Biomedical Engineering Reference
In-Depth Information
n
V
k
= N
k
n
k
, k = 1;:::;K, follow a multinomial law such that
Y
K
N
fEI(Y
i
2C
k
)g
N
k
:
Pr(fn
V
k
g) =
Q
K
k=1
N
k
!
k=1
Furthermore, the observations in the nonvalidation sample contribute the
following terms to the full-information likelihood function,
Y
K
Y
L
2
() =
f
(Y
j
)=fEI(Y
i
2C
k
)g
k=1
j2V
k
R
where the quantity f
(Y
j
) =
jx)dG
X
(x) is the contribution from a
non-validation set member which involves an unspecied G
X
.
Conditional on the observed size n
V
k
, the observations in the nonvalida-
tion sample are independent of the observations in the validation sample,
which contribute the terms inL
f
(Y
j
2
to the full-information likelihood. Thus,
after combining and simplifying these terms, the joint likelihood of the
two-stage study can be written as
Y
K
Y
Y
K
Y
L() =
f
(Y
i
jX
i
)g
X
(X
i
)
f
(Y
j
):
j2V
k
k=1
i2V
k
k=1
Obviously, direct maximizationL() is not possible since G
X
is unknown.
Recognizing the distribution of X can be written as
X
K
G
X
(x) = Pr(Xx) =
Pr(XxjY2C
k
)Pr(Y2C
k
);
k=1
therefore, a consistent estimator of G(x) has the form
X
K
G
k
(x)
N
k
G
X
(x) =
N
;
k=1
P
i2V
k
I(X
i
x)=(n
k
+ n
0;k
). Accordingly, a weighted
estimator for f
(Y
j
) is
G
k
(x) =
where
Z
X
K
X
N
k
(n
k
+ n
0;k
)N
f
(Y
j
jX
i
):
f
(Y
j
) =
f
(Y
j
jx)d G
X
(x) =
k=1
i2V
k
Then the estimated log likelihood function has the form
(
)
K
X
X
X
X
N
k
(n
k
+ n
0;k
)N
f
(Y
j
^
L() =
log f
(Y
i
jX
i
) +
log
jX
i
)
:
j2V
i2V
k=1
i2V
k
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