Abstract: Split-plot designs find wide applicability in multifactor experiments with randomization restrictions. Practical considerations often warrant the use of unbalanced designs. This study investigates randomization-based causal inference in split-plot designs that are possibly unbalanced. An extension of the balanced case yields an expression for the sampling variance of a treatment contrast estimator, as well as a conservative estimator of the sampling variance. However, the bias of this variance estimator does not vanish, even when the treatment effects are strictly additive. A careful and involved matrix analysis is employed to overcome this difficulty, resulting in a new variance estimator that becomes unbiased under milder conditions. We propose a construction procedure that generates such an estimator with a minimax bias. Empirical studies suggest the superiority of the proposed estimator with respect to bias uniformly across different populations. Furthermore, this superiority does not come at the cost of a large inflation of the mean squared error.

Key words and phrases: Bias, factorial experiment, finite population, minimaxity, treatment-effect additivity.