  Advances in wearables and digital technology now make it possible to deliver
behavioral mobile health interventions to individuals in their everyday life.
The micro-randomized trial (MRT) is increasingly used to provide data to inform
the construction of these interventions. In an MRT, each individual is
repeatedly randomized among multiple intervention options, often hundreds or
even thousands of times, over the course of the trial. This work is motivated
by multiple MRTs that have been conducted, or are currently in the field, in
which the primary outcome is a longitudinal binary outcome. The primary aim of
such MRTs is to examine whether a particular time-varying intervention has an
effect on the longitudinal binary outcome, often marginally over all but a
small subset of the individual's data. We propose the definition of causal
excursion effect that can be used in such primary aim analysis for MRTs with
binary outcomes. Under rather restrictive assumptions one can, based on
existing literature, derive a semiparametric, locally efficient estimator of
the causal effect. We, starting from this estimator, develop an estimator that
can be used as the basis of a primary aim analysis under more plausible
assumptions. Simulation studies are conducted to compare the estimators. We
illustrate the developed methods using data from the MRT, BariFit. In BariFit,
the goal is to support weight maintenance for individuals who received
bariatric surgery.
