Biomedical Engineering Reference
In-Depth Information
regression coecients
via a linear predictor combined with the baseline hazard
0
(tj) with parameter vector . We are interested in the eect of the covariates on
the survival function
S(tjX;;) = 1 F(tjX;;);
(11.2)
where F is the cumulative distribution function. Then
Z
t
0
(sj) exp
0
X
ds
S(tjX;;)
=
expf(tjX;;)g = exp
0
0
(sj)ds exp
0
X
Z
t
= S
0
(tj)
exp
[
0
X
]
: (11.3)
=
exp
0
where (tjX;;) is the cumulative hazard, which is the integral of the hazard
function (sjX;;) up to time t and S
0
(tj) is the baseline survival function, which
is independent of the covariates. Therefore,
= S(tjX;;) = S
0
(tj)
expf
0
Xg
[1 F(tjX;;)]
[1 F
0
(tj)]
expf
0
Xg
:
=
(11.4)
Thus, for n patients with observed interval data (ti1,
i1
;t
i2
), i = 1; ;n, the log-
likelihood (LL) function with regression parameter vector and the parameters
from the baseline distribution can be constructed as follows:
log
n
[1 F
0
(t
i1
j)]
exp(
0
X
i
)
[1 F
0
(t
i2
j)]
exp(
0
X
i
)
o
: (11.5)
X
n
LL(F
0
;;) =
i=1
In this semiparametric approach, this F
0
consists of piecewise constants as de-
scribed in Pan (1999). The maximum likelihood estimates (MLE) for the parameters
can be obtained by maximizing the log-likelihood function in Equation (11.5). Gen-
erally, there is no closed form for the MLE, and therefore an iterative numerical
search procedure is used to obtain the MLE. This approach is implemented in the
R package \IntCox".
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