Nearly random designs with greatly improved balance

We present a procedure that divides a set of experimental units into two groups that are similar on a prespecified set of covariates and are almost as random as with a complete randomization. Under complete randomization, the difference in the standardized average between treatment and control is $O_{\rm p}(n^{-1/2})$, which may be material in small samples. We present an algorithm that reduces imbalance to $O_{\rm p}(n^{-3})$ for one covariate and $O_{\rm p}\{n^{-(1 + 2/p)}\}$ for $p$ covariates, but whose assignments are, strictly speaking, nonrandom. In addition to the metric of maximum eigenvalue of allocation variance, we introduce two metrics that capture departures from randomization and show that our assignments are nearly as random as complete randomization in terms of all measures. Simulations illustrate the results, and inference is discussed. An R package to generate designs according to our algorithm and other popular designs is available.