Biomedical Engineering Reference
In-Depth Information
where we have dened
Y Q 2 ;
I(A(0) = a;A(1) = 1)
g 0 (a)g 1 (a)
D 2;a (O)
=
Q 2 Q 1 ;
I(A(0) = a)
g 0 (a)
D 1;a (O)
=
Q 1 (Q)
D 0;a (O)
=
and we have set g 0 (a)(O) = pr(A(0) = ajL(0)) and g 1 (a)(O) = pr(A(1) =
1jL(1), and A(0) = a;L(0)). A derivation is provided in Bang and Robins
(2005) and van der Laan and Gruber (2011). It is not dicult to show that
the fluctuation sub-models Q 2 ( Q 2 ) = f Q 2 () : jj < 1g, Q 1 ( Q 1 ) = f Q 1 () :
jj < 1g and Q 0 ( Q 0 ) = f Q 0 () : jj < 1g for each of
Q 2 ,
Q 1 and
Q 0
described, respectively, as
Q 2 + I(A(1) = 1;A(0) = a)
Q 2 ()
=
expit
logit
;
g 0 (a)g 1 (a)
Q 1 + I(A(0) = a)
Q 1 ()
=
expit
logit
;
g 0 (a)
expit logit
Q 0 +
Q 0 ()
=
indeed satisfy that
=0
=0
d
d L 2;a ( Q 2 ())
d
d L 1;a ( Q 1 (); Q 2 )
= D 2;a ;
= D 1;a
=0
d
d L 0;a ( Q 0 (); Q 1 )
= D 0;a ;
and
implying that the linear span of the fluctuation sub-model generalized scores
at = 0 contains the ecient inuence curve D a .
8.3.2.2
Description of Algorithm
Suppose that initial estimators Q 2;n , Q 1;n , and Q 0;n of Q 2 , Q 1 , and Q 0 , re-
spectively, are available, and that estimators g 0n (a) and g 1n (a) of the distri-
bution of the intervention nodes g 0 (a) and g 1 (a), respectively, have also been
constructed. All of these could have been obtained by simply fitting logistic
regression models, or may be the product of a more elaborate estimation algo-
rithm making use of machine learning, such as the super learner (see van der
 
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