Biomedical Engineering Reference
In-Depth Information
and
m+
X
X
m+
X
X
i
ir
[ G(s
r
) G(s
r
)]=
iu
[ G(s
u
) G(s
u
)] +
n
j
=
ij
:
r=j
i=1
u=1
i=1
It is easy to prove that the d
j
, n
j
, and elements of d
jl
and n
jl
will reduce to
the numbers of failures and risks in the case of right-censored data.
To estimate the covariance matrix of U
I
, Zhao and Sun (2004) proposed to
use the following multiple imputation approach (Rubin, 1987). Let M be the
number of imputations selected by the user. For each r (1 r M), rst
generate right-censored imputation sample f(T
i
;
i
; z
i
) ; i = 1;:::;ng, where
i
= 0 and T
i
= L
i
if
i
= 0 and, if
i
= 1, T
i
is a random number drawn
from the conditional probability function
G(s) G(s)
G(L
i
) G(R
i
)
f
i
(s) = PrfT
i
= sg =
;
where s represents s
j
's that belong to (L
i
;R
i
]. Then find the logrank test
statistic U
(r)
using the proposed formula for U
I
and its covariance esti-
mate V
(r)
for right-censored data. After analyzing M imputation samples,
the estimate of the covariance matrix of U
I
is given by combining the within-
imputation variability due to right-censored observations and the between-
imputation variability due to interval-censored observations,
P
r=1
[U
(r)
U][U
(r)
U]
T
M 1
X
1
M
1
M
)
V
(r)
+ (1 +
V
I
=
;
r=1
where U is the sample mean of the U
(r)
's. Then, H
0
can be tested using
I
=
U
T
I
V
I
U
I
, where V
I
is a generalized inverse of V
I
and
I
approximately
follows a
2
distribution with (k 1) degrees of freedom.
14.2.3
Generalized Logrank Test II (gLRT2)
This class of generalized logrank tests was proposed by Sun et al. (2005).
Assuming that Li
i
< Ri,
i
, the test statistic is given by
X
z
i
f G(L
i
)gf G(R
i
)g
G(L
i
) G(R
i
)
U
II
=
;
i=1
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