Biomedical Engineering Reference
In-Depth Information
1. A(0) and Y
a
are independent given L(0);
2. A(1) and Y
a
are independent given L(0), A(0) = a and L(1).
Based on the nonparametric structural equations model presented above, it
is clear that the sequential randomization assumption will be satisfied if, in
particular, (i) U
A
is independent of (U
Y
0
;U
0
;:::;U
Y
m
;U
M
m+1
;U
Y
m+1
) given
L(0), and (ii) U
m
and (U
M
m+1
;U
Y
m+1
) are independent given L(0), A(0) = a
and L(1). This assumption guarantees the absence of potential confounders
beyond recorded covariates and is crucial to untangle causal effects from ob-
served associations. The positivity assumption requires, on the other hand,
the existence of some > 0 such that pr(A(0) = a j L(0)) > and
pr(A(1) = 1 j Y
m
= 0;L(1);A(0) = a;L(0)) > with probability 1 for each
a 2 f0; 1g. This ensures that, even relying simply on chance, all considered
interventions are not impossible to observe in the sampled population, regard-
less of the observed past. This nonparametric system of structural equations
coupled with the sequential randomization assumption yields a nonparametric
statistical model, where by statistical model we mean the class of all poten-
tial probability measures for the data-generating distribution. As such, in this
chapter, estimation of the parameter (P) is performed within this nonpara-
metric statistical model restricted only by the positivity assumption made.
8.3
Estimation and Inference
8.3.1
Overview of Targeted Minimum Loss{Based Estima-
tion
Targeted minimum loss-based estimation is a general iterative estimation
framework first presented in van der Laan and Rubin (2006). While a brief
outline of targeted minimum loss-based estimation is provided below, we refer
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