# Causal Inference from Possibly Unbalanced Split-Plot Designs: A Randomization-based Perspective

## TL;DR

This paper develops new methods for causal inference in unbalanced split-plot designs, providing unbiased variance estimators and a construction procedure to improve inference accuracy in complex experimental setups.

## Contribution

It extends randomization-based causal inference methods to unbalanced split-plot designs, introducing a new unbiased variance estimator and a minimax bias construction procedure.

## Key findings

- Derived a sampling variance expression for treatment contrasts.
- Proposed a new unbiased variance estimator under milder conditions.
- Introduced a minimax bias construction procedure.

## Abstract

Split-plot designs find wide applicability in multifactor experiments with randomization restrictions. Practical considerations often warrant the use of unbalanced designs. This paper investigates randomization based causal inference in split-plot designs that are possibly unbalanced. Extension of ideas from the recently studied 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, which becomes unbiased under milder conditions. A construction procedure that generates such an estimator with minimax bias is proposed.