  Randomized block factorial experiments are widely used in industrial
engineering, clinical trials, and social science. Researchers often use a
linear model and analysis of covariance to analyze experimental results;
however, limited studies have addressed the validity and robustness of the
resulting inferences because assumptions for a linear model might not be
justified by randomization in randomized block factorial experiments. In this
paper, we establish a new finite population joint central limit theorem for
usual (unadjusted) factorial effect estimators in randomized block $2^K$
factorial experiments. Our theorem is obtained under a 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, numbers of blocks, block sizes, and propensity scores
across blocks. To improve the estimation and inference efficiency, we propose
four covariate adjustment methods. We 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 covariances to facilitate valid inferences. Simulation studies and a
clinical trial data analysis demonstrate the benefits of the covariate
adjustment methods.
