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Construction of just-in-time adaptive interventions, such as promptsdelivered by mobile apps to promote and maintain behavioral change, requiresknowledge about time-varying moderated effects to inform when and how wedeliver intervention options. Micro-randomized trials (MRT) have emerged as asequentially randomized design to gather requisite data for effect estimation.The existing literature (Qian et al., 2020; Boruvka et al., 2018; Dempsey etal., 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 potentialbetween-cluster treatment effect heterogeneity and within-cluster interferencein a sequential treatment setting for longitudinal binary outcomes. In thispaper, based on a cluster conceptualization of the potential outcomes, wedefine a larger class of direct and indirect causal excursion effects forproximal and lagged binary outcomes, and propose a new inferential procedurethat addresses effect heterogeneity and interference. We provide theoreticalguarantees of consistency and asymptotic normality of the estimator. Extensivesimulation studies confirm our theory empirically and show the proposedprocedure provides consistent point estimator and interval estimates with validcoverage. Finally, we analyze a data set from a multi-institution MRT study toassess the time-varying moderated effects of mobile prompts upon binary studyengagement outcomes.

Estimating Time-Varying Direct and Indirect Causal Excursion Effects with Longitudinal Binary Outcomes