Biomedical Engineering Reference
In-Depth Information
where = (
0
;
1
; ) is the (2K + 3)-vector of parameters for
0
=
(
00
;
01
;
02
),
1
= (
11
;:::;
1K
),
0j
P
j
k=1
1k
k
,
1j
=
00
+
=
01
P
j
k=1
1k
. The log-likelihood corresponding
to the EHR model (33) with the quadratic spline model (34) is
P
j
k=1
1k
k
, and
2j
=
02
+
2
X
n
`(; ; ) =
i
[x
i
+ log h(u
i
; )]
i=1
exp(x
i
())H(u
i
; )
;
(36)
R
u
0
h(t; )dt is a cumulative hazard function. Thus, once
the estimate ( ^ ;
^
;
^
) of (; ; ) is obtained by maximizing `(; ; ), we
can have the estimate of h(tjx). Let u
max
be the value of u = exp(x ^ )y
corresponding to the maximum observed time. To ensure that h(u; ) must
be nonnegative in [0;u
max
] while estimating (; ; ), the following con-
straints are needed: (i)
00
0; (ii) h(u
max
; ) > 0; (iii) h(
k
; )0,
k = 1;:::;K; (iv) if the jth polynomial piece of (35) has an extremum in
[
j
;
j+1
], then
where H(u; ) =
0j
(
1j
)
2
=4
2j
0;
(37)
otherwise h((
j
+
j+1
)=2; )0. Notice that (37) is the value of h(u; )
at the extremum of the jth polynomial piece, provided it falls in [
j
;
j+1
].
The subroutine GRG2 (Lasdon and Waren, Department of General
Business, University of Texas at Austin, 1982) can be used for the numeri-
cally constrained optimization while maximizing `(; ; ) (36) subject to
constraints (i) through (iv). The algorithm, which is based on the general-
ized reduced gradient method
3;89
, is considered one of the best in regard to
reliability and numerical stability of the solutions
123
. Within the framework
of the EHR model, the likelihood ratio test can be used to determine the
shape of the baseline hazard function, to determine the signicance of the
regression coecients, and to discriminate between AFT and PH. Although
several approaches to testing the PH assumption have been developed
11;142
,
the EHR model oers the unique advantage of permitting a comparison be-
tween PH and AFT.
3.5. Hazard Regression
Kooperberg, Stone, and Truong
82
developed an adaptive hazard regres-
sion (HARE) methodology to model the conditional log-hazard function
(tjx) = log h(tjx) as an alternative to the various aforementioned PH
models. Let x = (x
1
;:::;x
p
) range over the subsetX=X
1
X
p
of
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