Biomedical Engineering Reference
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rest of study cohort is also available. In CPP study, the additional infor-
mation about various health outcomes for the 44,075 subjects is known.
Weaver and Zhou
14
considered statistical inference for a two-stage ODS
design where in addition to the ODS data considered by Zhou et al.
15
some
information is available for the underlying population. These information
includes IQ, SES (socioeconomic status of the child's family), EDU (the
mother's education), SEX (the gender of the child) and RACE (the race of
the child) (i.e., everything but PCB).
Assume Y partitions the study population into K strata such that for
k = 1;:::;K thefY2C
k
gstratum has N
k
individuals. The total sample
size in is N =
P
K
k=1
N
k
. For each stratumfY2C
k
gof the rst stage, one
selects an outcome-dependent validation subsample, denoted as V
k
, of size
n
V
k
such that individuals in V
k
will have their true exposure variable X
observed besides their Y , while the remaining n
V
k
= N
k
n
V
k
individuals,
denoted as V
k
, have only their Y observed. For thefY2C
k
gstratum of
the study population, the date structure of two-stage sampling is
The rst stage:fY
i
g for i2V
k
+ V
k
The second stage:fX
i
jY2C
k
g for i2V
k
When data have been obtained through a two-stage design, it is easy
to see that conditional on the observed sizefn
V
k
g, the observations in the
non-validation sample are independent of the observations in the validation
sample.
Using the Bayes formula, Weaver and Zhou
14
showed that the likelihood
for the second stage observations can be shown to be
Y
K
Y
L
1
() =
f
(Y
i
jX
i
;Y
i
2C
k
)g
X
(X
i
)
k=1
i2V
k
K
Y
Y
I(Y
i
2C
k
)f
(Y
i
jX
i
)g
X
(X
i
)
EI(Y
i
2C
k
)
=
k=1
i2V
k
Y
K
Y
Y
K
2C
k
)g
n
V
k
:
=
f
(Y
i
jX
i
)g
X
(X
i
)
fEI(Y
i
k=1
i2V
k
k=1
They derived the likelihood function based on all N observations, both with
complete and incomplete information. Conditional on the component sizes
of the ODS being xed, the stratum sizes for the nonvalidation sample,
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