Targeted Minimum Loss-Based Estimation of a Causal E ect Using Interval-Censored Time-to-Event Data - Interval-Censored-Time-to-Event Data: Methods and Applications - page 204

statement is provided in the Appendix at the end of this chapter. Denote by

0 = (P O;U;0 ) the true value of the target parameter, that is, the value of
the target parameter at the true underlying distribution P O;U;0 .

8.2.2 Identifiability

Dene L(0) = M 0 , A(0) = A, L(1) = (Y 0 ; 0 ;M 1 ;Y 1 ; 1 ;:::;Y m ), A(1) = m ,
L(2) = M m+1 , and Y = Y m+1 . Components of L = (L(0);L(1);L(2)) consist
of recorded covariates, whereas components of A = (A(0);A(1)), consisting

of the baseline treatment assignment and the monitoring status for the last
monitoring time, can be considered intervention nodes in the causal framework

considered. Define the iterated means

Q 2

= E[YjA(1) = 1;L(1);A(0) = a;L(0)]

Q 1

E[ Q 2 jA(0) = a;L(0)]

=

Q 0

E[ Q 1 ] : =

Consider the statistical parameter = (P) of interest to be

(P) = Q 0 (P) Q 0 (P)

where P denotes the distribution for the observed data unit O. Denote by P 0
the true distribution of the observed data and by 0 = (P 0 ) the true value of
the statistical parameter. Because only depends on P through Q = (Q 0 ;Q 1 ),
where Q a = ( Q 0 ; Q 1 ; Q 2 ) for each a 2f0; 1g, (P) will sometimes be denoted
by (Q) for convenience. Statistical methodology will be developed to estimate

the parameter value 0 .
Under certain causal assumptions, this statistical parameter coincides pre-

cisely with the target parameter of interest, as defined above. Specifically,

provided all intervention nodes are sequentially randomized and satisfy the

positivity assumption, the law of total expectation can be used to show that

0 = 0 . On one hand, sequential randomization of all intervention nodes
stipulates that, for each a 2f0; 1g,