Cause-Specific Event Increments

From Event Counts to Cause-Specific Increments

At an event time, the risk set tells us how many individuals were still under observation and had not yet experienced an event. The event counts tell us how many of those individuals experienced each cause. Dividing a cause’s count by the size of that pre-event risk set gives a local, cause-specific event increment. This calculation is a building block for estimating event probabilities over time, but a single increment is not itself the probability of experiencing that cause by a later horizon.

The key is to use the same denominator for every cause occurring at the same time. If several people experience different causes at a tied event time, all those counts are measured against the risk set just before that time. Removing one tied event before calculating another cause’s increment would change the question and understate the increments.

Definition: At observed event time $t_j$, let $Y_j$ be the number of individuals at risk immediately before $t_j$, and let $d_{kj}$ be the number who experience cause $k$ at $t_j$. The estimated cause-specific event increment for cause $k$ at $t_j$ is $$ \Delta \widehat{A}_k(t_j)=\frac{d_{kj}}{Y_j}. $$ It is the observed fraction of the pre-event risk set experiencing cause $k$ at that event time. If $d_j=\sum_k d_{kj}$ is the total number of events from all causes at $t_j$, then the overall event increment is $d_j/Y_j$.

The increment is local to one time point. It does not accumulate the earlier experience of the risk set, and it does not by itself give a subject’s probability of experiencing cause $k$ by a specified horizon. That distinction matters because earlier events and censoring affect who remains at risk at later times. The cumulative incidence function combines information over time; its estimator is the subject of later material.

Why the Denominator Is the Pre-Event Risk Set

The numerator counts individuals who experience the cause at $t_j$; the denominator is the group that could experience an event at that time. Therefore, $Y_j$ includes individuals who were still event-free and under observation just before $t_j$, not everyone originally enrolled and not merely the individuals who had an event at $t_j$.

This ratio is a direct empirical conditional fraction: among the $Y_j$ individuals in the risk set, $d_{kj}$ had cause $k$ at that time. Its interpretation as evidence about a target population requires appropriate study conditions. In particular, for risk-set comparisons to represent the corresponding event-free population over follow-up, censoring must not selectively remove individuals in a way that carries unaccounted information about their subsequent event experience. The independent-censoring condition is a separate assumption from the arithmetic itself.

With competing risks, each individual can experience at most one first event, and each observed event is assigned one cause. Thus tied event counts for different causes refer to different individuals. At a given time, $d_j=\sum_k d_{kj}\leq Y_j$, so the cause-specific increments are nonnegative and sum to no more than one. Their sum equals the overall event increment $d_j/Y_j$.

Worked Example: Two Causes at the Same Event Time

In a hypothetical cohort, 20 individuals are in the risk set immediately before day 6. At day 6, three experience cause A and two experience cause B. No other cause occurs at that time. Calculate each cause-specific increment and the overall event increment.

For cause A, $d_{A}=3$ and $Y=20$: $$ \Delta \widehat{A}_{A}(6)=\frac{3}{20}=0.15. $$

For cause B, $d_{B}=2$ and the same pre-event risk set applies: $$ \Delta \widehat{A}_{B}(6)=\frac{2}{20}=0.10. $$

There are $d=3+2=5$ events in total. The overall event increment is $5/20=0.25$, and the cause-specific increments add to $0.15+0.10=0.25$. Thus, at this time, one quarter of the pre-event risk set experienced an event, with 0.15 of the risk set experiencing cause A and 0.10 experiencing cause B.

The denominator for cause B is not $20-3=17$. All five events occur at the same time, so neither cause’s count is removed before calculating the other’s increment.

Calculating the Risk Set Before Calculating Increments

The formula is simple once $Y_j$ is known. The harder practical step is often determining who is in that risk set. Start with the individuals under observation and event-free at the beginning of follow-up. Before a given event time, remove individuals who had an earlier event and those who were censored before that time. The remaining number is $Y_j$. An individual censored at an earlier time contributes to risk sets before that censoring time, but not to later risk sets.

Worked Example: Using Earlier Events and Censoring

In a hypothetical study, 30 individuals begin follow-up. Before day 5, two experience an event and three are censored. At day 5, one individual experiences cause A and three experience cause B. Find the risk-set size and the cause-specific increments.

The two earlier events and three earlier censorings each remove one individual from the risk set by day 5. Therefore, $$ Y(5)=30-2-3=25. $$

For cause A, the event count is one: $$ \Delta \widehat{A}_{A}(5)=\frac{1}{25}=0.0400. $$ For cause B, the event count is three: $$ \Delta \widehat{A}_{B}(5)=\frac{3}{25}=0.120. $$

The total event increment at day 5 is $(1+3)/25=4/25=0.160$, which equals $0.0400+0.120$. These are fractions of the 25 individuals at risk immediately before day 5. They are not fractions of the original 30-person cohort.

A Short Justification and an Important Interpretation

The ratio follows from the definition of an empirical conditional fraction. Consider the $Y_j$ individuals in the risk set immediately before $t_j$. For each such individual, record whether cause $k$ occurs at $t_j$. The number of “yes” outcomes is $d_{kj}$, so the observed proportion with that outcome is $d_{kj}/Y_j$. The denominator must be the size of the group about which that fraction is being calculated.

