Transcript 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 •

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:
Subject i
Subject j
Can we pool all the gap times to calculate a Kaplan-Meier
estimate?
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 usual right
censored times
• Risk set for right-truncated
gap 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)