Biostat/Stat 576

1. Biostat/Stat 576 Chapter 6Selected Topics on Recurrent Event Data Analysis

2. Introduction • Recurrent event data • Observation of sequences of events occurring as time progresses • Incidence cohort sampling • Prevalent cohort sampling • Can be viewed as point processes • Three perspectives to view point processes • Intensity perspective • Counting perspective • Gap time (recurrence) perspective

3. Data Structure • Prototype of observed data: • : ith individual, jth event • : ith censoring time • : last censored gap time:

4. Subject i Subject j Can we pool all the gap times to calculate a Kaplan-Meier estimate?

5. Subject i Subject j

6. Probability Structure • Last censored gap time: • Always biased • Example: • Suppose gap times are Bernoulli trials with success probability • Censoring time is a fixed integer • Observation of recurrences stops when we observe heads. • This means

8. Probability Structure • Example (Cont’d) • Suppose we have to include the last gap time to calculate the sample mean of recurrent gap times • Then its expected value would be always larger than , because we know

9. Probability Structure • Example (Cont’d) • But the estimator would be asymptotically unbiased, because additional one head and one additional one coin flip would not matter as sample size gets large • Reference: • Wang and Chang (1999, JASA)

10. Probability Structure • Complete recurrences • First recurrences • The complete recurrences are in fact sampled from the truncated distributions • The censoring time for jth complete gap time is

12. Probability Structure • Suppose underlying gap times follow exactly the same density functions, i.e., • Right-truncated complete gap times would be because

13. Probability Structure • Risk set for usual right censored times • Risk set for right-truncated gap times

14. Risk set for left-truncated and right-censored times • Need one more dimension about censoring time • Risk set for left-truncated times

15. Comparability of complete gap times • References • Wang and Chen (2000, Bmcs)

16. Probability Structure • Summary • Last censored gap time is always subject to intercept sampling • Reference: • Vardi (1982, Ann. Stat.) • First complete gap times are always subject to right-truncation • Reference: • Chen, et al. (2004, Biostat.)

17. Nonparametric Estimation (1) • Nonparametric of recurrent survival function: • Suppose observed data are

18. Then we re-define the recurrences by • Total mass of risk set at time t is

19. Those failed at time t is calculated by • A product-limit estimator is calculated as

20. Reference: • Wang and Chang (1999, JASA)

21. Nonparametric Estimation (2) • Total Times • Gap times • Data for two recurrences • Observed data

23. Distribution functions • Without censoring, consider • This would estimate • What if we have censoring? • Replace by

24. Then • Therefore • Now we can estimate H by

25. G(.) is estimated by Kaplan-Meier estimators based on censoring times • Assuming that censoring times are relatively long such that G(.) can be positively estimated for every subject • Inverse probability of censoring weighting (IPCW) • First derive an estimator without censoring • Then weighted by censoring probabilities • Censoring probabilities are estimated Kaplan-Meier estimates • Assume identical censoring distributions • Can be extended to varying censoring distributions by regression modeling • References • Lin, et al. (1999, Bmka) • Wang and Wells (1998, Bmka) • Lin and Ying (2001, Bmcs)

26. Nonparametric Estimation (3) • Nonparametric estimation of mean recurrences • Nelson-Aalen estimator for M(t) • Unbiased if • Assume that the censoring time (end-of-observation time) is independent of the counting processes • Reference • Lawless and Nadeau (1995, Technometrics)

27. Graphical Display • Rate functions • Example of recurrent infections

28. Estimation of rate functions • To estimate F-rate function • To estimate R-rate function • References • Pepe and Cai (1993)