Modeling Recurrent Event Data

What are Recurrent Events?

Recurrent events are, as the name suggests, events of interest that can happen repeatedly to the same individual over a period of observation. Analyzing only the first occurrence is often insufficient because it ignores the subsequent pattern of events, which may be clinically or scientifically important.

Describing and Visualizing Recurrent Data

In standard survival analysis, we use the Kaplan-Meier curve to visualize the probability of remaining event-free over time. However, since individuals can have multiple events, "event-free" is no longer the right concept. Instead, we need a way to visualize the average burden of events over time.

For this, we use the Mean Cumulative Function (MCF). The MCF at a given time $t$ represents the average number of events that a subject has experienced up to that time. It's a simple, powerful way to see how the event frequency evolves over the study period.

Structuring the Data: Different Views of Time

Before we dive into the models, it's crucial to understand that they require different ways of thinking about and structuring the event data. Your background in information systems gives you a good foundation for appreciating how data representation is key to the analytical method.

Let's look at a visual example. The image below shows recurrent upper respiratory infections (URI) in infants.

Let's break down what these panels show:

• Panel (a): This is the raw data, showing event times for 10 subjects along a continuous follow-up period.
• Panel (b) & (c) - Counting Process / Total Time: Here, the data is restructured. Each subject has multiple rows, one for each interval between events. The time scale is the total time from study entry. For example, for Subject 1, the first event is at day 28, the second at day 56, and so on. The data would be structured as intervals: (0, 28], (28, 56], etc. This is often called the "counting process" format.
• Panel (d) - Gap Time: In this view, the clock is reset to zero after each event. The focus is on the time between events. For Subject 1, the time to the first event is 28 days. The time between the first and second event is 56 - 28 = 28 days. The data for the second event would record a "gap time" of 28 days.

These different ways of measuring time—total time from the start versus gap time between events—are fundamental to the models we'll discuss next.

Modeling Approaches for Recurrent Events

There are two main families of models used for recurrent events.

1. Rate-based Models (Count Models): These models focus on the rate at which events occur. They are conceptually simpler and treat the outcome as a count of events over a period of time.
2. Time-to-Event Models (Cox Extensions): These models are extensions of the Cox proportional hazards model, adapted to handle repeated events. They focus on how covariates influence the instantaneous risk (or intensity) of an event at different points in time.

The following presentation slide provides a great overview of these categories.

1. Rate-Based Models

Poisson Regression

The simplest approach is to model the event count. Poisson regression is a well-known statistical method for modeling count data. The core idea is to assume that for a given individual, events occur at a constant rate over time. The model then relates this rate to the individual's covariates (e.g., age, treatment group).

A critical feature of this model in the context of recurrent events is the use of an offset. Since individuals are followed for different lengths of time, we can't just compare their raw event counts. The offset term mathematically accounts for this differential follow-up time, effectively turning the analysis into one of event rates (events per unit of time).

While simple, the assumption of a constant event rate can be quite strong and is often violated. For this reason, more flexible methods are often preferred. One such extension is Negative Binomial regression, which allows for "overdispersion"—more variability in counts than the Poisson model expects—by assuming that the event rate itself can vary between individuals.

2. Time-to-Event Models (Cox Extensions)

These models are generally more flexible and powerful than the rate-based models. They extend the Cox proportional hazards framework you are already familiar with.

The Andersen-Gill (AG) Model

Also known as the proportional intensity model, the AG model is the most direct extension of the Cox model. It uses the "counting process" data structure we saw in Panel (b) of the image.

• Time Scale: Total time from study entry.
• Risk Set: An individual is considered at risk for an event throughout their entire follow-up period, except for any gaps in observation. They are not removed from the risk set after an event occurs.
• Key Assumption: The baseline intensity function is the same for every event. This means the underlying risk of a third event is assumed to be the same as the risk of a first event, which may not be realistic. For example, the first infection might weaken a patient, increasing their risk for a second one.

The Prentice, Williams, and Peterson (PWP) Models

The PWP models were developed to relax the strong assumption of the AG model. They do this by stratifying the analysis by event number. In essence, they allow the baseline risk to be different for the 1st event, 2nd event, 3rd event, and so on. This is a much more realistic assumption. There are two versions, which differ only in their definition of the time scale.

a) PWP Total Time Model

• Time Scale: Total time from study entry (same as the AG model).
• Risk Set: For the k-th event, the risk set consists only of those individuals who have already experienced k-1 events. For example, only patients who have had at least one heart attack are in the risk set for a second heart attack.
• Advantage: Allows each sequential event to have a different baseline risk profile, and also allows the effect of a covariate (like a treatment) to differ for the first vs. subsequent events.

b) PWP Gap Time Model

• Time Scale: Gap time. The clock is reset to zero after each event. This model analyzes the time between events.
• Risk Set: Similar to the PWP total time model (stratified by event number), but the time scale is different. When constructing the risk set, it compares individuals who have had the same number of prior events and have been "at risk" for the same amount of time since their last event.
• Advantage: This is particularly useful if the research question is about what predicts a faster recurrence after an event has already occurred.

Summary and Model Choice

You've now seen the main approaches. How do you choose between them? The choice depends on the assumptions you are willing to make and the specific research question you want to answer.

• Poisson Regression: Simplest, but makes strong assumptions. Good for a quick look at overall event rates if you believe the rate is relatively constant.
• Andersen-Gill Model: A standard approach, but its assumption of a common baseline intensity for all events is often unrealistic.
• PWP Total Time Model: A robust and flexible choice. It models the overall process from study entry while allowing risk to change after each event. Often a preferred method.
• PWP Gap Time Model: Another robust choice. Use this when the focus is specifically on the time between events and factors that influence the duration of these event-free periods.

The following reading provides an excellent summary comparing these models and giving practical advice on selection.

Conclusion

This lesson concludes our module on advanced survival topics by introducing the methods for analyzing recurrent events. We've seen that moving beyond time-to-first-event analysis can provide much deeper insights into processes characterized by repeated occurrences.

Here are the key takeaways:

• Recurrent events are repeated occurrences of the same event in an individual, and analyzing them requires specialized methods.
• The Mean Cumulative Function (MCF) is the appropriate descriptive tool, showing the average number of events per person over time.
• Modeling approaches can be broadly divided into rate-based models (like Poisson regression) and more flexible Cox-style extensions.
• The main Cox-style models are the Andersen-Gill (AG) model, which assumes a common baseline risk for all events, and the Prentice, Williams, and Peterson (PWP) models, which relax this assumption.
• The PWP models come in two flavors based on the time scale: Total Time (time from study start) and Gap Time (time since the last event), allowing you to tailor the analysis to your research question.

This also marks the end of the theoretical portion of our course. You have built a strong conceptual foundation, from basic epidemiological measures and study designs all the way to advanced survival models for competing risks and recurrent events.

In the next and final module, we will switch gears entirely. We will move from theory to practice, learning how to implement many of these analyses—from calculating odds ratios to fitting Kaplan-Meier curves and Cox models—using the R programming language.