Biomedical Engineering Reference
In-Depth Information
observed if Li
i
= R
i
, right-censored if Ri
i
= 1, and left-censored if L
i
= 0. It
is assumed that the mechanism generating interval-censoring is independent
of the Ti
i
.
Let G
l
(t) be the survival function of the survival time of the l-th treatment
group. Our goal is to test the hypothesis H
0
: G
1
(t) = = G
k
(t), which
implies that the k treatment groups have identical failure time distributions.
Let G(t) denote the common survival function of survival times Ti
i
and G the
nonparametric maximum likelihood estimate of G based on observed interval-
censored data under H
0
. To estimate G, let 0 = s
0
< s
1
< < s
m+1
= 1
denote the ordered distinct time points of fL
i
;R
i
; i = 1;:::;ng at which G
has jumps. Also, define
ij
= I( s
j
2 (L
i
;R
i
] ), which indicates that subject
i is at risk at is
j
.
14.2.2
Generalized Logrank Test I (gLRT1)
This generalized logrank test for interval-censored data was proposed by Zhao
and Sun (2004). It allows data to have both censored and exactly observed
observations. Let
i
= 0 if observation on Ti
i
is right-censored and 1 otherwise.
Also, let
ij
= I(
i
= 0 ; L
i
s
j
), indicating the event that Ti
i
is right-
censored and subject i is still at risk at is
j
, i = 1;:::;n, j = 1;:::;m.
The statistic U
I
(k 1) for testing H
0
has the form
X
U
I
=
(d
jl
n
jl
d
j
=n
j
) ;
j=1
where
X
m+
X
z
i
i
ij
[ G(s
j
) G(s
j
)]=
iu
[ G(s
u
) G(s
u
)];
d
jl
=
i=1
u=1
m+
X
X
m+
X
X
z
i
i
ir
[ G(s
r
) G(s
r
)]=
iu
[ G(s
u
) G(s
u
)] +
n
jl
=
z
i
ij
;
r=j
i=1
u=1
i=1
X
m+
X
i
ij
[ G(s
j
) G(s
j
)]=
iu
[ G(s
u
) G(s
u
)] ;
d
j
=
i=1
u=1
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