Biomedical Engineering Reference
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variance-covariance matrix is U
1
(
^
) V (
^
) U
1
(
^
), where U and V are ob-
tained by replacing the large sample quantities in U and V with their
corresponding small sample quantities.
4.2. Semiparametric Estimated Likelihood
Wang and Zhou
12
extended the semiparametric estimated likelihood
method to allow some components of the covariate vector to be observed
for each individual in the population and there exists auxiliary information
for unobserved exposure variables. Especially, letfX;Zgbe the vector of
modeling covariates, where X is an exposure variable that is observed only
when an individual is selected into the second stage and Z is a vector of
additional covariates that are always observed. In addition, let W be a dis-
crete or continuous variable, which contains auxiliary information for X.
Wang and Zhou
13
considered a two-stage design in which the subsample
selection depends on both the outcome Y and a discrete variable C. Let
C be a discrete variable with L levels dened on the auxiliary covariate
W; C = h if W2(c
l1
;c
l
] for l = 1;:::;L. The lth interval, (c
l1
;c
l
],
is dened by a pair of ordered real values where c
0
=1and c
L
=1.
They assumed YC partitions the study cohort into a total KH strata
such that the stratumfY = k;C = lgcontains N
kl
subjects. The total
sample size in is N =
P
L
l=1
N
kl
. For each stratumfY = k;C = lg
of the rst stage, one selects an outcome-dependent validation subsample,
denoted as V
kl
, of size n
V
kl
such that individuals in V
kl
will have their true
exposure variable X observed besides their Y and W, while the remaining
n
V
kl
= N
kl
n
V
kl
individuals, denoted as V
kl
, have only their Y and W
observed. For thefY = k;C = lgstratum of the study population, the date
structure of two-stage sampling is
P
K
k=1
The rst stage:fY
i
;Z
i
;W
i
g for i2V
kl
+ V
kl
The second stage:fX
i
;Z
i
;W
i
jY = k;C = lg for i2V
kl
By the same argument as in last section, after combining and simplifying
these terms, the joint likelihood of the two-stage study can be written as
Y
K
Y
L
Y
Y
K
Y
L
Y
L() =
f
(Y
i
jX
i
;Z
i
)g(X
i
jZ
i
;W
i
)
f
(Y
j
jZ
j
;W
j
):
j2V
kl
k=1
l=1
i2V
kl
k=1
l=1
Obviously, direct maximizationL() is not possible. Note that some com-
ponents of (Z;W) may be uninformative with respect to the distribution of
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