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We develop a unified theory of designs for controlled experiments thatbalance baseline covariates a priori (before treatment and beforerandomization) using the framework of minimax variance and a new method calledkernel allocation. We show that any notion of a priori balance must go hand inhand with a notion of structure, since with no structure on the dependence ofoutcomes on baseline covariates complete randomization (no special covariatebalance) is always minimax optimal. Restricting the structure of dependence,either parametrically or non-parametrically, gives rise to certain covariateimbalance metrics and optimal designs. This recovers many popular imbalancemetrics and designs previously developed ad hoc, including randomized blockdesigns, pairwise-matched allocation and rerandomization. We develop a newdesign method called kernel allocation based on the optimal design whenstructure is expressed by using kernels, which can be parametric ornon-parametric. Relying on modern optimization methods, kernel allocation,which ensures nearly perfect covariate balance without biasing estimates undermodel misspecification, offers sizable advantages in precision and power asdemonstrated in a range of real and synthetic examples. We provide strongtheoretical guarantees on variance, consistency and rates of convergence anddevelop special algorithms for design and hypothesis testing.

Optimal A Priori Balance in the Design of Controlled Experiments