Time-to-First-Event vs. Recurrent Event Analysis

Introduction

Many clinical outcomes are not naturally limited to a single event. A patient may experience several hospitalizations, infections, disease exacerbations, bleeding episodes, fractures, or other clinically meaningful events during follow-up.

When this occurs, an investigator has an important design and analysis decision: should the primary analysis consider only the time to the first event, or should it use information from all relevant events?

These are not simply two statistical techniques for analyzing the same quantity. They generally answer different scientific questions.

Key idea: Time-to-first-event analysis asks how treatment affects the occurrence of the first event. Recurrent-event analysis asks a broader question about the patient's entire event history, such as the rate, timing, or cumulative burden of repeated events.

Choosing between them should therefore begin with the clinical estimand rather than with the availability of a particular statistical procedure.

The Basic Difference

Consider a study in which patients can experience repeated disease exacerbations.

Suppose three patients have the following event histories:

Patient	Event History	Total Events
A	No event during follow-up	0
B	Event at month 4	1
C	Events at months 2, 5, 8, and 11	4

A conventional time-to-first-event analysis treats Patients B and C as having the same basic event status once their first event has occurred.

Patient B has one event and Patient C has four events, but for a time-to-first analysis the distinction after the first event is largely irrelevant to the primary endpoint.

A recurrent-event analysis, by contrast, retains the subsequent events.

Information loss: If repeated events are clinically meaningful, restricting the analysis to the first event can discard potentially substantial information about treatment benefit.

Time-to-First-Event Analysis

In a time-to-first-event analysis, each patient contributes one event time at most. Let:

\(T_i\) = time to the first event for patient \(i\)
\(\delta_i\) = event indicator, equal to 1 if the first event is observed and 0 if the patient is censored

The observed data can therefore be represented as:

\[ Y_i=\min(T_i,C_i) \]

where \(C_i\) is the censoring time. The event indicator is:

\[ \delta_i=I(T_i\le C_i) \]

This creates a standard right-censored survival dataset.

Patient	First Event Time	Event Indicator
1	3.2 months	1
2	7.8 months	1
3	12.0 months	0
4	5.1 months	1

A Kaplan–Meier estimator or Cox proportional hazards model is often used for this type of endpoint.

What Does Time-to-First Event Estimate?

The scientific interpretation is straightforward: How does treatment affect the time until a patient's first occurrence of the specified event?

For a Cox model, the hazard can be written as:

\[ h_i(t)=h_0(t)\exp(\beta Z_i) \]

where \(Z_i\) is the treatment indicator. The hazard ratio is:

\[ HR=\exp(\beta) \]

A hazard ratio below 1 suggests that the treatment is associated with a lower instantaneous hazard of experiencing the first event, under the assumptions of the model.

Important: A first-event hazard ratio does not directly quantify the effect of treatment on the total number of events a patient experiences. A treatment could have little effect on the timing of the first event while substantially reducing subsequent events.

Why First-Event Analysis Can Be Attractive

Time-to-first-event analysis remains extremely useful because it has several advantages.

The endpoint is easy to define and communicate.
Each patient contributes a single event time.
Standard survival-analysis methods are readily available.
Model assumptions are relatively familiar.
Regulatory interpretation can be straightforward when the first event is the clinically relevant event.
The analysis avoids some of the complex dependence structures created by multiple events within the same patient.

For some clinical questions, this is exactly what is wanted.

First event is appropriate when: The first occurrence is itself the clinically important outcome, later events do not materially alter the scientific question, or the protocol explicitly defines treatment benefit in terms of delaying the first event.

The Problem: Patients Can Have Many Events

Suppose a clinical trial evaluates a treatment intended to reduce recurrent hospitalizations. Consider two hypothetical patients:

Patient	First Hospitalization	Subsequent Hospitalizations	Total
A	Month 3	Months 6, 9, 12	4
B	Month 3	None	1

A first-event analysis considers both patients to have their first hospitalization at month 3.

Yet their subsequent clinical experience is dramatically different.

If treatment meaningfully reduces repeated hospitalizations, an analysis limited to the first event may miss part of the treatment effect.

Recurrent Event Analysis

Recurrent-event analysis allows each patient to contribute multiple event times. For patient \(i\), define:

\[ T_{i1} where \(K_i\) is the number of observed events.

The resulting data are no longer one row per patient. Instead, patients may contribute multiple event records or multiple time intervals.

Patient	Start	Stop	Event	Event Number
101	0	2.4	1	1
101	2.4	5.8	1	2
101	5.8	9.1	1	3
101	9.1	12.0	0	4

The exact data structure depends on the recurrent-event model, but the key concept is that the sequence of events is retained.

