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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