Biomedical Engineering Reference
In-Depth Information
where
I() = K
X
X
L
@ 2 log f (YjX;Z)
@@ 0
kl E kl
k=1
l=1
@ 2 log f (YjZ;W)
@@ 0
+( kl
kl )E kl
kl () = Var X;Z;WjY2C k ;W2B l
(
)
X
K
X
L
1 k 1 l 1
k 1 l 1
k 1 l 1 (W i )E YjY2C k ;W2B l
fM X;Z;W (Y )g
k 1 =1
l 1 =1
and
M X;Z;W (Y ) = @f (YjX;Z)=@
@f (YjZ;W)=@
ff (YjZ;W)g 2
f (YjX;Z):
f (YjZ;W)
By the same argument as in last section, one can construct a consistent
estimator for .
5. Penalized Spline Estimated Likelihood
In this section, we consider nonlinear covariate eects in the ODS designs.
Most of the the results for ODS regression analysis are established in the
setting of linear regression. While in some applications, parametric models
are adequate to capture the underlying relationships between the response
variables and the associated covariates, most of the time they are chosen
simply for their convenience. For example, Zhou, You and Longnecker 16
found that the relationship between IQ and EDU may be not linear. In this
section, we review the partially linear regression analysis for a two-stage
outcome dependent sample in which one allow the relationship between the
response and exposure variable to be unspecied 16;7 . By combining the pe-
nalized splines 8 and the estimated maximum likelihood 14;16 proposed a pe-
nalized spline maximum likelihood estimation (PSMLE) for the parametric
and nonparametric components of a partially linear regression model under
the population based two component ODS sampling scheme.
Assume that the conditional density of Y i givenfX i ;Z i gbelongs to a
canonical exponential family, i.e.,
f (); (Y i jX i ;Z i ) = expf[Y i i b( i )]=a() + c(Y i ;)g;
where a(), b() and c(;) are all known functions, is a dispersion param-
eter and i is related to the X i and Z i by
i = (X i ) + 0 Z i :
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