Design-based causal inference for incomplete block designs

Summary Researchers often turn to block randomization to increase the precision of their inference or for practical reasons, such as in multi-site trials. However, if the number of treatments under consideration is large, it may not be feasible or practical to assign all treatments within each block. We develop novel inference results under the finite-population, design-based framework for natural alternatives to the complete block design that do not require reducing the number of treatment arms, namely the incomplete block design and the balanced incomplete block design. This includes deriving the properties of two design-based estimators, developing a finite-population central limit theorem and proposing conservative variance estimators. Comparisons between the design-based estimators and linear model-based estimators are also provided. Simulations and a data illustration further demonstrate the performance of incomplete block design estimators. This work highlights incomplete block designs as practical and currently underutilized.