Randomization-Based Causal Inference from Unbalanced 2^2 Split-Plot   Designs
  Given two 2-level factors of interest, a 2^2 split-plot design} (a) takes each of the $2^2=4$ possible factorial combinations as a treatment, (b) identifies one factor as `whole-plot,' (c) divides the experimental units into blocks, and (d) assigns the treatments in such a way that all units within the same block receive the same level of the whole-plot factor.   Assuming the potential outcomes framework, we propose in this paper a randomization-based estimation procedure for causal inference from 2^2 designs that are not necessarily balanced. Sampling variances of the point estimates are derived in closed form as linear combinations of the between- and within-block covariances of the potential outcomes. Results are compared to those under complete randomization as measures of design efficiency. Interval estimates are constructed based on conservative estimates of the sampling variances, and the frequency coverage properties evaluated via simulation. Asymptotic connections of the proposed approach to the model-based super-population inference are also established. Superiority over existing model-based alternatives is reported under a variety of settings for both binary and continuous outcomes. 
Anqi Zhao, Peng Ding, Tirthankar Dasgupta
stat.ME