  In this paper, we study the design and analysis of experiments conducted on a
set of units over multiple time periods where the starting time of the
treatment may vary by unit. The design problem involves selecting an initial
treatment time for each unit in order to most precisely estimate both the
instantaneous and cumulative effects of the treatment. We first consider
non-adaptive experiments, where all treatment assignment decisions are made
prior to the start of the experiment. For this case, we show that the
optimization problem is generally NP-hard, and we propose a near-optimal
solution. Under this solution, the fraction entering treatment each period is
initially low, then high, and finally low again. Next, we study an adaptive
experimental design problem, where both the decision to continue the experiment
and treatment assignment decisions are updated after each period's data is
collected. For the adaptive case, we propose a new algorithm, the
Precision-Guided Adaptive Experiment (PGAE) algorithm, that addresses the
challenges at both the design stage and at the stage of estimating treatment
effects, ensuring valid post-experiment inference accounting for the adaptive
nature of the design. Using realistic settings, we demonstrate that our
proposed solutions can reduce the opportunity cost of the experiments by over
50%, compared to static design benchmarks.