Different Recurrent-Event Questions

There is no single recurrent-event estimand. An investigator might ask:

Does treatment reduce the rate of recurrent events?
Does treatment delay each successive event?
Does treatment reduce the expected number of events over follow-up?
Does treatment reduce the cumulative burden of events?
Does treatment affect the risk of the second, third, or later event?
Does treatment change the probability of having repeated events?

These questions motivate different methods.

Time-to-First vs. Recurrent Event: Side-by-Side

Feature	Time-to-First Event	Recurrent Event Analysis
Events used	First event only	Multiple events
Typical data structure	One record per patient	Multiple records/intervals per patient
Primary question	When does the first event occur?	What happens across the event history?
Information retained	Limited after first event	Subsequent events retained
Common methods	Kaplan–Meier, Cox	Andersen–Gill, PWP, frailty, MCF
Within-patient dependence	Not applicable to repeated events	Must be addressed
Interpretation	Usually simple	Depends strongly on model

Event Independence Is a Major Issue

Repeated events from the same patient are generally not statistically independent.

A patient who experiences one exacerbation may be more likely to experience another because of underlying disease severity.

This creates within-patient dependence.

If a model incorrectly treats all event records as independent observations, standard errors can be too small and statistical inference can be misleading.

Core principle: Recurrent-event methods must account for the fact that multiple observations belong to the same patient.

The Andersen–Gill Model

The Andersen–Gill model extends the Cox proportional hazards framework to recurrent events using a counting-process representation.

Define:

\[ N_i(t)=\text{number of events experienced by patient }i\text{ by time }t \]

and let \(Y_i(t)\) indicate whether the patient is at risk at time \(t\). The intensity can be expressed as:

\[ \lambda_i(t)=Y_i(t)\lambda_0(t)\exp(\beta Z_i) \]

The treatment coefficient is interpreted through:

\[ HR=\exp(\beta) \]

The model can therefore estimate a treatment effect across recurrent events.

Andersen–Gill interpretation: The model is particularly natural when the recurrent events can be viewed as repeated occurrences governed by a common event process and the treatment effect is intended to describe the overall event rate or hazard across those events.

Why Robust Variance Matters

Because multiple events can come from the same patient, the usual independent observation variance calculation may not be appropriate.

A common approach is to use a robust sandwich variance estimator clustered by patient.

Conceptually:

\[ \widehat{\operatorname{Var}}_{\mathrm{robust}}(\hat{\beta}) = A^{-1}BA^{-1} \]

where the middle component accounts for within-patient clustering.

This allows the recurrent event records to contribute information while recognizing that observations from the same patient are correlated.

The Prentice–Williams–Peterson Model

The Prentice–Williams–Peterson, or PWP, framework takes a different approach. It recognizes the order of events.

For example:

First-event risk
Second-event risk
Third-event risk
Fourth-event risk

The model can therefore be useful when the event number itself has scientific meaning.

Total-Time PWP

In the total-time formulation, time is measured from study entry. A patient becomes eligible for the second-event stratum only after experiencing the first event.

Gap-Time PWP

In the gap-time formulation, the clock is reset after each event. The analysis therefore focuses on the time between successive events.

Approach	Time Scale	Useful Question
PWP Total Time	Time since study entry	How does treatment affect successive events over follow-up?
PWP Gap Time	Time since previous event	How does treatment affect the waiting time between events?

Frailty Models

Another approach is to introduce a patient-specific random effect, often called a frailty.

For example:

\[ h_i(t)=h_0(t)\exp(\beta Z_i+u_i) \]

where \(u_i\) represents unobserved patient-specific susceptibility.

Patients with larger frailty values may systematically experience events more frequently.

The frailty therefore induces dependence among repeated events from the same patient.

When frailty is attractive: A frailty model is particularly useful when patient-level heterogeneity is scientifically important and the investigator wants to model the dependence between recurrent events explicitly rather than relying only on robust standard errors.

Mean Cumulative Function

Not every recurrent-event analysis needs to be based on a hazard model. The mean cumulative function (MCF) describes the expected cumulative number of events by time \(t\).

Let \(N_i(t)\) denote the number of events for patient \(i\). Then the population mean cumulative function is conceptually:

\[ MCF(t)=E[N(t)] \]

For example, an estimated MCF of 1.8 at 12 months means that the estimated average cumulative number of events per patient by 12 months is approximately 1.8.

MCF curves can be especially intuitive for outcomes such as recurrent hospitalizations.

Why MCF is useful: The MCF directly reflects cumulative event burden rather than reducing the entire event history to a single first event.

