Randomization-based joint central limit theorem and efficient covariate adjustment in stratified $2^K$ factorial experiments

Stratified factorial experiments are widely used in industrial engineering, clinical trials, and social science to measure the joint effects of several factors on an outcome. Researchers often use a linear model and analysis of covariance to analyze experimental results; however, few studies have addressed the validity and robustness of the resulting inferences because assumptions for a linear model might not be justified by randomization. In this paper, we establish the finite-population joint central limit theorem for usual (unadjusted) factorial effect estimators in stratified $2^K$ factorial experiments. Our theorem is obtained under randomization-based inference framework, making use of an extension of the vector form of the Wald--Wolfowitz--Hoeffding theorem for a linear rank statistic. It is robust to model misspecification, arbitrary numbers of strata, stratum sizes, and propensity scores across strata. To improve the estimation and inference efficiency, we propose three covariate adjustment methods and show that under mild conditions, the resulting covariate-adjusted factorial effect estimators are consistent, jointly asymptotically normal, and generally more efficient than the unadjusted estimator. In addition, we propose Neyman-type conservative estimators for the asymptotic variances to facilitate valid inferences. Simulation studies demonstrate the benefits of covariate adjustment methods. Finally, we apply the proposed methods to analyze a real dataset from a clinical trial to evaluate the effects of the addition of bevacizumab and/or carboplatin on pathologic complete response rates in patients with stage II to III triple-negative breast cancer.