Biostat/Stat 576

Chapter 6Selected 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)