  Two-stage randomization is a powerful design for estimating treatment effects
in the presence of interference; that is, when one individual's treatment
assignment affects another individual's outcomes. Our motivating example is a
two-stage randomized trial evaluating an intervention to reduce student
absenteeism in the School District of Philadelphia. In that experiment,
households with multiple students were first assigned to treatment or control;
then, in treated households, one student was randomly assigned to treatment.
Using this example, we highlight key considerations for analyzing two-stage
experiments in practice. Our first contribution is to address additional
complexities that arise when household sizes vary; in this case, researchers
must decide between assigning equal weight to households or equal weight to
individuals. We propose unbiased estimators for a broad class of individual-
and household-weighted estimands, with corresponding theoretical and estimated
variances. Our second contribution is to connect two common approaches for
analyzing two-stage designs: linear regression and randomization inference. We
show that, with suitably chosen standard errors, these two approaches yield
identical point and variance estimates, which is somewhat surprising given the
complex randomization scheme. Finally, we explore options for incorporating
covariates to improve precision. We confirm our analytic results via simulation
studies and apply these methods to the attendance study, finding substantively
meaningful spillover effects.
