Mean Cumulative Function As An Estimand

Why Summarize Recurrent Events With a Mean?

When people can experience the same type of event repeatedly, a first-event summary may leave much of their experience unrepresented. For example, two patients might both have at least one hospitalization by a year, while one had one admission and the other had four. A summary of the probability of any hospitalization treats these histories alike. The mean cumulative function (MCF) instead describes the average number of events accumulated in the population by each time point.

The MCF is an estimand: it specifies the population quantity of interest before choosing an estimator or model. It does not require a parametric event-hazard model. Its interpretation depends on defining the event, the population, the time origin, and how follow-up-ending events are handled. In particular, it summarizes event counts, not event severity.

Definition: Let $N_i(t)$ be the number of events experienced by individual $i$ from the time origin through time $t$. For a specified target population, the mean cumulative function is $$ M(t)=E\{N_i(t)\}, \qquad t\geq 0. $$ It is the population mean number of events per individual accumulated by time $t$. Its units are events per individual; it is not a probability and can exceed 1.

The event count is nondecreasing: if $s\leq t$, then $N_i(s)\leq N_i(t)$, so $M(s)\leq M(t)$. The MCF starts at zero when the event-counting process starts at zero. It may increase in jumps as events accrue. It does not say that every individual experiences the same count, nor does it describe how counts are distributed around their mean.

What the Estimand Means

The time argument matters. $M(6)$, for example, is the expected number of defined events per member of the target population during the first six time units. The units must be specified: if time is measured in months, the estimand refers to six months. The population and event definition must also be clear. A count of symptom episodes is not interchangeable with a count of hospital admissions, even if both are measured in the same people.

The MCF is marginal over the population: it averages the individual event counts, including variation in patients’ characteristics and event histories. It is not conditional on having had a previous event. Nor is it the probability of a first event. A useful identity makes the distinction precise. For any nonnegative integer-valued event count,

E\{N_i(t)\}=\sum_{k=1}^{\infty} P\{N_i(t)\geq k\}.

Thus, the mean count combines the probabilities of having at least one event, at least two, at least three, and so on. A first-event probability uses only the first of these terms. The MCF can therefore carry information about repeated events that a first-event summary omits.

Worked Example: Calculating the MCF With Complete Follow-Up

Suppose, hypothetically, four patients are followed for six months, and every event is observed. Their cumulative event counts are as follows.

Patient	Count by month 2	Count by month 5
A	0	1
B	1	2
C	2	3
D	1	1

With complete follow-up, the empirical MCF at a given time is the average of the individual cumulative counts. At month 2, the total is $0+1+2+1=4$, so

\widehat M(2)=\frac{4}{4}=1.00\text{ event per patient}.

At month 5, the total is $1+2+3+1=7$, giving

\widehat M(5)=\frac{7}{4}=1.75\text{ events per patient}.

These are averages, not claims that each patient had exactly 1.00 or 1.75 events. The increase from month 2 to month 5 reflects additional accumulated events in the observed group.

Estimating the MCF With Right Censoring

If follow-up ends for some individuals before the time of interest, simply averaging their observed counts generally understates the target population’s event burden. An individual censored early has less opportunity for events to be observed. Under suitable assumptions, a risk-set estimator accounts for the amount of follow-up contributed by each individual at each event time.

Suppose $n$ independent individuals enter follow-up at time zero. Let $C_i$ be individual $i$’s censoring time, and define $Y(u)$ as the number still under observation immediately before time $u$. Let $dN_i(u)$ be the number of that individual’s events at $u$, and let $dN_{\mathrm{all}}(u)=\sum_{i=1}^{n}dN_i(u)$ be the total number of events at that time. The risk-set estimator is

\widehat M(t)=\sum_{u\leq t}\frac{dN_{\mathrm{all}}(u)}{Y(u)}.

At each event time, the estimator adds the number of events divided by the number still under observation just before that time. If multiple events occur at the same time, they are all included in $dN_{\mathrm{all}}(u)$. Individuals remain under observation after recurrent events unless their follow-up actually ends; an event does not, by itself, remove someone from the risk set.

Assumptions for the risk-set estimator: Individuals are sampled from the target population, and censoring is independent of the future event process (or is independent conditional on variables appropriately accounted for). Follow-up begins at the stated time origin, the event and censoring times are recorded consistently, and there is adequate follow-up support through the times being estimated. The estimator does not impose a parametric model for the event process, but it does rely on these observation and censoring conditions.

The justification is that, under independent censoring, the individuals still observed just before $u$ represent the target population with respect to their event increments at $u$. In a small time interval around $u$, the expected number of observed events is approximately the number under observation multiplied by the population’s mean event increment. Dividing the observed event count by $Y(u)$ estimates that increment; summing the increments estimates the cumulative mean count. More formally, if $G(u)=P(C_i\geq u)$ and censoring is independent of the event process, then

E\{I(C_i\geq u)dN_i(u)\} =G(u)E\{dN_i(u)\}.

The expected number under observation is $nG(u)$, while the expected total observed event increment is $nG(u)E\{dN_i(u)\}$. Their ratio is the population mean increment $E\{dN_i(u)\}$. This argument explains why the risk-set contribution is useful; consistency additionally requires adequate sample size and follow-up support, as well as the stated censoring assumptions.

