Biomedical Engineering Reference
In-Depth Information
for given Zi,
i
, where
0
(t) is an arbitrary baseline hazard function. This also
implies that
(t; Z
i
) =
0
(t) exp(Z
0
i
);i = 1;:::;n:
This section discusses the maximum likelihood estimators of and
0
(t).
Assume that T and C are independent given Z; then the likelihood function
is proportional to
Y
[1 S(C
i
)]
i
[S(C
i
)]
1
i
L(;
0
)
=
i=1
h
1 exp(
0
(C
i
)e
Z
0
i
)
i
i
h
(1
i
)
0
(C
i
)e
Z
0
i
i
Y
=
exp
i=1
n
1 [S
0
(C
i
)]
exp(Z
0
i
)
o
i
[S
0
(C
i
)]
(1
i
) exp(Z
0
i
)
:
Y
=
i=1
Thus the log-likelihood can be written as
i
log
n
i
log
n
1 [S
0
(C
i
)]
exp(Z
0
i
)
o
X
l(;S
0
) =
i=1
+(1
i
) exp(Z
0
i
) log[S
0
(C
i
)]g:
Because the likelihood function depends on only the values of S
0
(t) at
C
i
's, one can maximize L(;S
0
) over the right-continuous nonincreasing step
function with jump points at Ci
i
with values S
0
(t) in terms of S
0
(t). Suppose
there are k distinct observed time points, 0 < t
1
< t
2
< ::: < t
k
. Because
the survival function is always nonincreasing, we focus on the functions of the
form
S
0
(t
j
) = exp(exp(Z
0
)x
j
);j = 1;:::;k;
where x
1
< x
2
< ::: < x
k
. Now we reparameterize the space as x
j
=
0
(t
j
) =
P
l=1
exp(
l
);j = 1;:::;k, to make the new parameters
j
's no restrictions.
Then the baseline survival function follows the form
S
0
(t) = e
P
j:t
j
t
exp(
j
)
=
Y
j:t
j
t
e
exp(
j
)
:
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