Biomedical Engineering Reference
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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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