27
3

A general characterization of optimal tie-breaker designs

Abstract

In a regression discontinuity design, subjects with a running variable xx exceeding a threshold tt receive a binary treatment while those with xtx\le t do not. When the investigator can randomize the treatment, a tie-breaker design allows for greater statistical efficiency. Our setting has random xFx\sim 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 xx, to express the strength of a preference for treating subjects with higher xx. Under these conditions we show that there always exists an optimal design with treatment probabilities piecewise constant in xx. It is natural to require these treatment probabilities to be non-decreasing in xx; under this constraint, we find an optimal design requires just two probability levels, when FF is continuous. By contrast, a typical tie-breaker design as in Owen and Varian (2020) uses a three level design with fixed treatment probabilities 00, 0.50.5 and 11. 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 FF is not symmetric. Our methods easily extend to the fixed xx 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.

View on arXiv
Comments on this paper