A Simple Worked Example

Suppose a randomized trial evaluates a treatment intended to reduce recurrent disease exacerbations. There are 200 patients:

Group	Patients	Total Events	Average Follow-up
Placebo	100	180	11.5 months
Treatment	100	125	11.7 months

A simple event-count summary suggests fewer events under treatment. However, the total number of events alone does not establish the treatment effect because follow-up time, censoring, event timing, and patient-level dependence must be considered.

What Happens if We Analyze Only the First Event?

Suppose the first-event counts are:

Group	Patients	Patients with ≥1 Event
Placebo	100	75
Treatment	100	72

The difference between groups now appears much smaller than the difference in total event counts.

This illustrates a critical issue: the treatment effect on first-event risk can differ substantially from the treatment effect on recurrent-event burden.

A treatment might not prevent the first event but could reduce the number of subsequent events.

A Patient-Level Illustration

Consider four hypothetical patients followed for one year.

Patient	Group	Events
1	Placebo	0
2	Placebo	4
3	Treatment	1
4	Treatment	1

Both groups have two patients. The average number of events is:

\[ \bar{N}_{P}=\frac{0+4}{2}=2 \]

and:

\[ \bar{N}_{T}=\frac{1+1}{2}=1 \]

A first-event analysis, however, sees one event in each group.

The recurrent-event analysis sees a difference in cumulative event burden.

Lesson: The choice of estimand determines whether this distinction matters. If the scientific goal is to delay the first exacerbation, first-event analysis may be ideal. If the goal is to reduce repeated exacerbations, the recurrent-event estimand may be more appropriate.

Time-to-First Analysis Can Be More Conservative

There are situations in which first-event analysis deliberately sacrifices information.

Once a patient experiences the first event, later events are no longer relevant to the first-event endpoint.

This can be entirely appropriate when the first event represents an important clinical milestone. Examples include:

Time to first myocardial infarction
Time to first stroke
Time to first disease progression
Time to first treatment failure
Time to first hospitalization when the first hospitalization is the primary clinical milestone

The fact that subsequent events exist does not automatically mean that they should be included in the primary analysis.

When Recurrent Events Are More Appropriate

Recurrent-event analysis becomes particularly compelling when:

Repeated events are common.
Each event represents meaningful morbidity.
Preventing additional events is part of the treatment objective.
First-event analysis would discard a large fraction of observed events.
Patients can experience several events during the study.
The protocol or estimand explicitly targets event burden or event rate.

A Useful Decision Framework

Define the clinical question. Are you interested in delaying the first event or reducing the overall event burden?

Examine the event process. Determine how frequently repeated events occur and whether subsequent events remain clinically meaningful.

Define the estimand. Specify exactly what treatment effect should be estimated.

Select the analysis framework. Choose first-event survival analysis, a recurrent-event model, or complementary analyses.

Address dependence and censoring. Repeated events from the same patient require appropriate statistical handling.

Perform sensitivity analyses. Evaluate whether conclusions are robust to alternative recurrent-event assumptions and analysis choices.

How Censoring Changes the Analysis

Censoring is particularly important in recurrent-event settings because patients may contribute different amounts of observation time.

For example, one patient may be followed for 12 months while another withdraws at month 5.

A recurrent-event analysis must appropriately account for the amount of time each patient is observed.

This is one reason event rates are often expressed relative to exposure time.

\[ \text{Event rate} = \frac{\text{Number of events}} {\text{Total patient-time at risk}} \]

However, a simple event rate is not automatically a substitute for a time-to-event model.

Important: Patient-years or exposure-adjusted event rates are useful descriptive summaries, but the primary inferential method should be aligned with the prespecified estimand and censoring mechanism.

Terminal Events Create an Additional Complication

Some recurrent-event settings include a terminal event such as death.

For example, consider recurrent hospitalizations followed by death. A patient who dies can no longer experience future hospitalizations.

This means that death may not be ordinary non-informative censoring.

The treatment can affect both:

the recurrent event process, and
the probability and timing of the terminal event.

In such settings, the analysis should explicitly consider whether death is part of the estimand, a terminal event that truncates recurrent-event follow-up, or a competing process requiring a joint or composite framework.

Do not automatically censor death as if it were an ordinary dropout. When death prevents future recurrence by definition, the scientific interpretation of the recurrent-event estimand needs to account for that fact.

