Biomedical Engineering Reference
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structure not captured by the recorded variables. The time-ordering of the
various components of the underlying and observed data structures are ex-
plicitly acknowledged in these nonparametric structural equations. A typical
probability measure associated to the distribution of (O;U) is denoted by
P
O;U
. This framework facilitates the elaboration of a clear and unambiguous
definition for the target parameter. It is also useful for determining conditions
under which this target parameter can be expressed as a statistical parameter,
that is, as a parameter of the observed data distribution. In other words, it
allows a transparent investigation of assumptions required to ensure the target
parameter is identifiable in the given data setting.
Suppose our interest lies in the absolute difference between the probability
of surviving until end-of-study without having experienced the event of interest
in the treatment and control groups. As such, our target parameter =
(P
O;U
) consists of
(P
O;U
) = E
Y
m+1
Y
m+1
where
Y
m+1
= if
Y
m+1
(M
0
;a;Y
0
;
0
;M
1
;:::;M
m
;Y
m
; 1;M
m+1
;Y
m+1
;U
Y
m+1
)
for a 2 f0; 1g is the counterfactual of the end-of-study outcome Y
m+1
ob-
tained by fixing A at value a and setting
m
equal to 1 in the nonparametric
structural equations model. On one hand, the assignment A = a determines
the treatment allocated to the individual. On the other hand, the assignment
m
= 1 ensures that the individual is monitored at end-of-study, so that the
outcome of interest can indeed be recorded. This intervention on
m
might be
perceived as unnecessarily stringent, as in any case where the event of interest
is observed to occur prior to end-of-study, the outcome Y
m+1
is known even
when
m
= 0. In fact, for this reason, one may deem more appropriate the
stochastic intervention on
m
consisting of setting
m
= 1 if Y
m
= 0 and
refraining from any intervention on
m
otherwise. In reality, however, both
these interventions yield identical parameters in this setting; the proof of this
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