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

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

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

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

Subject i Subject j

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 –

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

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)

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

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

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

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

Comparability of complete gap times References –Wang and Chen (2000, Bmcs)

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.)

Nonparametric Estimation (1) Nonparametric of recurrent survival function: –Suppose observed data are

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

–Those failed at time t is calculated by –A product-limit estimator is calculated as

–Reference: Wang and Chang (1999, JASA)

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

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

Then Therefore Now we can estimate H by

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)

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)

Graphical Display Rate functions – Example of recurrent infections

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