Common Recurrent-Event Models

Method	Main Idea	Typical Use
Andersen–Gill	Counting-process extension of Cox	Overall recurrent-event rate/hazard
PWP Total Time	Stratifies by event order using study time	Ordered recurrent events
PWP Gap Time	Models time between events	Inter-event waiting times
Frailty model	Patient-specific random effect	Unobserved heterogeneity
Mean cumulative function	Expected cumulative number of events	Event-burden description
Poisson model	Models event counts/rates	Rate-based analyses
Negative binomial	Allows overdispersion in counts	Recurrent event counts

Andersen–Gill vs. PWP

A common question is whether an investigator should use Andersen–Gill or PWP. The distinction is primarily scientific.

Andersen–Gill treats recurrent events as repeated occurrences within a common event process.

PWP explicitly conditions on event order.

Question	Preferred Concept
Is the overall recurrent event process the primary target?	Andersen–Gill may be appropriate
Does the first, second, and third event have different meanings?	PWP may be attractive
Is time since the previous event important?	PWP gap-time formulation may be useful
Is patient heterogeneity itself important?	Frailty model may be useful
Is cumulative event burden the main descriptive target?	MCF can be highly informative

A Practical Analysis Strategy

For many clinical trials, it is useful to distinguish the primary estimand from supportive analyses.

For example, a study might prespecify:

Primary analysis: time to first hospitalization using a Cox model.

Key supportive analysis: recurrent hospitalizations using an Andersen–Gill model with robust variance.

Descriptive analysis: mean cumulative number of hospitalizations over time.

Sensitivity analysis: alternative recurrent-event model accounting for event order or patient frailty.

This can provide a much richer understanding of treatment effects than either analysis alone.

Do Not Assume the Recurrent-Event Analysis Must Replace the First-Event Analysis

The choice does not always have to be either/or.

A trial may reasonably report both:

time to first event, and
recurrent event burden.

The key requirement is that each analysis has a clearly defined role and interpretation.

Best practice: Do not report several recurrent-event models merely because they are available. Each analysis should correspond to a prespecified clinical question or sensitivity objective.

Example: Why the Hazard Ratio May Tell Only Part of the Story

Imagine a treatment produces a first-event hazard ratio of:

\[ HR_{\text{first}}=0.95 \]

This suggests little difference in time to first event.

Suppose, however, that the recurrent-event analysis produces:

\[ HR_{\text{recurrent}}=0.70 \]

These results are not necessarily contradictory.

They could indicate that treatment has relatively little effect on whether or when the first event occurs but substantially reduces the occurrence of additional events after the first event.

This is precisely why the estimand must be defined before the statistical method is selected.

Common Mistakes

Automatically analyzing only the first event. This may discard substantial information when recurrent events are central to the clinical question.
Automatically switching to a recurrent-event model. The presence of repeated events does not by itself establish that recurrent events should be the primary estimand.
Treating recurrent event records as independent. Multiple events from one patient are correlated.
Ignoring event order. If first, second, and later events have different clinical meanings, an unordered recurrent-event model may not answer the intended question.
Ignoring death. Death may prevent future events and therefore change the interpretation of the recurrent-event process.
Reporting only an event count. A raw count does not account for different follow-up durations or censoring.
Interpreting a recurrent-event hazard ratio as a first-event hazard ratio. The estimands are different.
Choosing the model before defining the estimand. The clinical question should determine the statistical method, not the reverse.
Failing to prespecify handling of multiple events. The protocol should define which events qualify, how events are separated, and how recurrent events are analyzed.
Using several recurrent-event models without a clear purpose. Additional analyses should have a defined scientific or sensitivity-analysis role.

What Should Be Defined in the Protocol?

A recurrent-event endpoint requires considerably more detail than simply writing "analyze recurrent events."

The protocol or statistical analysis plan should specify:

Definition of the qualifying event
Rules for multiple events occurring close together
Whether all events or only confirmed events are included
Whether events are classified by severity
Whether event order is relevant
Time origin
Risk intervals
Censoring rules
Handling of treatment discontinuation
Handling of study withdrawal
Handling of death
Primary recurrent-event estimand
Primary statistical model
Variance estimation
Handling of within-patient dependence
Sensitivity analyses

Reporting a Time-to-First Analysis

A clear first-event analysis report should include:

Component	Example
Endpoint	Time to first confirmed exacerbation
Time origin	Randomization
Event	First qualifying exacerbation
Censoring	Last adequate assessment or end of follow-up
Model	Cox proportional hazards model
Effect measure	Hazard ratio with confidence interval

Reporting a Recurrent-Event Analysis

A recurrent-event analysis should be even more explicit. For example:

Endpoint:
All confirmed disease exacerbations occurring after randomization.

Time origin:
Randomization.

Recurrent events:
Each qualifying exacerbation contributes an event.

