Biomedical Engineering Reference
In-Depth Information
is proportional to
Y
m+
X
L =
ij
[S(s
j1
jz
i
) S(s
j
jz
i
)]:
i=1
j=1
Under the PH model,
Y
m+
X
ij
[S(s
j1
)
e
0
z
i
S(s
j
)
e
0
z
i
]:
L = L(;S(s
1
); ;S(s
m
)) =
i=1
j=1
The restricted MLE S(s
j
) for S(s
j
);j = 1; ;m; can be obtained using
the self-consistency algorithm (Turnbull, 1976) by setting = 0. Let g
j
=
S(s
j1
) S(s
j
) and p
j
= S(s
j
)= S(s
j1
). Based on @ log L=@ evaluated at
= 0, the score test statistic is given by
f
z
i
log p
j
P
m+1
X
m+
X
r=j
ir
g
r
P
l
il
g
l
z
i
log p
j
1 p
j
ij
g
j
P
l
il
g
l
U
IV
=
g:
i=1
j=1
Note that U
IV
is equivalent to U
II
with link function (x) = x log x when
there is no exactly observed observation.
To compare difference survival functions, let V
IV
be the asymptotically
unbiased estimate of the covariance of U
IV
after evaluating the covariance
matrix at = 0 and replacing S by the restricted MLEs S: Then H
0
or the
hypothesis that = 0 can be tested using
IV
= U
T
IV
V
IV
U
IV
, which has
asymptotically a
2
-distribution with (k 1) degrees of freedom.
14.3
Software: glrt Package
Published in September 2010 by Zhao and Sun, the R package glrt version
1.0 implements the four types generalized logrank tests introduced in Section
14.2 and conducts a k-sample treatment comparison of survival functions for
k 2. This package depends on the package Icens, which helps estimate the
common survival function in methods gLRT1-gLRT4 assuming no difference
among treatment groups.
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