A general characterization of optimal tie-breaker designs
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In a regression discontinuity design, subjects with a running variable x exceeding a threshold t receive a binary treatment while those with x≤ t do not. When the investigator can randomize the treatment, a tie-breaker design allows for greater statistical efficiency. Our setting has random x∼ F, a working model where the response satisfies a two line regression model, and two economic constraints. One constraint is on the expected proportion of treated subjects and the other is on how treatment correlates with x, to express the strength of a preference for treating subjects with higher x. Under these conditions we show that there always exists an optimal design with treatment probabilities piecewise constant in x. It is natural to require these treatment probabilities to be non-decreasing in x; under this constraint, we find an optimal design requires just two probability levels, when F is continuous. By contrast, a typical tie-breaker design as in Owen and Varian (2020) uses a three level design with fixed treatment probabilities 0, 0.5 and 1. We find large efficiency gains for our optimal designs compared to using those three levels when fewer than half of the subjects are to be treated, or F is not symmetric. Our methods easily extend to the fixed x design problem and can optimize for any efficiency metric that is a continuous functional of the information matrix in the two-line regression. We illustrate the optimal designs with a data example based on Head Start, a U.S. government early-childhood intervention program.
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