can be suitable if the target is the combined population and the pooled risk sets represent that target. It does not provide separate group estimates, and differing event experience or censoring patterns between groups can make the pooled curve a poor description of either group. Another approach is to estimate cause-specific hazards and combine them to obtain risks; that requires modeling choices beyond the separate nonparametric group calculations here. A cause-specific Kaplan-Meier calculation that treats competing failures as censoring targets a different quantity, often called a net risk, and is not a substitute for observed cumulative incidence in the presence of competing causes.
Before interpreting differences, check that each group’s event counts and risk sets are internally consistent: risk sets should not increase over time unless there is delayed entry, and a failure or censoring should be removed from later risk sets. Confirm that every event has one valid cause and that each group has adequate follow-up through the horizon of interest. Estimates near the end of follow-up can be unstable when few individuals remain at risk. Uncertainty should be reported using an appropriate group-specific variance or resampling method; the point estimates alone do not show how precisely the groups’ probabilities are estimated.
A common mistake is to calculate a group’s cause-specific increment as \(d_{gk}(t_j)/Y_g(t_j)\) and then add those fractions over time. That ignores the probability of reaching each time event-free. The correction is to multiply each fraction by \(\widehat S_g(t_j-)\) before summing. A second common mistake is to use the overall risk set for each group. Each increment must use that group’s own \(Y_g(t_j)\), event-free estimate, and cause-specific event count.
Check Your Understanding
Answer each question using the group-specific definitions and calculations above. Solutions follow the questions.
- In one group, \(Y=20\), 2 individuals fail from cause 1, and 3 fail from cause 2 at the first event time. What is the event-free estimate immediately afterward, and what is the cause-1 incidence increment?
- Why does a censoring at time 1 not contribute directly to a cause-specific cumulative incidence increment, yet still affect an estimate at a later time?
- At a later time in a group, \(\widehat S(t-)=0.6\), \(Y(t)=10\), and 2 individuals fail from cause 2. Find the increment in cause-2 incidence.
- State the main censoring assumption needed for a nonparametric estimate within a group.
- Explain why a pooled cumulative incidence estimate should not automatically be reported as the estimate for each group.
Solutions. (1) The total failure count is \(2+3=5\), so the event-free estimate becomes \(1-5/20=0.75\). The cause-1 increment is \(1(2/20)=0.10\). (2) Censoring is not a failure from either cause, so it adds no event increment. The censored individual is removed from subsequent risk sets, however, reducing later denominators. (3) The increment is \(0.6(2/10)=0.12\). (4) Within the group, censoring must be independent of the future event process, so censored individuals do not have systematically different subsequent event experience from those remaining under observation. (5) The pooled estimate uses pooled event counts and risk sets and describes the pooled target population; group-specific event experience and censoring can differ, so it need not describe either group separately.