Biomedical Engineering Reference
In-Depth Information
K+1+i
=
K+1
+ i, i = 1; 2; 3. Using the log-likelihood (7) corresponding
to the model (10), one can have the log-likelihood `(a; K) as follows:
n
o
X
n
X
K
`(a; K) =
i
log(h(t; a))
exp(a
k
) [IB
k
(y
i
)IB
k
(y
min
)]
; (11)
i=1
k=3
where
k+4
X
(
`
t)
4
+
IB
k
(t) =
k+4
k
4
Y
:
(
m
`
)
`=k
m6=`
m=k;:::;k+4
For a given value of K, one can obtain an ML estimate a of a, hence the cu-
bic B-spline estimates h(t; a) of h(t) and S(t; a) = exp(
R
t
y
min
h(u; a)du)
of S(t) by maximizing `(a; K) (11). Within the model framework, three
methods (i.e., delta-method, prole likelihood, and bootstrap) can be used
to calculate condence intervals of h(t) and S(t). To avoid numerical dif-
culties when it occurs that a
k
!1for some k, Rosenberg suggested
adding a penalty term10
5
P
K
k=3
(10a
k
)
3
+
to `(a; K) (11). He also
developed an automatic knot selection procedure by choosing the kth knot
corresponding to the k=(K + 1) quantile of the uncensored failure times.
The nal model is the one that maximizes the Akaike information criterion
5
,
AIC(K) =2`(a; K)2(K + 4).
Cai, Hyndman, and Wand
27
proposed a linear spline model
K
X
(t;
1
;
2
) =
10
+
11
t +
2k
(t
k
)
+
(12)
k=1
for the log-hazard function (t) = log h(t), where
1
= (
10
;
11
)
T
and
2
= (
21
;:::;
2K
)
T
. The implementation chooses the kth interior knot
k
approximately corresponding to the k sample quantile of the unique
observed times y
i
and sets K = min(bn=4c; 30), wherebacis the greatest
integer less than or equal to a. See Ruppert (2002) for further reference
on the selection of K. To remedy the situation that the estimate of (t)
will be a somewhat wiggly piecewise linear function, Cai, Hyndman, and
Wand treated the
2k
s as random eects and assumed they were inde-
pendent and normally distributed with zero mean and nite variance
2
,
whose reciprocal acts as a smoothing parameter controlling the amount
of smoothing. Let Z
1
= [1;y
i
]
1in
and Z
2
= [(y
i
k
)
+
]
1in;1kK
.
LetH(
1
;
2
) be the sum of cumulative hazards evaluated at the y
i
s.
Let
`(
1
;
2
;) =
T
(Z
1
1
+ Z
2
2
)H(
1
;
2
)
1
2
2
2
2
, where =
Search WWH ::
Custom Search