Biomedical Engineering Reference
In-Depth Information
i
th individual failing at
t
i
given that the individual is from the risk set
R
(
t
i
) (i.e.,
R
(
t
i
)=
{
j
|
t
j
≥
t
i
}
) is [10]:
h
0
(
t
)exp(
x
i
β
)
∈
R
(
t
i
)
h
0
(
t
i
)exp(
x
β
)
.
(2
.
11)
Assuming that there are
K
failures. The partial likelihood function is then:
K
exp(
x
i
β
)
∈
R
(
t
i
)
h
0
(
t
i
)exp(
x
β
)
.
(2
.
12)
i
=1
Recalling the definition of
δ
i
at the beginning of this section, the partial
likelihood function can be expressed as:
δ
i
n
exp(
x
i
β
)
j
=1
L
(
β
)=
,
(2
.
13)
y
j
(
t
)exp(
x
j
β
)
i
=1
where
y
j
(
t
)=0when
t
t
j
,otherwise
y
j
(
t
) = 1. Equation (2.13) can be
written in another way to remove the expression of
δ
i
:
L
(
β
)=
iuncensored
≤
.
exp(
x
i
β
)
j
=1
(2
.
14)
y
j
(
t
)exp(
x
j
β
)
For a sample of size
n
, the log partial likelihood for expression (2.14) is
n
l
(
β
)=log
L
(
β
)=
iuncensored
x
i
β
−
log
y
j
(
t
)exp(
x
j
β
)
.
(2
.
15)
j
=1
The maximum partial likelihood estimation of
β
can be obtained as a so-
lution to the equation
∂l
(
β
)
∂β
=0
,
and thus,
j
=1
n
y
j
(
t
)
x
j
exp(
x
j
β
)
j
=1
x
i
−
=0
.
(2
.
16)
y
j
(
t
)exp(
x
j
β
)
iuncensored
Cox and others have shown that this partial log-likelihood can be treated
as an ordinary log-likelihood to derive valid (partial) MLE of
β
. Therefore,
we can estimate hazard ratios and confidence intervals using maximum
likelihood techniques whose principal will be discussed in the next section.
To avoid the baseline hazard, estimates are based on the partial as opposed
to the full likelihood.
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