Optimal Experimental Design for Staggered Rollouts

Experimentation has become an increasingly prevalent tool for guiding decision-making and policy choices. A common hurdle in designing experiments is the lack of statistical power. In this paper, we study the optimal multi-period experimental design under the constraint that the treatment cannot be easily removed once implemented; for example, a government might implement a public health intervention in different geographies at different times, where the treatment cannot be easily removed due to practical constraints. The treatment design problem is to select which geographies (referred by units) to treat at which time, intending to test hypotheses about the effect of the treatment. When the potential outcome is a linear function of unit and time effects, and discrete observed/latent covariates, we provide an analytically feasible solution to the optimal treatment design problem where the variance of the treatment effect estimator is at most 1+O(1/N^2) times the variance using the optimal treatment design, where N is the number of units. This solution assigns units in a staggered treatment adoption pattern -- if the treatment only affects one period, the optimal fraction of treated units in each period increases linearly in time; if the treatment affects multiple periods, the optimal fraction increases non-linearly in time, smaller at the beginning and larger at the end. In the general setting where outcomes depend on latent covariates, we show that historical data can be utilized in designing experiments. We propose a data-driven local search algorithm to assign units to treatment times. We demonstrate that our approach improves upon benchmark experimental designs via synthetic interventions on the influenza occurrence rate and synthetic experiments on interventions for in-home medical services and grocery expenditure