Biomedical Engineering Reference
In-Depth Information
demonstrate such an extension to target parameters of the form
Z
X
E (Y
a
jV = v) log m
(a;v)
(P) = argmax
h(a;v)
v2V
a2f0;1g
(1 E (Y
a
jV = v)) log (1 m
(a;v))
Q
V
(dv)
where V is a subset of the baseline covariates L(0) = M
0
, V is a collection of
values for V , h is some weight function, m
is a function of (a;v) indexed by
a nite-dimensional parameter = (
1
;
2
;:::;
R
) for R 2Nand such that
m
(a;v)
1 m
(a;v)
X
log
=
r
z
r
(a;v)
r=1
for some real functions z
1
;z
2
;:::;z
R
of (a;v), and Q
V
is the probability dis-
tribution of V . The function m
represents a working model for the coun-
terfactual mean E(Y
a
jV = v), where we have assumed the causal framework
of Section 8.4. Under causal assumptions previously stated, the parameter
may be written as a statistical parameter depending on P through Q =
( Q
0
m+2
; Q
0
m+1
;:::; Q
0
; Q
1
m+2
; Q
1
m+1
;:::; Q
0
;Q
V
), where, for k 2f1; 2;:::;m+2g,
Q
k
is defined as in Section 8.4, and we have redefined
Q
0
as the function
Q
0
(v) = E( Q
1
jV = v) :
Specifically, we may consider the statistical parameter
Z
X
Q
0
(v) log m
(a;v)
1 Q
0
(v)
log (1 m
(a;v))
(Q) = argmax
h(a;v)
v2V
a2f0;1g
Q
V
(dv)
which will agree with the target parameters under the causal assumptions
imposed and at the true data-generating distribution.
In order to construct a targeted minimum loss-based estimator of
0
=
(Q
0
), where Q
0
= Q(P
0
), we require the specification of appropriate loss
functions and fluctuation sub-models. The loss functions defined in Section 8.4
will be utilized in this section as well. However, for each k 2f0; 1;:::;m + 1g,
we define the sum loss function
X
L
k;a
( Q
k
;
Q
k+1
)
L
k
(Q
k
; g
k
) =
a2f0;1g
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