Biomedical Engineering Reference
In-Depth Information
1.3
Nonparametric Estimation
In this section, we consider nonparametric estimation of a survival function
S(t) = P(T > t). It is well-known that for the analysis of a survival study,
the first task is often to estimate the survival function S(t) or the cumulative
distribution function (CDF) F(t) = 1 S(t) of the failure time variable of
interest. It is apparent that estimating one is equivalent to estimating the
other and, in general, one usually focuses on S(t) as it is more meaningful to
practitioners such as medical investigators.
Consider a survival study that involves n independent subjects and yields
only interval-censored data. By using the notation defined above, interval-
censored data can usually be represented by fI i g i=1 , where I i = (L i ;R i ]
is the interval known or observed to contain the unobserved T associated
with the ith subject. Let ft j g m+1
j=0 denote the unique ordered elements of
f0;fL i g i=1 ;fR i g i=1 ;1g, that is, 0 = t 0 < t 1 < ::: < t m < t m+1 = 1, ij the
indicator of the event (t j1 ;t j ] I i , and p j = S(t j1 ) S(t j ). Then under
the independent assumption, the likelihood function for p = (p 1 ;:::;p m+1 ) 0 is
proportional to
h
i
Y
Y
m+ X
L S (p) =
S(L i ) S(R i )
=
ij p j :
i=1
i=1
j=1
The problem of finding the nonparametric maximum likelihood estimator
(NPMLE) of S becomes that of maximizing L S (p) under the constraints that
P m+1
j=1 p j = 1 and p j 0 (j = 1;:::;m + 1) (Groeneboom and Wellner (1992);
Gentleman and Geyer (1994); Li et al. (1997); Turnbull (1976)). Obviously,
the likelihood function L S depends on S only through the values fS(t j )g j=1 .
Thus the NPMLE of S, which we denote by S, can be uniquely determined
only over the observed intervals (t j1 ;t j ] and the behavior of S within these
intervals will be unknown. Conventionally, however,
S(t) is often taken to be
S(t) =
S(t j1 ) when t j1 t < t j .
a right continuous step function. That is,
 
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