Recurrent Events From Routine-Care Records

Why Recurrent Events Need Their Own Definition

Many clinical outcomes can occur more than once for the same patient: hospital admissions, exacerbations, infections, or rescue-medication episodes. A time-to-first-event analysis discards later events, while a simple total count ignores how long each patient could be observed. Comparing trial patients with routine-care external controls therefore requires both a prespecified event definition and an appropriate measure of observation time.

The basic rate is events divided by person-time at risk. This handles unequal observation duration at the group level, but does not by itself make trial and external-control populations comparable. Differences in prognosis, care pathways, event recording, or eligibility can still distort an external comparison. The earlier material on outcome regression, standardization, overlap, and external-control bias applies to recurrent-event rates as well as to other endpoints.

The event definition must specify what counts as a new event, whether a minimum event-free interval is required, how transfers or duplicate records are handled, and when observation begins and ends. For example, two emergency visits on consecutive days may represent either two events or one continuous episode, depending on the clinical definition. Follow-up should contribute only time during which the patient is eligible to experience the endpoint and the record system can plausibly capture it. Death, loss to follow-up, treatment discontinuation, and database exit may end or alter observation, but their handling must be stated rather than inferred from the event count.

A patient can contribute many events but only one accumulated person-time value. If event occurrence itself changes subsequent eligibility or risk time, the convention must be explicit. In particular, a rate based on time until treatment discontinuation estimates a different quantity from one based on continued follow-up after discontinuation.

Person-Time Rates and a Bayesian Count Model

For group \(g\), let \(D_g=\sum_i N_i\) be the total event count and \(T_g=\sum_i Y_i\) the total person-time. The observed rate is \(D_g/T_g\). A group with fewer events can have a higher rate if it contributed substantially less observation time, so counts alone are not a fair rate comparison.

A simple model treats the group count as Poisson conditional on its total person-time: \(D_g\mid \lambda_g,T_g\sim \operatorname{Poisson}(\lambda_gT_g)\), where \(\lambda_g\) is the event rate. This model assumes a common rate within each group over the analyzed follow-up, conditionally independent event increments, and no extra variation in counts beyond Poisson variation. It also requires comparable event definitions and sufficiently complete ascertainment. The model is a useful starting point, not a guarantee that recurrent events are independent within patients.

The prior parameters have rate units: \(b_g\) is measured in person-time, so \(b_g+T_g\) is also person-time. The shape parameter behaves like an event-count contribution in the posterior update. Priors should be chosen on a clinically sensible rate scale and examined for sensitivity; a prior that seems numerically small in one unit of time may not be small after changing years to days.

For independent group-specific priors and likelihoods, the posterior distributions for \(\lambda_T\) and \(\lambda_C\) are independent, where \(T\) denotes trial and \(C\) denotes external control. A central comparative quantity is the rate ratio \(\theta=\lambda_T/\lambda_C\). Its posterior summarizes uncertainty in the relative event rate, conditional on the model, prior, observed data, and comparison assumptions.

Why the Posterior Update Has This Form

The likelihood for a group count \(D\), given person-time \(T\), is proportional as a function of \(\lambda\) to \(\lambda^D e^{-\lambda T}\); factors not involving \(\lambda\) can be omitted. Multiplying by a Gamma prior kernel gives

This is the kernel of a Gamma distribution with shape \(a+D\) and rate \(b+T\), which establishes the posterior update. The same result also shows why person-time, not merely the number of patients, enters the rate update.

Adjustment and a Common Target Population

A crude rate comparison can be misleading if trial and external-control patients differ in prognostic characteristics. Adjustment can be performed by estimating rates within covariate strata and standardizing them to a shared target distribution. This requires adequate overlap and measured covariates sufficient for the intended comparison; it does not eliminate bias from unmeasured or poorly measured differences.

One transparent rate-standardization calculation uses target weights \(w_j\), where \(w_j\geq0\) and \(\sum_j w_j=1\), across strata \(j\). If \(r_{gj}\) is the estimated rate for group \(g\) in stratum \(j\), then the standardized rate is \(\sum_j w_jr_{gj}\). The weights define the target population and must be the same for both groups. A Bayesian analysis can carry uncertainty in the stratum-specific rates through to the standardized rates and their ratio.

Diagnostics, Alternatives, and Limitations

Before interpreting a rate ratio, examine whether the count model and data construction are credible. Useful checks include plotting cumulative events against person-time, comparing rates over calendar time and follow-up intervals, inspecting patient-level event-count distributions, and checking whether a small number of patients contribute a large share of events. Compare mean and variance of counts across clinically meaningful groups; substantial excess variation is a warning that a simple Poisson model may understate uncertainty.

Routine-care records can miss events, duplicate encounters, or change coding practices over time. Unequal sensitivity of event capture between the trial and external data source can bias a rate ratio even when person-time is calculated perfectly. Also inspect how observation ends. If loss to follow-up is related to event risk, treating the remaining observed time as representative may be misleading. Calendar-time drift, selection into the external cohort, and residual confounding remain concerns from external-control analyses generally.

The aggregate Gamma-Poisson model is attractive for its clear person-time interpretation and conjugate update. It is limited when rates vary substantially across patients or time, recurrent events are strongly dependent, or event intensity changes after a prior event. In those settings, a hierarchical model, negative binomial model, or recurrent-event model may better represent the data. Model complexity does not solve poor endpoint harmonization or external-control confounding, and alternative models should be justified by the estimand and data structure rather than chosen only to obtain a preferred result.

A further interpretive pitfall is to describe a rate ratio below one as a reduction in the probability that a patient has any event. A rate ratio compares event rates per person-time, not the proportion of patients who ever experience an event. Recurrent events can also make a group’s total count sensitive to a few patients with many episodes. Report the event definition, total events, person-time, rate units, model, uncertainty, and the limitations of the external comparison together.