Estimating The Mean Cumulative Function

What the Mean Cumulative Function Measures

A recurrent-event history records not only whether a person has an event, but also how many events have occurred by each time. Examples include repeated hospital admissions or recurrent infections. A useful population summary is the mean cumulative function (MCF): the average number of events accumulated per person by time $t$. Unlike a probability of having an event, this mean can exceed one.

Right censoring complicates estimation because some participants stop being observed before the time horizon of interest. Simply averaging the event counts that happened to be observed can give too much weight to people with longer follow-up. The estimator below adjusts each observed event increment using the number of participants still under observation at that time.

Definition: Let $N_i(t)$ be the number of recurrent events for person $i$ by time $t$, with $N_i(0)=0$. For a population of individuals sampled from the target population, the mean cumulative function is $$ M(t)=E\{N_i(t)\}. $$ It is the expected cumulative event count per person by time $t$, not the probability of at least one event.

Suppose person $i$ is followed until right-censoring time $C_i$. Events are observed up to that time, and their observed history is denoted $N_i^{\mathrm{obs}}(t)$. At a time $u$, define $Y_i(u)=1$ if person $i$ is still under observation immediately before $u$, and $Y_i(u)=0$ otherwise. The aggregate risk set size is $Y.(u)=\sum_iY_i(u)$. Each person who remains under observation stays in this risk set after a recurrence: a prior event does not remove that person from follow-up.

Estimator: At an observed event time $u$, let $dN_i^{\mathrm{obs}}(u)$ be the number of events for person $i$ at that time. The nonparametric MCF estimator is $$ \widehat M(t)=\sum_{u\le t:\,Y.(u)>0} \frac{\sum_{i=1}^n dN_i^{\mathrm{obs}}(u)}{Y.(u)}. $$ Each jump is the number of events observed at that time divided by the number of people still under observation immediately before it.

This is a cumulative estimator: its increments are nonnegative, so the estimated curve never decreases. Its units are events per person. It does not require a proportional recurrent-event effect or a specified parametric distribution for event times.

Assumptions and Why the Risk-Set Adjustment Works

The basic estimator targets the population MCF under the following conditions. Individuals are independent and identically distributed draws from the target population. Their recurrent-event histories are fully observed until censoring. Most importantly, censoring is independent of the complete recurrent-event process: $C_i$ is independent of $N_i(\cdot)$. There must also be a positive chance of remaining observed over the time interval being estimated. This last condition is called positivity of follow-up; an interval beyond which nobody remains observed cannot be estimated by this formula.

The independence assumption means that, at a given time, people still observed are not systematically more or less likely to have event increments than the target population simply because they remain observed. A short justification makes the adjustment clear. For an interval beginning at $u$, write $\Delta N_i$ for the number of events in that interval and $Y_i$ for the indicator that person $i$ is observed through the interval. Independence gives

E(Y_i\Delta N_i)=E(Y_i)E(\Delta N_i).

If $E(Y_i)>0$, the ratio of these expectations is $E(\Delta N_i)$. With independent individuals, sample averages of $Y_i\Delta N_i$ and $Y_i$ converge to their expectations, so their ratio estimates the population mean increment. Summing estimated increments over time gives the MCF estimator. This is an asymptotic justification; a finite-sample ratio is not generally exactly unbiased.

In practice, ties between censoring and event times require a consistent convention for whether a person is at risk at the tied time. With continuous event and censoring times, exact ties have probability zero. If times are recorded on a coarse scale, the analysis convention should be specified and used consistently.

Worked Example: Building an MCF Curve From Censored Histories

Consider four hypothetical participants. Their event times and censoring times (in months) are: A has events at 1 and 3 and is censored at 5; B has an event at 2 and is censored at 4.2; C has events at 1.5 and 4 and is censored at 6; D has no events and is censored at 2.5. Assume censoring is independent of each participant’s recurrent-event history and there are no tied censoring and event times.

At month 1, all four participants are still under observation and one event occurs. The jump is $1/4=0.25$ events per person. The same risk-set size applies at months 1.5 and 2, so each of those one-event jumps is also $1/4=0.25$. The running estimate at month 2 is therefore $0.25+0.25+0.25=0.75$.

Participant D is censored at month 2.5, so three people remain under observation at month 3. A has an event then, giving a jump of $1/3=0.3333$. At month 4, A, B, and C are still under observation immediately before the event; C has an event, giving another $1/3=0.3333$. Thus $$ \widehat M(4)=\frac14+\frac14+\frac14+\frac13+\frac13=1.4167 $$ events per person. Under the stated assumptions, the estimate describes an average of about 1.42 recurrent events per person by month 4 in the target population.

