Biomedical Engineering Reference
In-Depth Information
number of exactly observed and interval-censored (including right-censored)
observations, respectively. Also, let N
1
=
P
l=1
n
l1
denote the total number of
exactly observed observations and N
2
=
P
l=1
n
l2
the total number of nonex-
actly observed observations in the study. Define diagonal coecient matrices
A = Diag(
N
1
n
l1
) and B = Diag(
N
2
n
l2
);l = 1; ;k. The test statistic U
III
for
testing H
0
is given by
X
z
i
e
i
f G(R
i
)gf G(R
i
)g
U
III
= A
G(R
i
) G(R
i
)
i=1
(14.1)
X
z
i
(1 e
i
)
f G(L
i
)gf G(R
i
)g
G(L
i
) G(R
i
)
+B
;
i=1
where is a link function dened earlier in gLRT2. Note that the rst term
on the right-hand side is for exactly observed and the second is for interval-
censored observations. Special cases include
•
When there is no exact observed observation (N
1
= 0), U
III
reduces to
U
II
in gLRT2.
•
When an observation in the data is either exactly observed or right-
censored, we have right-censored data. If (x) = xlog(x), U
III
, as dis-
cussed in Peto and Peto (1972), is the logrank test statistic for right-
censored data.
Zhao et al. (2008) showed that, as n !1 and under H
0
and some minor
conditions, U
III
=
p
n follows an asymptotic normal distribution with mean 0
and a covariance matrix that can be consistently estimated by V
III
= (v
lr
)
kk
,
where
8
<
n
l1
1)P
N
1
( if
G
) +
N
2
n
(
N
2
n
l2
1)Q
N
2
( K
G
);
N
1
n
(
N
1
if l = r;
v
lr
=
:
N
1
n
P
N
1
( if
G
)
N
2
n
Q
N
2
( K
G
);
otherwise;
where
X
(1 e
i
)[
f G(L
i
)gf G(R
i
)g
G(L
i
) G(R
i
)
1
N
2
Q
N
2
( K
G
) =
]
2
;
i=1
Search WWH ::
Custom Search