Rerandomization in $2^K$ Factorial Experiments

TL;DR

This paper investigates rerandomization in $2^K$ factorial experiments, demonstrating its advantages in covariate balance and estimator concentration without distributional assumptions, and provides methods for confidence set construction.

Contribution

It develops a theoretical framework for rerandomization in factorial experiments, analyzing its asymptotic properties and advantages over classical methods.

Findings

Rerandomization leads to more concentrated estimators around true effects.

The joint asymptotic distribution under rerandomization is symmetric and unimodal.

Conservative confidence sets for factorial effects are constructed with large-sample validity.

Abstract

With many pretreatment covariates and treatment factors, the classical factorial experiment often fails to balance covariates across multiple factorial effects simultaneously. Therefore, it is intuitive to restrict the randomization of the treatment factors to satisfy certain covariate balance criteria, possibly conforming to the tiers of factorial effects and covariates based on their relative importances. This is rerandomization in factorial experiments. We study the asymptotic properties of this experimental design under the randomization inference framework without imposing any distributional or modeling assumptions of the covariates and outcomes. We derive the joint asymptotic sampling distribution of the usual estimators of the factorial effects, and show that it is symmetric, unimodal, and more "concentrated" at the true factorial effects under rerandomization than under the classical factorial experiment. We quantify this advantage of rerandomization using the notions of "central convex unimodality" and "peakedness" of the joint asymptotic sampling distribution. We also construct conservative large-sample confidence sets for the factorial effects.