Reading the Estimate and Checking the Calculation

The curve is a step function, with a jump at each observed event time. The jump size depends on the observed event count and the risk set at that time, not on the number of people who have remained event-free. If several events occur at the same recorded time, their total count goes in the numerator. The risk set includes all individuals still under observation immediately before that time, including those with earlier recurrences.

The MCF is not bounded above by one. For example, an estimate of $1.42$ events per person by month 4 does not mean that 142% of people had an event. It is a mean count, allowing some people to experience multiple events. Nor does the value itself identify the proportion who experienced at least one event: different distributions of event counts can have the same mean.

Worked Example: The Complete-Follow-Up Check

Suppose four hypothetical participants are all followed through month 3. A has events at months 0.5 and 2; B has one event at month 1; C has no events; and D has events at months 0.7, 2.5, and 3. There is no censoring before month 3.

The risk set has size four at each of the six event times. Each event therefore contributes $1/4=0.25$, and $$ \widehat M(3)=6\left(\frac14\right)=1.5 $$ events per person. Directly averaging the complete event counts gives $(2+1+0+3)/4=1.5$, the same result. This equality is a useful check: with complete follow-up for everyone, the estimator at a time is just the sample mean of the cumulative counts by that time.

Worked Example: Accounting for Early Censoring

Consider five hypothetical participants followed toward month 4. A has events at months 1 and 3 and is censored at 5; B has an event at month 2 and is censored at 4.2; C has an event at month 3.5 and is censored at 5; D has no events and is censored at 2.5; and E has an event at month 4 and is censored at 5. Assume independent censoring and no event–censoring ties.

The first two events, at months 1 and 2, each occur with all five participants under observation, contributing $1/5$ each. By month 3, D has been censored, so the event for A contributes $1/4$. The events at months 3.5 and 4 also each have four people at risk and contribute $1/4$ each. Therefore, $$ \widehat M(4)=\frac15+\frac15+\frac14+\frac14+\frac14=1.15 $$ events per person. The observed counts through month 4 sum to five, or one per enrolled participant on a raw average. That raw average does not adjust for D’s shorter observation. The estimator instead uses the people still observed at each event time to estimate the population mean increment.

Diagnostics, Alternatives, and Limitations

The estimate should be examined alongside the risk-set sizes. A curve that continues to rise when only a few people remain under observation can be highly sensitive to individual events; at times with no one at risk, the formula cannot estimate further increments. Report the time range supported by follow-up and avoid presenting a long tail as if it were equally well supported throughout. Also check that event times, censoring times, and the definition of a recurrent event are recorded consistently.

Independent censoring is a substantive assumption, not something guaranteed by applying the formula. It is implausible, for example, if worsening health both raises recurrence frequency and causes participants to drop out. Comparing observed censoring patterns across groups or baseline characteristics can reveal concerns, but cannot prove independence from unobserved future event histories. If censoring is conditionally independent only after accounting for measured covariates, an unconditional MCF estimate may not suffice; an appropriately adjusted analysis is needed.

The estimator is a descriptive, marginal summary of cumulative burden. Rate models instead specify or estimate event rates and can relate those rates to covariates; their assumptions and coefficients answer different questions. A raw average of observed counts is simpler but fails to account for unequal follow-up. A survival estimator for time to first event answers a different question, because it discards recurrences after the first event. None of these alternatives is automatically interchangeable with the MCF.

The basic estimator also does not resolve every design issue. It assumes a common time origin and does not by itself handle delayed entry, informative terminal events such as death, or censoring that depends on event history. The point estimate alone also does not quantify sampling uncertainty; uncertainty requires inference that respects the fact that repeated events from the same person are clustered. Those limitations matter when interpreting or reporting a curve.

Common mistake: After a participant’s first recurrence, remove that person from the risk set as though the target were time to first event. Correction: For the MCF, the person remains under observation and at risk for further recurrences until censoring. Removing them would estimate a different quantity and would omit later event increments.

Check Your Understanding

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

What does an MCF estimate of 1.8 events per person by month 6 mean, and why is it not a probability?
At an event time, 12 people are under observation and 3 events are recorded. What is the MCF jump?
Should someone leave the MCF risk set after their first recurrence if they remain under observation? Explain.
What key assumption is threatened if people with rapidly recurring events are more likely to drop out?

Solutions. (1) It estimates an average of 1.8 cumulative events per person by month 6; multiple events per person are allowed, so the value can exceed one and is not a probability. (2) The jump is $3/12=0.25$ events per person. (3) No. The person remains under observation and can contribute later recurrent events until censoring. (4) Independent censoring is threatened because censoring is associated with the recurrent-event process; the risk-set adjustment may then be biased for the target population.

Estimating The Mean Cumulative Function

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