Biomedical Engineering Reference
In-Depth Information
where D
= D
V
+
P
m+2
k=0
D
k
. As such, it is clear that the ecient influ-
ence curve of is a member of the linear span of the generalized scores at
(
0
;
1
;:::;
m+2
;
V
) = (0; 0;:::; 0), implying eciency of any associated tar-
geted minimum loss-based estimators under regularity conditions. An itera-
tive procedure may be implemented as in Sections 8.3 and 8.4, resorting, in
particular, to relatively simple offset logistic regressions to obtain updated
estimates. The targeted estimate of the distribution Q
V;0
= Q
V
(P
0
) of the
baseline covariate vector V will be the empirical distribution Q
V;n
of observed
baseline covariates. Once the targeted estimator
Q
n
=
Q
0;
0;n
; Q
0;
1;n
;:::; Q
0;
m+2;n
; Q
1;
0;n
; Q
1;
1;n
;:::; Q
1;
m+2;n
;Q
V;n
of Q
0
= Q(P
0
) is obtained, the targeted estimator
n
of
0
, and of
0
= (P
0
)
under causal assumptions, will simply be given by
X
X
Q
a;
(Q
n
) = argmax
I(v
i
2V)h(a;v
i
)
0;n
(v
i
) log m
(a;v
i
)
i=1
a2f0;1g
1 Q
a;
0;n
(v
i
)
log (1 m
(a;v
i
))
:
This estimator can therefore be obtained as the estimated coecient vector
n
using weighted maximum likelihood from the multivariable logistic regression
based on the fabricated data set
Outcome
Covariate 1
Covariate 2 Covariate R
Weight
Q
0;
0;n
(v
i
1
)
z
1
(0;v
i
1
)
z
2
(0;v
i
1
) z
R
(0;v
i
1
)
h(0;v
i
1
)
Q
1;
0;n
(v
i
1
)
z
1
(1;v
i
1
)
z
2
(1;v
i
1
) z
R
(1;v
i
1
)
h(1;v
i
1
)
Q
0;
0;n
(v
i
2
)
z
1
(0;v
i
2
)
z
2
(0;v
i
2
) z
R
(0;v
i
2
)
h(0;v
i
2
)
Q
1;
0;n
(v
i
2
)
z
1
(1;v
i
2
)
z
2
(1;v
i
2
) z
R
(1;v
i
2
)
h(1;v
i
2
)
Q
0;
0;n
(v
i
d
)
z
1
(0;v
i
d
)
z
2
(0;v
i
d
) z
R
(0;v
i
d
)
h(0;v
i
d
)
Q
1;
0;n
(v
i
d
)
z
1
(1;v
i
d
)
z
2
(1;v
i
d
) z
R
(1;v
i
d
)
h(1;v
i
d
)
where (i
1
;i
2
;:::;i
d
) (0 d n) is the vector of indices of all observations sat-
isfying the condition v
i
2V. Asymptotic properties of the resulting estimator
can be obtained as described in Section 8.3, for example.
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