Geometric Insights and Empirical Observations on Covariate Adjustment and Stratified Randomization in Randomized Clinical Trials

The statistical efficiency of randomized clinical trials can be improved by incorporating information from baseline covariates (i.e., pre-treatment patient characteristics). This can be done in the design stage using a covariate-adaptive randomization scheme such as stratified (permutated block) randomization, or in the analysis stage through covariate adjustment. This article provides a geometric perspective on covariate adjustment and stratified randomization in a unified framework where all regular, asymptotically linear estimators are identified as augmented estimators. From this perspective, covariate adjustment can be viewed as an effort to approximate the optimal augmentation function, and stratified randomization aims to improve a given approximation by projecting it into an affine subspace containing the optimal augmentation function. The efficiency benefit of stratified randomization is asymptotically equivalent to making full use of stratum information in covariate adjustment, which can be achieved using a simple calibration procedure. Simulation results indicate that stratified randomization is clearly beneficial to unadjusted estimators and much less so to adjusted ones and that calibration is an effective way to recover the efficiency benefit of stratified randomization without actually performing stratified randomization. These insights and observations are illustrated using real clinical trial data.