We consider a potential outcomes model in which interference may be present between any two units but the extent of interference diminishes with spatial distance. The causal estimand is the global average treatment effect, which compares counterfactual outcomes when all units are treated to outcomes when none are. We study a class of designs in which space is partitioned into clusters that are randomized into treatment and control. For each design, we estimate the treatment effect using a Horovitz-Thompson estimator that compares the average outcomes of units with all neighbors treated to units with no neighbors treated, where the neighborhood radius is of the same order as the cluster size dictated by the design. We derive the estimator's rate of convergence as a function of the design and degree of interference and use this to obtain estimator-design pairs in this class that achieve near-optimal rates of convergence under relatively minimal assumptions on interference. We prove that the estimators are asymptotically normal and provide a variance estimator. Finally, we discuss practical implementation of the designs by partitioning space using clustering algorithms.

翻译：我们考虑一个潜在结果模型,其中两个单元之间可能存在干扰,但干扰的程度会降低空间距离。因果估计效应是全球平均处理效果,当所有单位被处理为结果时,将反事实结果与无结果作比较。我们研究空间被分割成集群的一类设计,这些集群被随机地分成为处理和控制组。对于每一种设计,我们使用Horovitz-Thompson估计器来估计处理效果,该模型将单位的平均结果与所有被处理的邻居的单位相比较,而没有邻居被处理的单位,其周边半径与设计所要求的集群大小的顺序相同。我们用该估计的趋同率作为设计和干扰程度的函数,并用这个模型来获取在相对最低的干扰假设下达到近于最佳的趋同率的估测配方。我们证明估量器与不同时正常,并提供差异估测器。最后,我们讨论通过利用组合算法分割空间来实际实施设计。