Biomedical Engineering Reference
In-Depth Information
j = 1;:::;p. The conditional hazard model
h(tjx) = h(t) exp( (x; G))
(25)
is referred to as an additive PH model. Let = ( (x
1
; G);:::; (x
n
; G))
T
.
Then, the partial likelihood corresponding to the model (25) with Peto's
(1972) approximation for ties is
P
i2D
r
(x
i
; G)
Y
m
exp
PL() =
d
r
;
(26)
P
i2R
r
exp( (x
i
; G))
r=1
where D
r
is the set of indices of failures at t
(r)
. To estimate g
j
(), Hastie
and Tibshirani maximized the penalized log partial likelihood
Z
X
p
`
p
() = `()
1
2
g
00
j
(x)
2
dx;
j
(27)
j=1
where `() = log PL() and
j
0 are smoothing parameters. The rst
term in (27) measures the closeness of the t to the data, and the second
term penalizes the curvature of the tted functions. One can establish the
existence of a unique solution to this problem under certain conditions by
using the arguments of O'Sullivan
107
extended to the additive model by
Buja, Hastie, and Tibshirani
25
. Given that a unique solution exists, it can
be seen that the solution must be a cubic spline for each j.
One can restrict the innite-dimensional problem to a nite one by
choosing a suitable basis. A convenient basis can result from considering
the evaluations of the cubic splines g
j
() at the observed points x
1j
;:::;x
nj
.
The penalized log partial likelihood (27) then can be rewritten as
p
X
`
p
() = `()
1
2
j
g
j
K
j
g
j
;
(28)
j=1
where K
j
are symmetric penalty matrices, and g
j
= (g
j
(x
1j
);:::;g
j
(x
nj
))
T
is a vector of the values of g
j
at x
1j
;:::;x
nj
. One may obtain (28) as the log
posterior from a Bayesian model with independent priors g
j
N(0;K
j
);
the additive function solution lies in a reproducing kernel Hilbert space with
P
p
j=1
K
j
equal to the reproducing kernel evaluated at x
1j
;:::;x
nj
. The
curves g
j
maximizing `
p
() can be obtained by using the Newton-Raphson
algorithm with the \Gauss-Seidel" method. In the statistical literature, the
Gauss-Seidel method has become known as \backtting," which was rst
proposed on more heuristic grounds using nonlinear smoothers
54
. Note that
in the algorithm the functions are standardized to have a mean of zero,
Search WWH ::
Custom Search