  Given an (optimal) dynamic treatment rule, it may be of interest to evaluate
that rule -- that is, to ask the causal question: what is the expected outcome
had every subject received treatment according to that rule? In this paper, we
study the performance of estimators that approximate the true value of: 1) an
$a$ $priori$ known dynamic treatment rule 2) the true, unknown optimal dynamic
treatment rule (ODTR); 3) an estimated ODTR, a so-called "data-adaptive
parameter," whose true value depends on the sample. Using simulations of
point-treatment data, we specifically investigate: 1) the impact of
increasingly data-adaptive estimation of nuisance parameters and/or of the ODTR
on performance; 2) the potential for improved efficiency and bias reduction
through the use of semiparametric efficient estimators; and, 3) the importance
of sample splitting based on CV-TMLE for accurate inference. In the simulations
considered, there was very little cost and many benefits to using the
cross-validated targeted maximum likelihood estimator (CV-TMLE) to estimate the
value of the true and estimated ODTR; importantly, and in contrast to non
cross-validated estimators, the performance of CV-TMLE was maintained even when
highly data-adaptive algorithms were used to estimate both nuisance parameters
and the ODTR. In addition, we apply these estimators for the value of the rule
to the "Interventions" Study, an ongoing randomized controlled trial, to
identify whether assigning cognitive behavioral therapy (CBT) to criminal
justice-involved adults with mental illness using an ODTR significantly reduces
the probability of recidivism, compared to assigning CBT in a
non-individualized way.
