Neymanian inference in randomized experiments

In his seminal work in 1923, Neyman studied the variance estimation problem for the difference-in-means estimator of the average treatment effect in completely randomized experiments. He proposed a variance estimator that is conservative in general and unbiased when treatment effects are homogeneous. While widely used under complete randomization, there is no unique or natural way to extend this estimator to more complex designs. To this end, we show that Neyman's estimator can be alternatively derived in two ways, leading to two novel variance estimation approaches: the imputation approach and the contrast approach. While both approaches recover Neyman's estimator under complete randomization, they yield fundamentally different variance estimators for more general designs. In the imputation approach, the variance is expressed as a function of observed and missing potential outcomes and then estimated by imputing the missing potential outcomes, akin to Fisherian inference. In the contrast approach, the variance is expressed as a function of several unobservable contrasts of potential outcomes and then estimated by exchanging each unobservable contrast with an observable contrast. Unlike the imputation approach, the contrast approach does not require separately estimating the missing potential outcome for each unit. We examine the theoretical properties of both approaches, showing that for a large class of designs, each produces conservative variance estimators that are unbiased in finite samples or asymptotically under homogeneous treatment effects.