1. Topic: Several Approaches to Modeling Recurrent Event Data Presenter: Yu Wang

2. 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)

3. 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).

4. 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.

5. Characteristic of a recurrent event We observe the same event in a subject multiple times during the follow-up period.

6. 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. Counting process formulation (Anderson 1993) 2. Conditional model A (Prentice 1981). 3. Conditional model B (Prentice 1981). 4. Marginal event-specific model (Lin 1989)

7. 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.

8. Assumptions 1. Assumptions: Assume a sufficient number of subjects had four recurrent events to allow modeling four recurrent events. 2. There may be add assumptions in a specific model. (case by case)

9. Counting Process Formulation Follow-up time is broken into segments defined by the events. Events are assumed to be independent. Subject 1Subject 2 ModelTime Interval Event Indicator StratumTime Interval Event Indicator Stratum Counting Process (0, 9]11(0, 10]11 (9, 13]11(10, 15]11 (13, 28]11 (28, 31]01

10. 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.

11. 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.

12. 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 having. Subject 1Subject 2 ModelTime Interval Event Indicator StratumTime Interval Event Indicator Stratum Condition al A (0, 9]11(0, 10]11 (9, 13]12(10, 15]12 (13, 28]13 (28, 31]04

13. Conditional B The time since the previous event, “reset the clock”. Subject 1Subject 2 ModelTime Interval Event Indicator StratumTime Interval Event Indicator Stratum Condition al B (0, 9]11(0, 10]11 (0, 4]12(0, 5]12 (0, 15]13 (0, 3]04

14. 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

15. 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.

16. 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.

17. Marginal Event-specific Subject 1Subject 2 ModelTime Interval Event Indicator StratumTime Interval Event Indicator Stratum Marginal (0, 9]11(0, 10]11 (0, 13]12(0, 15]12 (0, 28]13(0, 15]03 (0, 31]04(0, 15]04

18. 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 follow- up times to all possible recurrent events, whether they experienced that particular recurrent event or not.

19. 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.

20. Thank You!