Some theoretical foundations for the design and analysis of randomized experiments

Neyman106's seminal work in 1923 has been a milestone in statistics over the century, which has motivated many fundamental statistical concepts and methodology. In this review, we delve into Neyman106's groundbreaking contribution and offer technical insights into the design and analysis of randomized experiments. We shall review the basic setup of completely randomized experiments and the classical approaches for inferring the average treatment effects. We shall in particular review more efficient design and analysis of randomized experiments by utilizing pretreatment covariates, which move beyond Neyman's original work without involving any covariate. We then summarize several technical ingredients regarding randomizations and permutations that have been developed over the century, such as permutational central limit theorems and Berry-Esseen bounds, and elaborate on how these technical results facilitate the understanding of randomized experiments. The discussion is also extended to other randomized experiments including rerandomization, stratified randomized experiments, matched pair experiments, cluster randomized experiments, etc.