  Micro-randomized trials are commonly conducted for optimizing mobile health
interventions such as push notifications for behavior change. In analyzing such
trials, causal excursion effects are often of primary interest, and their
estimation typically involves inverse probability weighting (IPW). However, in
a micro-randomized trial additional treatments can often occur during the time
window over which an outcome is defined, and this can greatly inflate the
variance of the causal effect estimator because IPW would involve a product of
numerous weights. To reduce variance and improve estimation efficiency, we
propose a new estimator using a modified version of IPW, which we call
"per-decision IPW". It is applicable when the outcome is binary and can be
expressed as the maximum of a series of sub-outcomes defined over sub-intervals
of time. We establish the estimator's consistency and asymptotic normality.
Through simulation studies and real data applications, we demonstrate
substantial efficiency improvement of the proposed estimator over existing
estimators (relative efficiency up to 1.45 and sample size savings up to 31% in
realistic settings). The new estimator can be used to improve the precision of
primary and secondary analyses for micro-randomized trials with binary
outcomes.
