We establish the inferential properties of the mean-difference estimator for the average treatment effect in randomized experiments where each unit in a population is randomized to one of two treatments and then units within treatment groups are randomly sampled. The properties of this estimator are well-understood in the experimental design scenario where first units are randomly sampled and then treatment is randomly assigned, but not for the aforementioned scenario where the sampling and treatment assignment stages are reversed. We find that the inferential properties of the mean-difference estimator under this experimental design scenario are identical to those under the more common sample-first-randomize-second design. This finding will bring some clarifications about sampling-based randomized designs for causal inference, particularly for settings where there is a finite super-population. Finally, we explore to what extent pre-treatment measurements can be used to improve upon the mean-difference estimator for this randomize-first-sample-second design. Unfortunately, we find that pre-treatment measurements are often unhelpful in improving the precision of average treatment effect estimators under this design, unless a large number of pre-treatment measurements that are highly associative with the post-treatment measurements can be obtained. We confirm these results using a simulation study based on a real experiment in nanomaterials.

中文翻译：

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潜在结果框架下基于抽样的因果推断随机设计

我们为随机实验中的平均治疗效果建立了均值差估计量的推论性质，其中群体中的每个单位随机分配到两种治疗之一，然后随机抽取治疗组内的单位。该估计量的属性在第一个单元随机抽样然后随机分配处理的实验设计场景中很好理解，但不适用于上述抽样和处理分配阶段颠倒的场景。我们发现，在此实验设计方案下的均值差估计量的推论性质与更常见的样本先随机化第二设计下的推论性质相同。这一发现将为因果推断的基于抽样的随机设计带来一些澄清，特别是对于超级人口有限的环境。最后，我们探讨了在多大程度上可以使用预处理测量来改进这种随机化第一个样本第二个设计的平均差估计量。不幸的是，我们发现在这种设计下，治疗前测量通常无助于提高平均治疗效果估计量的精度，除非可以获得大量与治疗后测量高度相关的治疗前测量。我们使用基于纳米材料真实实验的模拟研究证实了这些结果。我们发现，在这种设计下，治疗前测量通常无助于提高平均治疗效果估计量的精度，除非可以获得大量与治疗后测量高度相关的治疗前测量。我们使用基于纳米材料真实实验的模拟研究证实了这些结果。我们发现，在这种设计下，治疗前测量通常无助于提高平均治疗效果估计量的精度，除非可以获得大量与治疗后测量高度相关的治疗前测量。我们使用基于纳米材料真实实验的模拟研究证实了这些结果。