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 211

Laan et al. (2007) and van der Laan and Rose (2011)), for example. The

fluctuation sub-models described above thus become

Q 2;n + I(A(1) = 1;A(0) = a) Q 2;n ()

= expit logit ;

g 0n (a)g 1n (a)

Q 1;n + I(A(0) = a)
g 0n (a)

Q 1;n ()

= expit logit ;

expit logit

Q 0;n + : Q 0;n ()

=

One step of targeted minimum loss-based estimation results in revised

Q a; Q 2;n ( a; Q 2 , Q a; Q 1;n ( a; Q 1 , and Q a;

estimates

2;n = 2;n ) of 1;n = 1;n ) of 0;n =

Q 0;n ( a;

0;n ) of Q 0 , where the optimal fluctuation parameters are given by

X

a; L 2;a ( Q 2;n ())(O i ) ;

2;n =

argmin

i=1

X

L 1;a ( Q 1;n ();

Q a;

a;

1;n =

argmin

2;n )(O i ) ;

i=1

X

L 0;a ( Q 0;n (); Q a; a;

0;n =

argmin

1;n )(O i ) :

i=1

A clear advantage of the choice of loss function and fluctuation sub-model

above is that rather than requiring a possibly cumbersome use of general-

purpose optimization routines to obtain the optimizers above, widely avail-

able statistical software may be easily utilized instead. Indeed, the minimizer
a;

2;n can be obtained as the estimated coecient in a logistic regression of the
binary outcome Y on covariate [g 0n (a)g 1n (a)] 1 with offset logit Q 2;n , tted
on the subset of data for which A(1) = 1 and A(0) = a. Subsequently, the
minimizer a;

1;n is the estimated coecient in a logistic regression of the out-

come Q a;

2;n on covariate g 0n (a) 1 with offset logit Q 1;n , fitted on the subset of
data for which A(0) = a. Finally, the minimizer a;

0;n consists of the estimated
coecient in a logistic regression of the outcome Q a;

1;n on a constant predictor

with offset logit Q 0;n , fitted on all available data.
In particular, it is easy to verify that

P i=1

Q a; Q a;

1
n

0;n = 1;n (O i ), and there-