Within-patient dependence:
Robust sandwich variance clustered by patient.

Primary model:
Andersen–Gill recurrent-event model.

Treatment effect:
Hazard ratio with two-sided 95% confidence interval.

Supportive analysis:
Mean cumulative number of exacerbations over follow-up.

Terminal event:
Death handled according to the prespecified recurrent-event strategy.

Practical SAS Structure

A counting-process representation is commonly used when fitting recurrent-event Cox-type models.

proc phreg data=recurrent;
    model (start, stop)*event(0) = treatment age baseline_severity
        / ties=efron;
    id subject_id;
    assess ph / resample;
run;

The precise SAS specification depends on the chosen recurrent-event framework, variance approach, stratification, and study design.

For an Andersen–Gill analysis, the start and stop times define the risk intervals for each event record.

Practical R Structure

In R, a counting-process formulation can similarly be represented using start-stop survival data.

library(survival)

fit <- coxph(
  Surv(start, stop, event) ~ treatment + age + baseline_severity +
    cluster(subject_id),
  data = recurrent
)

summary(fit)

The cluster() term is one common way to obtain a robust variance estimate that accounts for repeated observations within a patient.

Software is not the design. The same dataset can often be analyzed with several recurrent-event models. The software command should follow the prespecified estimand and model, not determine them.

First-Event vs. Recurrent Event: A Decision Table

If Your Main Question Is...	Consider...
When does the first event occur?	Time-to-first-event analysis
Does treatment reduce the overall rate of repeated events?	Andersen–Gill or another recurrent-event rate model
Does treatment delay successive events?	PWP or related ordered-event approach
How long is the interval between events?	Gap-time recurrent-event analysis
Does unobserved patient susceptibility matter?	Frailty model
How many events accumulate over time?	Mean cumulative function

The Most Important Concept

The most important point is that time-to-first-event and recurrent-event analysis are not competing versions of the same endpoint.

They generally represent different estimands.

Time-to-first-event analysis asks:

\[ \text{How does treatment affect time to the first event?} \]

Recurrent-event analysis may instead ask:

\[ \text{How does treatment affect the complete sequence or burden of events?} \]

Those questions can produce different answers even when they are based on the same patients.

Bottom line: Use time-to-first-event analysis when the first event is the clinically meaningful milestone. Consider recurrent-event analysis when repeated events are themselves important and treatment benefit is expected to extend beyond the first event. When recurrent events are used, define the estimand first, then select an appropriate framework such as Andersen–Gill, PWP, frailty, or mean cumulative function analysis. Always account for within-patient dependence, censoring, and the role of terminal events such as death.

Practical Workflow

Define the clinical event precisely.

Determine whether the first event or repeated events are clinically important.

Define the treatment-effect estimand.

Assess how frequently recurrent events occur.

Specify the time origin and follow-up period.

Define censoring and treatment-discontinuation rules.

Determine whether death is a terminal event.

Choose first-event or recurrent-event methodology.

Account for within-patient dependence when modeling recurrent events.

Perform clinically meaningful sensitivity analyses.

Report the estimand and effect measure in terms that are clinically interpretable.

Final Comparison

Characteristic	Time-to-First Event	Recurrent Event Analysis
Primary unit of information	First event	Entire event history
Subsequent events	Not used for primary endpoint	Retained
Typical model	Cox	Andersen–Gill / PWP / frailty
Within-patient correlation	Not relevant	Must be addressed
Event burden	Poorly captured	Directly incorporated
Interpretability	Usually straightforward	Depends on estimand/model
Best suited for	Clinically important first milestone	Repeated clinically meaningful events

References

Andersen, P.K. & Gill, R.D. (1982). Cox's regression model for counting processes: A large sample study. Annals of Statistics, 10, 1100–1120.
Prentice, R.L., Williams, B.J. & Peterson, A.V. (1981). On the regression analysis of multivariate failure time data. Biometrika, 68(2), 373–379.
Wei, L.J., Lin, D.Y. & Weissfeld, L. (1989). Regression analysis of multivariate incomplete failure time data by modeling marginal distributions. Journal of the American Statistical Association, 84, 1065–1073.
Cook, R.J. & Lawless, J.F. (2007). The Statistical Analysis of Recurrent Events. Springer.
Andersen, P.K., Borgan, Ø., Gill, R.D. & Keiding, N. (1993). Statistical Models Based on Counting Processes. Springer.
Kelly, P.J. & Lim, L.L.Y. (2000). Survival analysis for recurrent event data: An application to childhood infectious diseases. Statistics in Medicine, 19, 13–33.