  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.
