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 226

iterative convex minorant algorithm for computing the nonparametric maxi-
mum likelihood estimator based on interval-censored data with two monitoring
times is also provided in Groeneboom and Wellner (1992). The nonparamet-
ric maximum likelihood estimator for a general marginal interval-censored
data structure allowing for multiple monitoring times is studied in Huang and
Wellner (1997). Regression methods under interval-censoring have also been
proposed in, for example, Rabinowitz et al. (1995) and Huang and Wellner
(1997). Some, including Sparling et al. (2006), have discussed estimation with
interval-censored data in the presence of time-varying covariates through the
use of fully parametric models. Previous work on estimating equation method-
ology for a particular extended interval-censored data structure incorporating
baseline and time-dependent covariates have been carried out in van der Laan
and Hubbard (1997) and in Chapter 6 of van der Laan and Robins (2003),
and, in the context of optimal treatment regimes, in Robins et al. (2008). This
chapter considered this same data structure but made use of state-of-the-art
targeted minimum loss-based estimation instead to estimate causal effects.
Appendix: Equivalence of Candidate Interventions on m
We consider two interventions in the setting of Section 8.3. The first interven-
tion is a static rule and consists of setting A(1) = 1. The second is a stochastic
intervention that sets A(1) = 1 if Y m = 0 and does not intervene on A(1) oth-
erwise. For each a 2f0; 1g, dene the counterfactual Y a by setting A(0) = a
and A(1) = 1 in the system of nonparametric structural equations, and Y a
by setting A(0) = a and fixing A(1) = 1 only if Y m = 0, with no intervention