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Biostat/Stat 576
Chapter 6Selected Topics on Recurrent
Event Data Analysis
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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
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Data Structure
• Prototype of observed data: – : ith individual, jth event – : ith censoring time– : last censored gap time:
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Can we pool all the gap times to calculate a Kaplan-Meier estimate?
Subject i
Subject j
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Subject i
Subject j
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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
–
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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
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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)
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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
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Probability Structure
– Suppose underlying gap times follow exactly the same density functions, i.e.,
– Right-truncated complete gap times would be
because
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Probability Structure
• Risk set for right-truncated gap times
• Risk set for usual right censored times
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• Risk set for left-truncated times • Risk set for left-truncated and right-censored times
– Need one more dimension about censoring time
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• Comparability of complete gap times
• References– Wang and Chen (2000, Bmcs)
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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.)
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Nonparametric Estimation (1)
• Nonparametric of recurrent survival function:
– Suppose observed data are
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– Then we re-define the recurrences by
– Total mass of risk set at time t is
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– Those failed at time t is calculated by
– A product-limit estimator is calculated as
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– Reference: • Wang and Chang (1999,
JASA)
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Nonparametric Estimation (2)
• Total Times
• Gap times
• Data for two recurrences
• Observed data
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• Distribution functions
• Without censoring, consider
• This would estimate
• What if we have censoring?– Replace by
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• Then
• Therefore
• Now we can estimate H by
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• 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)
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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)
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Graphical Display
• Rate functions– Example of recurrent infections
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• Estimation of rate functions– To estimate F-rate function
– To estimate R-rate function
• References– Pepe and Cai (1993)