Kernel regression analysis of tie-breaker designs

Tie-breaker experimental designs are hybrids of Randomized Controlled Trials (RCTs) and Regression Discontinuity Designs (RDDs) in which subjects with moderate scores are placed in an RCT while subjects with extreme scores are deterministically assigned to the treatment or control group. In settings where it is unfair or uneconomical to deny the treatment to the more deserving recipients, the tie-breaker design (TBD) trades off the practical advantages of the RDD with the statistical advantages of the RCT. The practical costs of the randomization in TBDs can be hard to quantify in generality, while the statistical benefits conferred by randomization in TBDs have only been studied under linear and quadratic models. In this paper, we discuss and quantify the statistical benefits of TBDs without using parametric modelling assumptions. If the goal is estimation of the average treatment effect or the treatment effect at more than one score value, the statistical benefits of using a TBD over an RDD are apparent. If the goal is nonparametric estimation of the mean treatment effect at merely one score value, we prove that about 2.8 times more subjects are needed for an RDD in order to achieve the same asymptotic mean squared error. We further demonstrate using both theoretical results and simulations from the Angrist and Lavy (1999) classroom size dataset, that larger experimental radii choices for the TBD lead to greater statistical efficiency.