Worked Example: Risk-Set Calculation With Censoring

Consider five hypothetical patients. Their observed events occur at months 1 (one event), 2 (two events), and 4 (one event). One patient is censored at month 3, and the other censoring times are after month 4.5. Assume censoring is independent of the event process and there is no censoring tied with an event time.

All five patients are under observation immediately before month 1, so the contribution at month 1 is $1/5=0.200$. All five are also under observation immediately before month 2; two events occur then, contributing $2/5=0.400$. The patient censored at month 3 is no longer in the risk set at month 4, leaving four under observation just before month 4. The event at month 4 contributes $1/4=0.250$. Therefore,

\widehat M(4)=\frac{1}{5}+\frac{2}{5}+\frac{1}{4} =0.200+0.400+0.250 =0.850\text{ events per patient}.

This estimate adjusts for the shorter observation of the patient censored at month 3. It is not the raw observed total divided by five, which would be $4/5=0.800$. The risk-set estimate is meaningful as an estimate of the target population MCF only if the censoring and follow-up assumptions are credible.

Interpretation, Alternatives, and Limits

The MCF is useful when the scientific question concerns cumulative recurrent-event burden. It puts event histories with different numbers of recurrences on a common scale and gives a population average at every time point. It can be compared across populations or treatment groups, provided the event definitions, time origins, follow-up rules, and target populations are made comparable. An observed difference describes a difference in mean cumulative counts; by itself, it does not establish a causal effect.

The MCF answers a different question from a first-event probability or time-to-first-event analysis. A first-event analysis concerns whether, or when, the first event occurs. It discards subsequent event counts. Conversely, the MCF does not directly report the proportion with any event, the distribution of individual counts, or how severe the events were. A rate is another possible summary, but it has a different interpretation and should not be substituted for the cumulative mean without a clear scientific reason.

Worked Example: Same Mean, Different Event Experience

At a hypothetical 12-month assessment, suppose two groups each contain six patients with complete follow-up. Group A has cumulative counts $0,0,0,0,3,3$. Group B has counts $0,1,1,1,1,2$. For group A, the mean is

\widehat M_A(12)=\frac{0+0+0+0+3+3}{6}=1.00\text{ event per patient}.

For group B,

\widehat M_B(12)=\frac{0+1+1+1+1+2}{6}=1.00\text{ event per patient}.

The MCF values are equal, but the proportions with at least one event differ: group A has $2/6=0.333$, while group B has $5/6=0.833$. The groups have the same average count at 12 months but very different distributions of counts. The mean is a useful burden summary, not a complete description of individual experience.

Diagnostics and Common Pitfalls

Before interpreting an estimated MCF, examine the follow-up pattern and the number still under observation over time. At late times, a small $Y(u)$ can make individual contributions $dN_{\mathrm{all}}(u)/Y(u)$ large and the estimate unstable. A curve that becomes erratic near the end of follow-up may reflect sparse risk sets rather than a sudden population-wide change in burden. Report the time range with adequate support, and avoid interpreting the tail as reliable when few individuals remain under observation.

Also check whether censoring could depend on event history or health status. If people with many events are more likely to drop out, the remaining risk set may not represent the population’s future event increments. The simple risk-set estimator can then be biased. Conditional censoring adjustments may be needed, but they require measured predictors and appropriate methods; unmeasured dependence cannot be diagnosed away by a plot. The definition of follow-up also matters: if a terminal event such as death prevents later recurrences, clarify whether the target counts events only while individuals are alive or concerns a different hypothetical setting.

Mistake: treating the MCF as a probability. An estimate of 1.4 events per patient does not mean a 140% chance of an event. It is an average count and may exceed 1.
Mistake: averaging observed counts despite unequal censoring. People with shorter follow-up contribute fewer observable events. Use a censoring-adjusted estimator when its assumptions are credible.
Mistake: removing someone from observation after their first recurrence. Recurrent events do not end follow-up unless the study protocol or a terminal event does so.
Mistake: assuming equal MCF values imply identical experiences. Different count distributions can have the same mean, as the example shows.

Key takeaway: The MCF $M(t)=E\{N_i(t)\}$ is the population’s expected cumulative event count per individual by time $t$. Its definition requires no event-hazard model. Estimation with right censoring, however, requires appropriate censoring and follow-up assumptions.

Check Your Understanding

Answer each question, then compare with the solutions below.

In a population, the probabilities of at least one, two, and three events by time $t$ are 0.50, 0.20, and 0.05, with zero probability of four or more. What is $M(t)$?
At an event time, three events occur while 12 people are under observation. What contribution does this time make to the risk-set estimate?
Why can a raw average of observed counts be misleading when some individuals are censored early?
Can two populations have the same MCF at a particular time but different probabilities of at least one event? Explain briefly.

Solutions. (1) By the tail-sum identity, $M(t)=0.50+0.20+0.05=0.75$ events per individual. (2) The contribution is $3/12=0.250$ events per individual. (3) Early-censored individuals have less time in which events can be observed; their observed counts may not represent their counts through the target time. A risk-set adjustment can address this under independent censoring and adequate follow-up. (4) Yes. The MCF averages all event-count thresholds, whereas the probability of at least one event uses only the first threshold. Different count distributions can therefore share a mean while having different proportions with any event.