Biomedical Engineering Reference
In-Depth Information
8.4
Extension to the Case of Right-Censored Lifetimes
8.4.1
Causal Framework
In most applications, it is generally the case that individuals may be right-
censored at some point in time before the event of interest is observed to
have occurred. While the underlying data structure X remains unchanged,
the observed data structure can, in this case, be written as
O = (M
0
;A;Y
0
;
0
;C
0
;M
1
;Y
1
;
1
;C
1
;:::;M
m
;Y
m
;
m
;C
m
;M
m+1
;Y
m+1
)
where M
k
=
k1
C
k1
M
k
+ (1
k1
C
k1
)M
k1
and Y
k
=
k1
C
k1
Y
k
+
(1
k1
C
k1
)Y
k1
for k 1, where we dene C
k
as a binary indicator of
not having yet been lost to follow-up. Once more, we construct a system of
nonparametric structural equations given by
M
k
= g
M
k
(pa(M
k
);U
M
k
) , k = 0; 1;:::;m + 1
A = g
A
(M
0
;U
A
)
Y
k
= g
Y
k
(pa(Y
k
);U
Y
k
) , k = 0; 1;:::;m + 1
k
= g
k
(pa(
k
);U
k
) , k = 0; 1;:::;m
C
k
= g
C
k
(pa(C
k
);U
C
k
) , k = 0; 1;:::;m
where g
M
k
, k = 0; 1;:::;m + 1; g
A
; g
Y
k
, k = 0; 1;:::;m + 1; g
k
, k = 0; 1;:::;m;
and g
C
k
, k = 0; 1; 2;:::;m, are unspecied functions. We may dene the coun-
terfactuals
Y
m+1
= g
Y
m+1
(M
0
;a;Y
0
;
0
; 1;M
1
;Y
1
;
1
; 1;:::;M
m
;Y
m
; 1; 1;M
m+1
;Y
m+1
)
obtained by assigning the patient to a specific treatment group by setting
A = a, by requiring the monitoring mechanism to include an observation
at the last study visit by enforcing
m
= 1, and by ensuring the patient
is not lost to follow-up at any time during the study time frame by fixing
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