Biomedical Engineering Reference
In-Depth Information
4. ODS with an Ordinal Outcome Variable and Auxiliary
Covariate
In this section, we consider inference for ODS design with ordinal outcomes.
For example, in CPP one ordinary outcome variable is preterm birth where
preterm is dened as delivery that occurs before 37 completed weeks of
gestation.
In this section, we will review some semiparametric methods for the
ODS scheme in which the outcome variable is ordinal.
4.1. Semiparametric Empirical Likelihood
Wang and Zhou
12
extended the semiparametric empirical likelihood method
developed by Zhou, et al.
15
to the case that there exists auxiliary covariate
information. They focused on the ordinal outcome variable.
Let Y be a categorical disease outcome with possible values 1;:::;J,
and X be a vector of covariates, X be a vector of covariates, X can consist
of either continuous or discrete variable. f
(YjX) is the conditional density
function of Y given X. Let W be an auxiliary covariate for X with possible
values 1;:::;K. Wang and Zhou
12
considered a sampling scheme in which
the subsamples in the two-component study are observed from the K strata
dened by W. From each of the stratafk : W = kgin the study population,
one observes a SRS subsample, denoted as V
0k
, of size n
0k
. In addition, one
observes an ODS subsample from each of the stratafj;k : Y = j;W = kg
denoted as V
jk
, having sizes n
1;k
;:::;n
jk
, respectively. Then the likelihood
of the two-component study with auxiliary information is
Y
K
Y
Y
K
L=
f
(Y
i
jX
i
)dG(X
i
jW
i
= k)
k=1
i2V
0k
k=1
J
Y
Pr(Y
i
= jjX
i
)
Pr(Y
i
= jjW
i
= k)
dG(X
i
jW
i
= k)
j=1
where G(xjw) is the cumulative distribution of X given W. Because of the
constraint Pr(Y
i
= jjw) =
R
Pr(Y = jjx)dG(xjw), the above likelihood
involves G(xjw).
For ease of presentation, Wang and Zhou
12
considered the case of a
binary outcome. Let Y be a binary disease outcome with 1 for the positive
outcome and 2 for the negative outcome. Without losing generality, let
W = 1; 2(K = 2) be a binary auxiliary covariate. Dene V
k
=[
j=0
V
jk
with
size n
k
=
P
P
2
2
k=1
n
k
,
j=0
n
jk
and V =[
k=1
V
k
with the total study size n =
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