Biomedical Engineering Reference
In-Depth Information
6.1
Introduction
Interval-censored data occur naturally in many fields, such as AIDS clinical
trials and other follow-up medical, epidemiological, and sociological studies.
In such studies, subjects are either not under continuous observation, or the
detection of the failure event of interest is available only at some specific times.
For example, HIV status is only detectable by some laboratory examination
when patients visit clinics. Thus, the HIV infection time is only known to fall
between the last visit time with negative result and the first visit time with
positive result. In the tumor studies conducted by the National Toxicology
Program, the status of nonlethal tumors of rats or mice can only be determined
when the rats (or mice) die or are sacrificed. The onset times of different
tumors are only known to be larger or smaller than the death or sacrifice
time. In both examples, the failure time of interest is not able to be observed
exactly but is known to fall within some interval. General interval-censored
data or Case 2 interval-censored data usually refer to the data sets that are a
mixture of left-, interval-, and right-censored observations.
Let T i denote the failure time of interest and xi i a p 1 covariate vector
for subject i. Let Li i ;R i ] denote the observed interval for subject i. If Li i = 0,
subject i is left-censored; if Ri i = 1, subject i is right-censored; otherwise,
subject i is interval-censored. Assume Ti i F(jx i ), the cumulative distribu-
tion function (CDF) given covariate xi. i . Conditional on the covariates and the
observed intervals, the observed likelihood can be written as
Y
L =
[F(R i jx i ) F(L i jx i )]:
i=1
This likelihood is a reduced full likelihood under the assumption of noninfor-
mative censoring (Sun, 2006). One general case of noninformative censoring is
that the failure time is independent of the observation process given covariates.
To distinguish different types of censoring, we can rewrite the above like-
 
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