= Mean cumulative postoperative event count
The \b[mean cumulative postoperative event count] is the expected number of specified postoperative episodes accumulated per operated patient by a stated time. It measures repeated event burden, rather than only whether a patient had any event. An event definition must distinguish a new episode from continued observation of an existing complication.
Let $N_i(t)$ be patient $i$'s <counting process> for qualifying events up to postoperative time $t$. The population target is
$$
\mu(t)=\operatorname E[N_i(t)].
$$
With complete observation of $n$ independent patients through $t$, its empirical estimate is
$$
\widehat\mu(t)=\frac1n\sum_{i=1}^nN_i(t).
$$
This can exceed one. In contrast, the probability of at least one event is $p(t)=\Pr(N_i(t)\ge1)$ and cannot exceed one. The inequality $\mu(t)\ge p(t)$ follows because each patient with any event contributes at least one to the count.
For an original five-patient example, counts $(0,0,0,0,5)$ give $\widehat\mu=1$ and an observed any-event fraction $1/5$. Counts $(1,1,1,1,1)$ also give mean count $1$, but any-event fraction $1$. The same mean burden can describe very different distributions across patients. Neither count summary encodes the severity of each episode.
Uncertainty should respect the patient as the sampling unit. At fixed $t$, write $K_i=N_i(t)$ and calculate
$$
s_K^2=\frac1{n-1}\sum_i(K_i-\overline K)^2,\qquad
\widehat{\operatorname{SE}}(\overline K)=\frac{s_K}{\sqrt n}.
$$
The first example has $s_K^2=5$ and estimated standard error $1$. Treating its five events as five unrelated patient outcomes would discard the observed clustering. A small-sample estimated standard error is not itself a validated confidence interval.
Death and incomplete follow-up require explicit definitions. For accumulated burden per original operated patient, events cease at death but events already experienced remain counted; the denominator is not silently restricted to survivors. <D. Ghosh> and <D. Y. Lin> studied the marginal mean cumulative number of recurrent events in the presence of a terminal event, including nonparametric estimation and comparison.<Nonparametric analysis of recurrent events and death by D. Ghosh and D. Y. Lin>[Abstract, mean frequency function, terminal event and estimator]{ref}
Loss to follow-up hides later events, whereas death prevents them. Simply averaging the available patients at each time can change the target population. Suitable observation-process assumptions and a censoring-aware method are needed for incomplete data. Comparisons should also show mortality: fewer accumulated events can accompany shorter survival, rather than reduced morbidity among otherwise comparable survivors.
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