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A general characterization of optimal tie-breaker designs

25 February 2022
Harrison H. Li
Art B. Owen
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Abstract

In a regression discontinuity design, subjects with a running variable xxx exceeding a threshold ttt receive a binary treatment while those with x≤tx\le tx≤t do not. When the investigator can randomize the treatment, a tie-breaker design allows for greater statistical efficiency. Our setting has random x∼Fx\sim Fx∼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 xxx, to express the strength of a preference for treating subjects with higher xxx. Under these conditions we show that there always exists an optimal design with treatment probabilities piecewise constant in xxx. It is natural to require these treatment probabilities to be non-decreasing in xxx; under this constraint, we find an optimal design requires just two probability levels, when FFF is continuous. By contrast, a typical tie-breaker design as in Owen and Varian (2020) uses a three level design with fixed treatment probabilities 000, 0.50.50.5 and 111. 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 FFF is not symmetric. Our methods easily extend to the fixed xxx 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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