Biomedical Engineering Reference
In-Depth Information
where X is the p-vector of covariates. Let (y
i
;x
i
;
i
) be observed data. Let
t
(1)
< < t
(m)
be m ordered uncensored failure times and d
j
be the
number of observed failures at time t
(j)
,R
j
=fi : Y
i
t
(j)
gthe risk set at
time t
(j)
, just prior to time t
(j)
, j = 1;:::;m.
3.1. Local Polynomial PH Models
Assume that the functional form of the covariate eects (x) (i.e., the
logarithm of the relative risk) in the PH model is unspecied. Then, we
refer to the conditional hazard model
h(tjx) = h(t) exp( (x))
(15)
as a nonparametric PH model. For simplicity of discussion, for the moment,
we consider the univariate case; thus, the nonparametric PH model (15)
becomes
h(tjx) = h(t) exp( (x)):
(16)
The log-likelihood corresponding to the model (16) is
X
n
`(h; ) =
f
i
[log h(Y
i
) + (X
i
)]H(Y
i
) exp[ (X
i
)]g:
(17)
i=1
To estimate (x), Fan, Gijbels, and King
50
used the local polynomial regres-
sion technique
130;131;132;32;46;47;48;121
. They assumed that the pth derivative
of (X) at the point x
0
exists and, by a Taylor's expansion, they modeled
(X) as
(X)X
; (18)
where X
= (1;Xx
0
;:::; (Xx
0
)
p
), and
= (
0
;:::;
p
)
T
=
( (x
0
);:::;
(p)
(x
0
)=p!)
T
. We refer to the conditional hazard model
h(tjx) = h(t) exp(x
)
(19)
as a local polynomial PH model.
To estimate
, Fan, Gijbels, and King considered two caseswhen
the baseline hazard function is parameterized and when it is not. We will
focus on the latter case. When h(t) is not parameterized, they used the local
polynomial model (18) with a local version of the log partial likelihood to
nd the
that maximizes the local log partial likelihood
m
n
X
fjg
log
h
X
io
X
exp(X
i
)K
b
(X
i
x
0
)
K
b
(X
fjg
x
0
)
; (20)
j=1
i2R
j
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