Topic:
Several Approaches to Modeling
Recurrent Event Data
Presenter: Yu Wang
References:
On the Regression Analysis of Multivariate
Failure Time data
L. Prentice; B. Williams; A. Peterson(1981)
Regression Analysis of Multivariate
Incomplete Failure Time Data by
Modeling Marginal Distributions
L. Wei; D. Lin; L. Weissfeld (1989)
What is recurrent event data?

An event of interest may occur multiple
times in the course of the follow-up of a
subject. Data of this type are referred to as
recurrent event data (RED).
Examples of RED
A subject’s cancer may be treated and then
recur sometime later, and this process may
be repeated.
 Recording the interval of time between
heart attacks in a group of subjects at high
risk for this event.
 Cars break, are repaired, and break again all
the time.

Characteristic of a recurrent event

We observe the same event in a subject
multiple times during the follow-up period.
How to model RED?
There are several approaches, among them a
number of proportional hazard-type
models have been proposed for use with
RED:
1.
2.
3.
4.
Counting process formulation (Anderson 1993)
Conditional model A (Prentice 1981).
Conditional model B (Prentice 1981).
Marginal event-specific model (Lin 1989)
Data Example
The data for two hypothetical subjects among
n subjects:
Subject 1: Experienced the event at 9, 13 and
28 months of follow-up, study ended at 31 month.
Subject 2: Experienced the event at 10 and 15
months, and follow-up ended at the second event.
Assumptions
1.
2.
Assumptions: Assume a sufficient number
of subjects had four recurrent events to
allow modeling four recurrent events.
There may be add assumptions in a
specific model. (case by case)
Counting Process Formulation
Follow-up time is broken into segments
defined by the events. Events are assumed
to be independent.
Subject 1
Model
Counting
Process
Subject 2
Event
Event
Time
Stratum Time
Stratum
Interval Indicator
Interval Indicator
(0, 9]
1
1
(0, 10]
1
1
(9, 13]
1
1
(10, 15]
1
1
(13, 28]
1
1
(28, 31]
0
1
Counting Process (cont.)

Subject 1 will be in the risk set for any event
occurring between 0 and 31 months.
 Under assumption of no tied event times, the
subject contributes the event defining the risk set
at 9, 13 and 28.
 The data for sub1 could be described as data for
four different subjects. The first begin follow-up at
0 and has event at 9, the second has delayed entry
at 9 and is followed until 13.
 From the way the data are constructed, the model
treats the events as being independent and does
not different the first with the second.
Two Conditional Models

Conditional means a subject is assumed not to be
at risk for a subsequent event until a prior event
has occurred.
 A stratum variable is used to keep track of the
event number
 Difference between the two: time scale used. A
uses time defined by the beginning of the study
while B uses time since the previous event.
Conditional Model A
Time defined by the beginning of the study
The stratum variable indicates the specific
event number the subject is at risk of
Subject 2
having. Subject 1
Model
Condition
al A
Event
Event
Time
Stratum Time
Stratum
Interval Indicator
Interval Indicator
(0, 9]
1
1
(0, 10]
1
1
(9, 13]
1
2
(10, 15]
1
2
(13, 28]
1
3
(28, 31]
0
4
Conditional B

The time since the previous event, “reset the
clock”.
Subject 1
Model
Condition
al B
Subject 2
Event
Event
Time
Stratum Time
Stratum
Interval Indicator
Interval Indicator
(0, 9]
1
1
(0, 10]
1
1
(0, 4]
1
2
(0, 5]
1
2
(0, 15]
1
3
(0, 3]
0
4
Conditional (cont.)
Under assumption that all the covariates are
fixed at the beginning of the study, the
proportional hazards function for the sth
event under conditional model A is:
Blackboard
Conditional (cont.)

Parameter estimates for either model may
be obtained by using the stratified partial
likelihood with data in the form of the
tables.
 Event-specified parameter estimates are
obtained by including stratum by covariate
interactions in the model.
 Time-varying covariates can also be used.
Marginal Event-specific model
Marginal means that each event is considered
as a separate process. Time for each event
starts at the beginning of follow-up for each
subject.
All subjects are considered to be at risk for all
events, regardless of how many events they
actually had.
Marginal Event-specific
Subject 1
Model
Marginal
Subject 2
Time
Interval
Event
Indicator
Stratum
Time
Interval
Event
Indicator
Stratum
(0, 9]
1
1
(0, 10]
1
1
(0, 13]
1
2
(0, 15]
1
2
(0, 28]
1
3
(0, 15]
0
3
(0, 31]
0
4
(0, 15]
0
4
Marginal (cont.)

The fourth interval for subject 1 records the
“marginal” time the subject was at risk for
the fourth event.
 All subjects in this study contribute followup times to all possible recurrent events,
whether they experienced that particular
recurrent event or not.
Method for adjusting the estimates of the
variance of the coefficients to account for the
correlation among the observations on an
individual subject.
Lin and Wei (1989) proposed an extension of
White’s (1980, 1982) robust variance
estimator to the proportional hazards model
setting.
Thank You!