This reasoning also verifies the sum across causes. Because each individual can have at most one first event, the cause-specific event groups at $t_j$ do not overlap. Their total size is $d_j=\sum_k d_{kj}$. Consequently, $$ \sum_k \Delta \widehat{A}_k(t_j) =\sum_k\frac{d_{kj}}{Y_j} =\frac{\sum_k d_{kj}}{Y_j} =\frac{d_j}{Y_j}. $$ Since $d_j\leq Y_j$, this sum cannot exceed one.

For event times in a continuous-time setting, the probability of an event at one exact instant may be zero. In that setting, these count-over-risk-set quantities are conventionally understood as estimates of local hazard increments, rather than literal positive probabilities at an exact instant. The simple conditional-fraction interpretation applies directly to observed event-time data, especially when times are recorded in discrete units. In either setting, the calculation remains $d_{kj}/Y_j$; interpretation should match the time scale and model.

Worked Example: Why an Event-Only Denominator Gives a Different Quantity

At day 9 in a hypothetical cohort, 12 individuals are at risk. Two experience cause A and one experiences cause B. Compare the cause-specific increments with the shares of events attributed to each cause.

The cause-specific increments use the full pre-event risk set: $$ \Delta \widehat{A}_{A}(9)=\frac{2}{12}=0.167,\qquad \Delta \widehat{A}_{B}(9)=\frac{1}{12}=0.0833. $$ The overall event increment is $3/12=0.250$.

If the denominator is instead the total number of events, the event shares are $2/3=0.667$ for cause A and $1/3=0.333$ for cause B. Those shares answer a different question: among the three events observed at day 9, what fraction had each cause? They do not describe the fractions of the 12 people at risk who experienced each cause at that time.

The event shares add to one because they divide up the events. The cause-specific increments add to 0.250 because only three of the 12 at-risk individuals had an event at day 9. The two calculations are both valid, but they have different denominators and interpretations.

Diagnostics, Alternatives, and Limitations

Several checks can reveal calculation or interpretation errors:

Check the risk-set count. Confirm that it is the number at risk just before the event time, after removing earlier events and censorings.
Check the event counts. Each cause count must be nonnegative, and the total across causes cannot exceed the risk-set size.
Check the sum. Cause-specific increments at a tied time should sum to the total event count divided by the risk-set size.
Check the time ordering. Do not remove individuals with events at $t_j$ until after calculating every cause’s increment at $t_j$.
Check the interpretation. A local increment is not a cumulative incidence by a later time and is not an event share among those who experienced events.

The most tempting alternative denominator is the total number of events at that time. It is appropriate for describing the distribution of causes among those events, but not for estimating the event increment among the risk set. Another tempting approach is to treat competing events as ordinary censoring and then calculate a single-cause survival complement. Competing events are outcomes, not loss to follow-up; that approach does not produce the cause-specific event increment defined here. Standard survival methods remain relevant for their intended estimands, but their interpretation must not be substituted for this risk-set calculation.

The count-over-risk-set ratio is a local summary. It does not account, by itself, for the full history of the risk set or provide uncertainty estimates, and sparse risk sets can make it unstable. Misclassified causes also affect the cause-specific numerator. Finally, independent censoring and appropriate observation of event times matter for extending sample calculations to population quantities. These considerations do not change the formula, but they limit what can be concluded from it.

Common Mistake

Common mistake: “There are four cause A events and two cause B events at this tied time, so I will remove the four cause A events first and use a smaller denominator for cause B.” This treats tied events as if they occurred in a known sequence. The correction is to use the same pre-event risk-set size for every cause at that time: $\Delta\widehat{A}_k(t_j)=d_{kj}/Y_j$. Sequential denominators are not the cause-specific increments defined here.

Key takeaway: At each observed event time, divide each cause’s event count by the number at risk immediately before that time. Use that same denominator for all causes tied at the time, then verify that their increments sum to the overall event increment.

Check Your Understanding

Try each question, then compare your reasoning with the solutions.

A pre-event risk set contains 40 individuals. At that time, four experience cause A and two experience cause B. Calculate both cause-specific increments and the overall event increment.
At a tied event time, should cause B’s denominator be reduced by the number who experienced cause A at that same time? Explain.
At a time with 18 individuals at risk, one has cause A and two have cause B. What are the cause-specific increments, and what do they sum to?
Five events occur at a time, of which three are cause A. What does $3/5$ describe, and why is it not necessarily cause A’s event increment?

Solutions. (1) Cause A: $4/40=0.100$; cause B: $2/40=0.0500$; total: $6/40=0.150$. (2) No. All causes at the tied time use the same pre-event risk set; removing tied events first imposes an unsupported sequence. (3) Cause A: $1/18=0.0556$; cause B: $2/18=0.111$; together they sum to $3/18=0.167$, the overall event increment. (4) $3/5$ is the share of observed events at that time attributed to cause A. The cause A increment divides its event count by the pre-event risk-set size, not by the event count across causes.