Biomedical Engineering Reference
In-Depth Information
Considering the boundary condition
S
(0) = 1 as we mentioned before, we
have
C
= 0, and thus
t
S
(
t
)=exp
{−
h
(
z
)
dz
}
.
(2
.
7)
0
Combining (2.7) with (2.4), we obtain
t
f
(
t
)=
h
(
t
)
S
(
t
)=
h
(
t
)exp
{−
h
(
z
)
dz
}
.
(2
.
8)
0
A recent survey on dynamic mortality modeling in actuarial mathematics
is given in [17].
2.2.
Cox Regression
A
Cox model
is a well-recognized statistical technique for exploring the re-
lationship between the survival of patient and a set of explanatory variables
(see [1], [16] for example). We call these explanatory variables
covariates
.
Suppose that we have collected
n
patients with lung cancer. For the
i
th patients, let (
t
i
;
δ
i
) be the observed phenotype, where
t
i
is the failure
time (in other words, when death occurs) when
δ
i
= 1, and is the censoring
time (e.g., last time known of a patient being cancer-free) when
δ
i
=0.Let
x
i
,x
ip
) be the vector of
p
covariates for the
i
th sample taken
from the
i
th patient. We assume that a general Cox model with the hazard
function for the
i
th patient is modeled as
=(
x
i
1
,
···
h
(
t
|
x
i
)=
h
(
t
)exp(
f
(
x
i
))
,
(2
.
9)
0
where
h
0
(
t
) is called the
baseline hazard function
. Although
f
(
x
i
)may
assume many formats, the most popular and also the simplest model for
f
(
x
)is
f
(
x
i
)=
x
i
·
···
β
=
x
i
1
β
1
+
+
x
i
p
β
p
,
(2
.
10)
where
β
is a column vector of coecients. In this equation, it is assumed that
the effects of the different covariates on survival are constant over time and
are addictive in a particular scale. The Cox model makes no assumptions
about the form of
h
0
(
t
), but assumes the parametric form for the effect of
the covariates (predictors) on the hazard. In this sense, the Cox model is a
semi-parametric model.
The vector
β
of parameters can be estimated by the partial likelihood
method. Let the observed follow up time of the
i
th individual be
t
i
with
corresponding covariates
x
i
,
i
=1
, .., n
. The conditional probability for the
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