Recurrent-event data record a sequence \[ 0<T_{i1}<T_{i2}<\cdots \] rather than one terminal time. The primary target may be an event intensity, a marginal mean count, an event-number-specific hazard, or a gap-time distribution. These are not interchangeable, especially when a terminal event stops recurrence.
Counting-process targets
Definition. Let \[ N_i(t)=\sum_{k\ge1}1\{T_{ik}\le t\} \] be the recurrent-event count and \(Y_i(t)\) the observation indicator. Its conditional intensity is \[ \lambda_i(t\mid\mathcal F_{t-}) =\lim_{h\downarrow0} \frac{E\{N_i(t+h)-N_i(t)\mid\mathcal F_{t-}\}}{h}. \] The marginal mean function is \[ \mu(t)=E\{N_i(t)\}. \]
Proposition. If observation is complete and \(E\int_0^t\lambda_i(u)\,\mathrm{d}u<\infty\), then \[ \mu(t)=E\int_0^t\lambda_i(u\mid\mathcal F_{u-})\,\mathrm{d}u. \]
Proof. The Doob-Meyer decomposition gives \[ N_i(t)=M_i(t)+\int_0^t\lambda_i(u)\,\mathrm{d}u. \] Taking expectations and using \(EM_i(t)=0\) yields the identity.
Mean cumulative function
Definition. Under independent observation censoring, the Nelson-type mean cumulative function estimator is \[ \widehat\mu(t) =\int_0^t\frac{\mathrm{d}N(u)}{Y(u)}, \qquad N=\sum_iN_i,\quad Y=\sum_iY_i. \] Unlike a survival cumulative hazard, \(\widehat\mu(t)\) may exceed one.
Theorem. If subjects are independent, censoring is noninformative, and the limiting observation probability \(y(t)\) is bounded away from zero, then \[ \widehat\mu(t)-\mu(t) =\frac1n\sum_{i=1}^n\phi_i(t)+o_P(n^{-1/2}) \] uniformly on compact intervals for a subject-level influence process \(\phi_i\). Consequently, \[ \sqrt n(\widehat\mu-\mu) \rightsquigarrow G \] for a centered Gaussian process whose covariance is estimated by empirical cluster influence products.
Proof outline. Write aggregate event increments as conditional mean plus subject martingales. Expand the inverse random risk proportion \(1/\{Y(t)/n\}\) around \(1/y(t)\). The leading term is a sum of independent subject-level martingale and risk-process contributions. A functional central limit theorem gives the Gaussian limit; within-subject event dependence remains inside each \(\phi_i\), which is why variance is clustered by subject.
Andersen-Gill intensity model
Definition. The Andersen-Gill model is \[ \lambda_i(t\mid\mathcal F_{t-}) =Y_i(t)\lambda_0(t)e^{\beta^\top Z_i(t)}. \] After each event the subject may return immediately to the risk set. The Cox partial likelihood uses every recurrent event and calendar or total time as its clock.
Theorem. If the conditional intensity model is correct, the partial-likelihood score is a martingale and \[ \sqrt n(\widehat\beta-\beta_0) \Rightarrow N(0,A^{-1}BA^{-\top}), \] where \(A\) is limiting partial-likelihood curvature and \(B\) is the variance of the subject-level score. If the model also supplies the correct conditional covariance structure, \(A=B\); otherwise the subject-cluster sandwich remains valid under suitable moment conditions.
Proof. Each subject contributes \[ U_i(\beta_0) =\int\{Z_i(t)-e(t)\}\,\mathrm{d}M_i(t). \] Subjects are independent, but events within one subject need not be. Taylor expansion of the aggregate score gives \[ \sqrt n(\widehat\beta-\beta_0) =A^{-1}n^{-1/2}\sum_iU_i(\beta_0)+o_P(1). \] The ordinary independent-subject CLT gives covariance \(A^{-1}BA^{-\top}\). Estimating \(B\) by \(n^{-1}\sum_i\widehat U_i\widehat U_i^\top\) retains all within-subject dependence.
Event-number-specific risk sets
Definition (PWP models). For the \(k\)-th event, define \[ Y_{ik}(t) =1\{N_i(t-)=k-1,\text{ subject observed at }t\}. \] A Prentice-Williams-Peterson total-time model is \[ \lambda_{ik}(t) =Y_{ik}(t)\lambda_{0k}(t) e^{\beta^\top Z_i(t)}. \] The gap-time version resets the clock after event \(k-1\).
Only subjects who have experienced event \(k-1\) enter the \(k\)-th event risk set. This conditions on event history and estimates an event-order-specific effect. By contrast, a Wei-Lin-Weissfeld marginal model treats each event number as a separate process whose risk set may include everyone from baseline; robust covariance accounts for dependence across event types. The two approaches target different populations.
Example. In recurrent hospital admission, an intervention may strongly delay the first readmission but have little effect after several admissions. A common Andersen-Gill coefficient averages across event orders. A stratified PWP model can estimate \(\beta_1,\beta_2,\ldots\), but later-event risk sets are selected survivors with prior events and may be small.
Gap-time models
Definition. The \(k\)-th gap time is \[ G_{ik}=T_{ik}-T_{i,k-1},\qquad T_{i0}=0. \] A renewal model assumes the \(G_{ik}\)'s are identically distributed and conditionally independent given covariates or frailty. A semi-Markov model allows the next-event hazard to depend on elapsed time since the preceding event.
Censoring of gaps. If follow-up ends at calendar time \(C_i\), later gap times are censored by \(C_i-T_{i,k-1}\), a quantity depending on prior event history. Treating observed gaps as an ordinary independent survival sample induces selection. Calendar-time counting-process methods or explicit renewal likelihoods handle this dependence more transparently.
Terminal events
Definition. Let \(D_i\) be a terminal time, such as death, after which recurrence is impossible. The observed count is \[ N_i(t\wedge D_i). \] Death is not independent censoring for recurrence: it is part of the outcome process.
Proposition. The marginal mean count \[ \mu(t)=E\{N(t\wedge D)\} \] decomposes as \[ \mu(t) =\int_0^t P(D\ge u) E\{\lambda_N(u\mid\mathcal F_{u-})\mid D\ge u\} \,\mathrm{d}u. \] Thus it depends jointly on recurrence intensity and terminal-event survival.
Proof. The at-risk indicator for recurrence contains \(1\{D\ge u\}\). Taking expectation of its compensator gives \[ E\{1(D\ge u)\lambda_N(u\mid\mathcal F_{u-})\} =P(D\ge u) E\{\lambda_N(u\mid\mathcal F_{u-})\mid D\ge u\}. \] Integrate over time.
Possible targets include the composite burden \(\mu(t)\), a while-alive rate, joint recurrence-death probabilities, or a causal estimand under treatment. Joint frailty models specify correlated recurrence and death intensities; multistate models treat death as an absorbing transition; inverse-probability methods require a hypothetical estimand in which death is treated as censoring. The scientific question must choose among them.
Robustness and frailty
A sandwich covariance protects standard errors against some misspecified within-subject correlation while retaining the fitted mean or score target. A frailty model instead specifies a latent dependence mechanism and changes both likelihood and marginal interpretation. Robust covariance, frailty, and event-number stratification are therefore complementary modeling choices, not interchangeable variance options.