  While attractive from a theoretical perspective, finely stratified
experiments such as paired designs suffer from certain analytical limitations
not present in block-randomized experiments with multiple treated and control
individuals in each block. In short, when using an appropriately weighted
difference-in-means to estimated the sample average treatment effect, the
traditional variance estimator in a paired experiment is conservative unless
the pairwise average treatment effects are constant across pairs; however, in
more coarsely stratified experiments, the corresponding variance estimator is
unbiased if treatment effects are constant within blocks, even if they vary
across blocks. Using insights from classical least squares theory, we present
an improved variance estimator appropriate in finely stratified experiments.
The variance estimator is still conservative in expectation for the true
variance of the difference-in-means estimator, but is asymptotically no larger
than the classical variance estimator under mild conditions. The improvements
stem from the exploitation of effect modification, and thus the magnitude of
the improvement depends upon on the extent to which effect heterogeneity can be
explained by observed covariates. Aided by these estimators, a new test for the
null hypothesis of a constant treatment effect is proposed. These findings
extend to some, but not all, super-population models, depending on whether or
not the covariates are viewed as fixed across samples in the super-population
formulation under consideration.
