Cox Proportional Hazards Model: Example 3  Models for Recurrent Event
Data
Multiple event or recurrent event data is an extension of the single
event model. Cancer patients are an example of those subjects that can
experience multiple events, that is, the cancer goes into remission and
then at a later time can reoccur. In this example, we will look at the
different ways we can analyze recurrent event data in Statistica.
The data set for this example, Psoriasis.sta,
is taken from KabatZinn et al. (1998) and contains the results of a study
that investigated stress reduction techniques in the treatment of psoriasis.
There are many ways to analyze this type of data. We will discuss four
types of methods. In order to account for the correlation within subjects,
the Lin and Wei robust estimator of the covariance matrix is used.
Open the example data file Psoriasis.sta,
and start the Cox Proportional Hazards
module. Following are instructions to do this from the ribbon bar and
from the classic menus.
Ribbon
bar. Select the Home tab.
In the File group, click the Open arrow, and select Open
Examples to display the Open a
Statistica Data File dialog box. The data set is located in the
Datasets folder.
Next, on the Statistics tab,
in the Advanced/Multivariate group,
click the Advanced Models arrow
and select Cox Proportional Hazards
to display the Cox Proportional Hazards
Regression dialog box.
Classic
menus. From the File menu,
select Open Examples to display
the Open a Statistica Data File
dialog box. Psoriasis.sta is located
in the Datasets folder.
Next, from the Statistics  Advanced
Linear/Nonlinear submenu, select Cox
Proportional Hazards Models.
AG Model
The first model that is considered is the AG (AndersonGill) or counting
process model. This model treats events within a subject as independent
and does not distinguish between the first event versus subsequent events.
In the Cox Proportional Hazards Regression
dialog box, on the Quick tab,
in the Input type group box, select
the Counting process style of input (start,
stop, censor, covariates, factors) option button.
Click the Variables button,
and select variables as shown in the following image.
Click OK in the variable selection
dialog box.
Specify the codes for the complete and censored values. Enter a value
of 1 for the Code
for complete responses and a value of 0
for the Code for censored responses.
Select the Options tab. Select the Robust
variance estimator check box, and click the Subject
button. In the Select a subject variable
dialog box, select variable 1  ID.
Click OK in the variable selection
dialog box.
Click OK in the Cox
Proportional Hazards Regression dialog box to run the analysis
and display the Cox Proportional Hazards
Results dialog box.
On the Quick tab, click the
Parameter estimates button to
produce the coefficients for the AG model.
PWP Models
The next two models were suggested by Prentice, Williams, and Peterson
(1981). Both are referred to as conditional models. They are conditional
in the sense that a subject is not considered at risk for the k+1
th
event until the kth
event has occurred. In both models, subjects are stratified by event number.
The PWPCP (CP stands for conditional probability) model uses the actual
time the event occurs, whereas, the PWPGT (GT stands for gaptime) model
uses the time since the last event. Both models can be estimated using
the stratified proportional hazards model.
PWPCP Model
We will continue to use the Psoriasis.sta
data file. Return to the Cox Proportional Hazards
Regression dialog box (click the Modify
button in the Cox Proportional Hazards
Results dialog box).
In the Input type group box,
ensure that the Counting process style
of input (start, stop, censor, covariates, factors) option button
is selected.
Click the Variables button,
and select variable as shown in the next image. Note that, for the PWPCP
model, a strata variable needs to be selected. The strata variable keeps
track of the number of events that have occurred.
Click OK in the variable selection
dialog box.
Ensure that 1 is entered for
the Code for complete responses
and 0 for the Code
for censored responses.
Select the Options tab. Ensure that the Robust variance estimator check box
is selected. Click the Subject
button, and ensure that variable 1 
ID is selected. Click OK.
Click OK in the Cox
Proportional Hazards Regression dialog box to run the analysis
and display the Cox Proportional Hazards
Results dialog box. On the Quick
tab, click the Parameter estimates
button to produce the coefficients for the PWPCP model.
PWPGT
For the GT or GapTime model, we will model the time between two events
occurring. Use the same data set, and click Modify
in the Results dialog box to return
to the Cox Proportional Hazards Regression
dialog box.
In the Input type group box,
select the Survival time, covariates,
factors, censor option button.
Click the Variables button,
and select variable as shown in the next image.
Click OK.
Ensure that a value of 1 is
specified for the Code for complete responses
and a value of 0 for the Code for censored responses.
Select the Options tab. Ensure
that the Robust variance estimator
check box is selected. Click the Subject
button and ensure that variable 1 
ID is selected. Click OK.
Click OK in the Cox
Proportional Hazards Regression dialog box to run the analysis
and display the Cox Proportional Hazards
Results dialog box.
On the Quick tab, click the
Parameter estimates button
to produce the coefficients for the PWPGT model.
WLW Model
The next model considered is the model proposed
by Wei, Lin, and Weissfeld and is referred to as the WLW model. This model
is a marginal model that separately models each event as its own process.
For this model, we will need to use another
data set. Close all open files, open the Psoriasis2.sta
data file, and start the Cox Proportional Hazards module.
This data is the same as the previous except
that in the WLW model all subjects are considered at risk for all events.
In order to reflect this assumption in the data set, we need to have data
on all subjects for each event. As an example, suppose a subject experiences
2 events. The data for the subject might look like:

Time 
Event 
Stratum 
1 
10 
1 
1 
2 
15 
1 
2 
Suppose that in this data set, the maximum
number of events a subject experiences is 4. In the WLW model we would
add two rows to the subject as follows.

Time 
Event 
Stratum 
1 
10 
1 
1 
2 
15 
1 
2 
3 
15 
1 
3 
4 
15 
1 
4 
In the Cox
Proportional Hazards Regression dialog box, on the Quick
tab, select Survival time, covariates,
factors, censor as the Input type.
Click the Variables button,
and make the following variable selections.
Click OK.
Specify the codes for the complete and censored values. Enter a value
of 1 for the Code
for complete responses and a value of 0
for the Code for censored responses.
Select the Options tab. Select
the Robust variance estimator
check box. Click the Subject button
and select variable 1  ID. Click
OK.
Click OK to run the analysis
and display the Cox Proportional Hazards
Results dialog box. On the Quick
tab, click the Parameter estimates
button to produce the coefficients for the WLW model.