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
Search WWH ::
Custom Search