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.
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