Biomedical Engineering Reference
In-Depth Information
X
1
N
1
P
N
1
( f
G
) =
e
i
[ f
G
(R
i
)]
2
;
i=1
and
8
<
if G(R
i
) =
G(R
i
);
dx
(1 x)j
x=1 G(R
i
)
;
f
G
(R
i
) =
:
f G(R
i
)gf G(R
i
)g
G(R
i
) G(R
i
)
;
otherwise:
The test of the hypothesis H
0
can be conducted as follows:
1. If
n
l1
N
1
=
n
l2
N
2
for all l = 1; ;k, then the sum of scaled test statis-
tics
P
l=1
n
l1
N
1
U
III
[l] = 0. Let U
III;0
denote the rst (k 1) compo-
nents of U
III
and V
III;0
the matrix after deleting the last row and
column of V
III
. H
0
can then be tested using the statistic
III;1
=
U
T
III;0
V
1
III;0
U
III;0
=n, which asymptotically has a
2
-distribution with
(k 1) degrees of freedom.
2. if the condition in (1) does not hold, then H
0
can be tested using the
test statistic
III;2
= U
T
III
V
1
III
U
III
=n, which asymptotically has a
2
-
distribution with k degrees of freedom.
14.2.5
Generalized Logrank Test IV (gLRT4 or Score Test)
This generalized logrank test for interval-censored data was proposed by
Finkelstein (1986) when the covariates are treatment indicators. She took the
maximum likelihood approach by assuming a proportional hazards regression
model (Cox, 1972),
(tjz) =
0
(t) expf
0
zg;
where
0
(t) is an unspecied baseline hazard function and is the regression
coecient for covariates z. Note that testing H
0
is equivalent to testing the
hypothesis that = 0. Let S(tjz) denote the survival function for a subject
with covariates z and S(t) for a subject with covariates z = 0. The likelihood
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