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 Kabat-Zinn 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 (Anderson-Gill) 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 PWP-CP (CP stands for conditional probability) model uses the actual time the event occurs, whereas, the PWP-GT (GT stands for gap-time) model uses the time since the last event. Both models can be estimated using the stratified proportional hazards model.  

PWP-CP 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 PWP-CP 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 PWP-CP model.

PWP-GT

For the GT or Gap-Time 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 PWP-GT 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.