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 227

on A(1) otherwise. First, we can write that

E[Y a ]

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

= E [E [Y m + (1 Y m )

E [YjL(0);A(0) = a;L(1);A(1) = 1]jL(0);A(0) = a]] :

Now, dening the function d(y;) = (1 y) + y, we may write E[Y a ] as

" " X

E E E [YjL(0);A(0) = a;L(1);A(1) = d(Y m ;)]

##
: pr( m = jL(0);A(0) = a;L(1)) L(0);A(0) = a

Because we have that

E [YjL(0);A(0) = a;L(1);A(1) = d(Y m ;)] =

Y m + (1 Y m )E [YjL(0);A(0) = a;L(1);A(1) = 1] ;

irrespective of the value of , we can express E[Y a ] as

n
Y m + (1 Y m )E [YjL(0);A(0) = a;L(1);A(1) = 1]

o

E E

L(0);A(0) = a

X

pr( m = jL(0);A(0) = a;L(1))

= E [E [Y m + (1 Y m )

E [YjL(0);A(0) = a;L(1);A(1) = 1]jL(0);A(0) = a]] :

from which we conclude, as claimed, that E[Y a ] = E[Y a ].

Bibliography

Andrews, C., van der Laan, M., and Robins, J. (2005). Locally ecient es-

timation of regression parameters using current status data. Journal of

Multivariate Analysis 96, 332{351.