Balancing Covariates in Randomized Experiments with the Gram-Schmidt Walk Design

TL;DR

This paper introduces a new experimental design that balances covariate balance and robustness by formalizing the trade-off and providing a method to navigate it, improving the accuracy of treatment effect estimation.

Contribution

It formalizes the balance-robustness trade-off in experimental design and proposes a design that optimally navigates this trade-off using a robustness parameter.

Findings

Design bounds mean squared error via implicit ridge regression

Asymptotically balances all linear functions of covariates

Provides conditions for asymptotic normality and valid confidence intervals

Abstract

The design of experiments involves a compromise between covariate balance and robustness. This paper provides a formalization of this trade-off and describes an experimental design that allows experimenters to navigate it. The design is specified by a robustness parameter that bounds the worst-case mean squared error of an estimator of the average treatment effect. Subject to the experimenter's desired level of robustness, the design aims to simultaneously balance all linear functions of potentially many covariates. Less robustness allows for more balance. We show that the mean squared error of the estimator is bounded in finite samples by the minimum of the loss function of an implicit ridge regression of the potential outcomes on the covariates. Asymptotically, the design perfectly balances all linear functions of a growing number of covariates with a diminishing reduction in robustness, effectively allowing experimenters to escape the compromise between balance and robustness in large samples. Finally, we describe conditions that ensure asymptotic normality and provide a conservative variance estimator, which facilitate the construction of asymptotically valid confidence intervals.