Biomedical Engineering Reference
In-Depth Information
participant, instead, can be represented as
O = (M
0
;A;Y
0
;
0
;M
1
;Y
1
;
1
;:::;M
m
;Y
m
;
m
;M
m+1
;Y
m+1
) ;
where M
k
=
k1
M
k
+ (1
k1
)M
k1
and Y
k
=
k1
Y
k
+ (1
k1
)Y
k1
for k 0, and
1
is defined as 1. In particular, this data structure reflects
the fact that covariates and outcome statuses at a given time can only be ob-
served if the individual was monitored at that particular time. Otherwise, the
covariate and outcome processes are set arbitrarily at their previous observed
values. Of course, if Y
k
= 1, then it must be true that Y
r
= 1 for any r k
because then either death or the event of interest is known to have occurred in
the past; in other words, the process Y
k
, k = 0; 1;:::;m+ 1, is a binary process
with absorbing state 1.
To codify the causal assumptions under which the inference we draw will
have desirable causal interpretations, we resort to a system of nonparametric
structural equations, as formalized by Pearl (2000). Specifically, we suppose
that there exist functions f
M
k
, k = 0; 1;:::;m + 1, f
A
, f
Y
k
, k = 0; 1;:::;m + 1,
and f
k
, k = 0; 1;:::;m, such that
M
k
= f
M
k
(pa(M
k
);U
M
k
) , k = 0; 1;:::;m + 1
A = f
A
(M
0
;U
A
)
Y
k
= f
Y
k
(pa(Y
k
);U
Y
k
) , k = 0; 1;:::;m + 1
k
= f
k
(pa(
k
);U
k
) , k = 0; 1;:::;m
where, for any random vector W, pa(W) represents the parents of W, defined
as the set of all random elements collected before W in the time-ordering
explicitly used in O. In particular, we have that pa(M
0
) =?, pa(M
k
) =
(M
0
;A;Y
0
;
0
;M
1
;:::;
k1
), pa(
k
) = (M
0
;A;Y
0
;
0
;M
1
;:::;Y
k
), pa(Y
k
) =
(M
0
;A;Y
0
;
0
;M
1
;:::;M
k
), and
U = (U
M
0
;:::;U
M
m+1
;U
A
;U
Y
0
;:::;U
Y
m+1
;U
0
;:::;U
m
)
is a vector of exogenous errors encapsulating the randomness in the unit data
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