On the centrosymmetry of treatment effect estimators for stepped wedge and related cluster randomized trial designs

Stepped wedge and other longitudinal cluster randomized trials are being conducted with increasing frequency, and in recent years several new treatment effect estimators for the analysis of these trials have been proposed. In this paper we discuss a key property that treatment effect estimators should possess, related to the symmetry of the designs under reversing the order of the cluster-period cells. We define various forms of this symmetry: symmetry of designs, regression models, and treatment effect estimators. We prove that when designs and regression models possess these symmetries, the most efficient treatment effect estimator will have a similar symmetry. We apply our results to a recently-described estimator for stepped wedge designs.