  Construction of just-in-time adaptive interventions, such as prompts
delivered by mobile apps to promote and maintain behavioral change, requires
knowledge about time-varying moderated effects to inform when and how we
deliver intervention options. Micro-randomized trials (MRT) have emerged as a
sequentially randomized design to gather requisite data for effect estimation.
The existing literature (Qian et al., 2020; Boruvka et al., 2018; Dempsey et
al., 2020) has defined a general class of causal estimands, referred to as
"causal excursion effects", to assess the time-varying moderated effect.
However, there is limited statistical literature on how to address potential
between-cluster treatment effect heterogeneity and within-cluster interference
in a sequential treatment setting for longitudinal binary outcomes. In this
paper, based on a cluster conceptualization of the potential outcomes, we
define a larger class of direct and indirect causal excursion effects for
proximal and lagged binary outcomes, and propose a new inferential procedure
that addresses effect heterogeneity and interference. We provide theoretical
guarantees of consistency and asymptotic normality of the estimator. Extensive
simulation studies confirm our theory empirically and show the proposed
procedure provides consistent point estimator and interval estimates with valid
coverage. Finally, we analyze a data set from a multi-institution MRT study to
assess the time-varying moderated effects of mobile prompts upon binary study
engagement outcomes.
