Recurrent Event Analysis

How long does it take for an asthma patient to return to the emergency room after a first visit? If the same patient went to the ER three or more times in a year, can all of those visits be included in the analysis? Does the pattern of repeat visits differ by treatment method or air pollution level?

In this way, the method that analyzes the timing and pattern of repetition when the same type of event occurs several times to the same subject is recurrent event analysis. Unlike survival analysis that observes only the first event, it can follow the second, third, and later events.

In an ordinary Cox survival analysis, if only the first ER visit is designated as the event, information on later repeat visits is not included in the analysis. Recurrent event analysis uses models that can account for the gaps between repeat visits, the order of events, and the correlation of events within a patient. However, the Andersen–Gill model and the PWP model differ in how they handle risk sets and the order of events.

Recurrent event analysis is useful for data in which the same event occurs several times, such as ER visits, disease recurrence, repeated infections, frequent machine breakdowns, and repeat customer purchases. In this article, using the example of asthma patients, we look at how the data are structured, the main models, how to interpret hazard ratios, the assumptions of the analysis, and how to report results.

Recurrent event analysis uses the occurrence times of multiple events that happen to the same subject during the observation period. Here, an event is an outcome clearly defined in the study, such as an ER visit due to asthma.

For example, if patient A visited at 2, 5, and 9 months, each of the three visits is recorded. It is easy to assume that visits by different patients are independent, but repeated visits within one patient share past disease status and personal characteristics, so their correlation needs to be taken into account.

Suppose two asthma patients both made their first ER visit one month after follow-up began. After that, patient A never visited again, but patient B visited five more times. An analysis that targets only the first event can hardly tell apart how the two patients fared afterward.

Recurrent event analysis reflects these differences and examines how treatment method and environmental factors relate to the rate of repeat visits. However, you should choose the model according to whether the study's aim is the total number of visits, the time until the next visit, or the risk for each visit in order.

Visits can occur several times between the start and end of each patient's observation. You need to record the treatment start date, treatment changes during follow-up, visit dates, and the date follow-up stopped separately. It is also important to structure the data so that no time intervals overlap.

Suppose we follow 240 asthma patients for 12 months. We record the type of treatment, the patient's baseline severity, monthly air pollution exposure, and the dates of ER visits. The figures and tables below are hypothetical data for learning purposes.

Real ER visit data are affected by many factors, such as seasonality, disease severity, and missed visits to other hospitals. So you should first check when the explanatory variables were measured and the scope of the outcome records.

The table above is an illustrative example close to the start–stop format of the Andersen–Gill model. If you use the PWP model, you need to specify strata and risk sets according to event order, and also decide which time scale to use: total time or time elapsed since the previous event.

The Andersen–Gill (AG) model is a counting-process extension of the Cox model. It records the recurrent events during the observation period over time and estimates the effect of covariates on the event intensity. In its basic form, it does not use a separate baseline hazard for each event order.

Since events from the same patient are not independent, in practice robust (sandwich) standard errors clustered by individual are often used. This means you need to understand the estimation of the coefficients and the assessment of uncertainty in the standard errors as separate things.

The Prentice–Williams–Peterson (PWP) model takes the order of events into account. A second ER visit is possible only for patients who have already had a first visit, so a separate risk set is set up for each event order.

Using gap time in the PWP model lets you focus on how much time has passed since the previous event. Whichever model you use, check the actual research question and the event process before choosing.

Below is a hypothetical example presented for practice in reading model output tables. It is not a result obtained by fitting actual data on 240 patients.

An HR of 0.72 for intensified treatment means that, under this model and its assumptions, compared with standard treatment, the instantaneous intensity of repeat ER visits is about 28% lower. It does not mean that the probability of visiting the ER during the 12 months is 28 percentage points lower, or that the total number of visits necessarily falls by 28%.

The mean cumulative function expresses how many events have occurred per person, on average, up to a given time. The difference between the lines above is only an example; statistical significance must be assessed separately with confidence intervals and model results.

Correlation within the same patient

Since a patient's multiple visits can be related to each other, variance estimation that accounts for clustering by individual may be needed. Even with robust standard errors, problems such as missing severity information or dependence over time are not all solved.

Event definition and event order

Mixing ER visits with hospital admissions, or scheduled visits with emergency visits, makes the event definition unclear. When using the PWP model, check that there are enough events for each event order.

Proportional hazards and time-varying covariates

Check the proportional hazards assumption that the effect stays constant throughout the observation period, and when fine particulate exposure or treatment type changes, reflect the time of the change. When treatment changes right after an event, also consider the possibility of reverse causation.

Loss to follow-up and competing events

A patient's death or loss to follow-up can prevent observation of later visits. In particular, death can be a competing event that is hard to treat as simple independent censoring of ER visits, so consider an extended analysis suited to the situation.

The approach depends on whether timing information matters, whether the order of events within patients matters, and whether follow-up lengths differ. Simple count comparisons alone can hardly explain the gaps between events or the order of transitions.

You should report not only the number of study subjects but also the total number of events, the distribution of events per subject, the follow-up period, the event definition, censoring, the time scale, and the recurrent event model used. If the standard errors were adjusted for clustering by individual, state this as well.

In an actual paper, the results should reflect the exact numbers of observed events and censored observations, diagnostics of the model assumptions, and the content of sensitivity analyses.

When analyzing asthma patients' ER use, looking only at the first visit misses differences in later repeat visits. Recurrent event analysis uses the occurrence times of multiple visits to help estimate differences in event intensity by treatment type or environmental exposure. In particular, the Andersen–Gill model, which analyzes all repeat visits, and the PWP model, which distinguishes the order of events, can be chosen according to the research question.

However, it is important to properly handle the correlation of repeated events within the same patient, missing visit records, the end of follow-up, treatment changes over time, and risk sets by event order. The hazard ratio, too, should not be read directly as the probability of returning or as the causal effect of treatment. You can trust the results only after first checking how the event was defined and when it was observed.