Biomedical Engineering Reference
In-Depth Information
where
β
,β
p
are
p
unknown constant parameters which need to be
estimated. Denote
β
τ
=(
β
,
···
1
,
···
,β
p
). Conduct an experiment and obtain
1
N
independent observations,
x
,x
N
, which correspond in the case of
life data analysis to failure times. The likelihood function is given by
,
···
1
,β
p
)=Π
i
=1
L
=
L
(
x
,
···
,x
N
|
β
,
···
f
(
x
i
|
β
,
···
,β
p
)
.
(3
.
2)
1
1
1
The Logarithmic function is
N
l
=log
L
=
log
f
(
x
i
|
β
1
,
···
,β
p
)
.
(3
.
3)
i
=1
For the survival analysis, we assume (2.9) and (2.10). Then the
pdf
becomes
t
i
f
(
t
i
|
x
i
)=
h
(
t
i
|
x
i
)
S
(
t
i
|
x
i
)=
h
0
(
t
i
)exp
{
x
i
β
−
h
0
(
z
)exp(
x
i
β
)
dz
}
.
0
(3
.
4)
The log-likelihood function
l
(
β
) has the expression
t
i
N
x
i
)=
i
l
=
log
f
(
t
i
|
[log
h
0
(
t
i
)+(
x
i
β
−
h
0
(
z
)exp(
x
i
β
)
dz
)]
0
i
=1
t
i
=
N
log
h
0
(
t
i
)+
h
0
(
t
i
)+
i
x
i
β
−
h
0
(
z
)exp(
x
i
β
)
dz.
(3
.
5)
0
i
When taking partial derivatives with respect to
β
to maximize
l
(
β
), the
computation often becomes very dicult due to the presentation of
h
0
(
z
)
in the integration term. That is why a proportional hazard model is used
in the Cox models so that the term
h
0
(
z
) can be canceled out in MLE
calculation.
Recall (2.15), the MLE for
β
is
s
(
β
) = 0, where the score function is
∂l
(
β
)
∂β
1
...
∂l
(
β
)
∂β
p
s
(
β
)=
.
(3
.
6)
One of many nonlinear algorithms to compute this maximization is the
Newton-Raphson iteration. The Newton-Raphson algorithm for computing
β
starts with an initial guess
β
(0)
and then iteratively determines
β
(
m
)
from
the formula
β
(
m
)
=
U
−
1
(
β
(
m
−
1)
)
s
(
β
(
m
−
1)
)
,
(3
.
7)
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