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The expression "treatment effect" refers to the causal effect of a given treatment or intervention (for example, the administering of a drug) on an outcome variable of interest (for example, the health of the patient). In the [[Rubin causal model|Neyman-Rubin "potential outcomes framework"]] of [[causality]] a treatment effect is defined for each individual unit in terms of two "potential outcomes." Each unit has one outcome that would manifest if the unit were exposed to the treatment and another outcome that would manifest if the unit were exposed to the control. The "treatment effect" is the difference between these two potential outcomes. However, this individual-level treatment effect is unobservable because individual units can only receive the treatment or the control, but not both. [[Random assignment]] to treatment ensures that units assigned to the treatment and units assigned to the control are identical (over a large number of iterations of the experiment). Indeed, units in both groups have identical [[Probability distribution|distributions]] of [[covariate]]s and potential outcomes. Thus the average outcome among the treatment units serves as a [[Counterfactual conditional|counterfactual]] for the average outcome among the control units. The differences between these two averages is the ATE, which is an estimate of the [[central tendency]] of the distribution of unobservable individual-level treatment effects.<ref>{{cite journal |last=Holland |first=Paul W. |year=1986 |title=Statistics and Causal Inference |journal=[[Journal of the American Statistical Association|J. Amer. Statist. Assoc.]] |volume=81 |issue=396 |pages=945–960 |jstor=2289064 |doi=10.1080/01621459.1986.10478354}}</ref> If a sample is randomly constituted from a population, the sample ATE (abbreviated SATE) is also an estimate of the population ATE (abbreviated PATE).<ref>{{cite journal |last=Imai |first=Kosuke |first2=Gary |last2=King |first3=Elizabeth A. |last3=Stuart |year=2008 |title=Misunderstandings Between Experimentalists and Observationalists About Causal Inference |journal=[[Journal of the Royal Statistical Society, Series A|J. R. Stat. Soc. Ser. A]] |volume=171 |issue=2 |pages=481–502 |doi=10.1111/j.1467-985X.2007.00527.x |url=http://nrs.harvard.edu/urn-3:HUL.InstRepos:4142695 }}</ref>

The expression "treatment effect" refers to the causal effect of a given treatment or intervention (for example, the administering of a drug) on an outcome variable of interest (for example, the health of the patient). In the [[Rubin causal model|Neyman-Rubin "potential outcomes framework"]] of [[causality]] a treatment effect is defined for each individual unit in terms of two "potential outcomes." Each unit has one outcome that would manifest if the unit were exposed to the treatment and another outcome that would manifest if the unit were exposed to the control. The "treatment effect" is the difference between these two potential outcomes. However, this individual-level treatment effect is unobservable because individual units can only receive the treatment or the control, but not both. [[Random assignment]] to treatment ensures that units assigned to the treatment and units assigned to the control are identical (over a large number of iterations of the experiment). Indeed, units in both groups have identical [[Probability distribution|distributions]] of [[covariate]]s and potential outcomes. Thus the average outcome among the treatment units serves as a [[Counterfactual conditional|counterfactual]] for the average outcome among the control units. The differences between these two averages is the ATE, which is an estimate of the [[central tendency]] of the distribution of unobservable individual-level treatment effects.<ref>{{cite journal |last=Holland |first=Paul W. |year=1986 |title=Statistics and Causal Inference |journal=[[Journal of the American Statistical Association|J. Amer. Statist. Assoc.]] |volume=81 |issue=396 |pages=945–960 |jstor=2289064 |doi=10.1080/01621459.1986.10478354}}</ref> If a sample is randomly constituted from a population, the sample ATE (abbreviated SATE) is also an estimate of the population ATE (abbreviated PATE).<ref>{{cite journal |last=Imai |first=Kosuke |first2=Gary |last2=King |first3=Elizabeth A. |last3=Stuart |year=2008 |title=Misunderstandings Between Experimentalists and Observationalists About Causal Inference |journal=[[Journal of the Royal Statistical Society, Series A|J. R. Stat. Soc. Ser. A]] |volume=171 |issue=2 |pages=481–502 |doi=10.1111/j.1467-985X.2007.00527.x |url=http://nrs.harvard.edu/urn-3:HUL.InstRepos:4142695 }}</ref>

”治疗效果”一词是指某一特定治疗或干预(例如，给予某种药物)对有关结果变量(例如，病人的健康)的因果影响。在因果关系的 Neyman-Rubin“潜在结果框架”中，治疗效果被定义为每个单元的两个“潜在结果”每个单位都有一个结果，如果该单位暴露于治疗中，就会显现; 如果该单位暴露于控制中，就会显现另一个结果。“治疗效果”是这两种潜在结果之间的差异。然而，这种个体水平的治疗效果是不可观察的，因为个体单位只能接受治疗或控制，但不能同时接受两者。随机分配给处理确保分配给处理的单元和分配给控制的单元是相同的(经过大量的实验迭代)。事实上，两组中的单位在协变量和潜在结果上的分布是相同的。因此，治疗单位之间的平均结果与控制单位之间的平均结果相反。这两个平均值之间的差异是 ATE，这是一个估计的中心趋势分布的不可观测的个人水平的治疗效果。如果样本是从总体中随机构成的，那么样本 ATE (缩写为 sat)也是总体 ATE (缩写为 PATE)的估计值。

”治疗效果”一词是指某一特定治疗或干预(例如，给予某种药物)对有关结果变量(例如，病人的健康)的因果影响。在因果关系的 Neyman-Rubin“潜在结果框架”中，治疗效果被定义为每个单元的两个“潜在结果”每个单位都有一个结果，如果该单位暴露于治疗中，就会显现; 如果该单位暴露于控制中，就会显现另一个结果。“治疗效果”是这两种潜在结果之间的差异。然而，这种个体水平的治疗效果是不可观察的，因为个体单位只能接受治疗或控制，但不能同时接受两者。随机分配给处理确保分配给处理的单元和分配给控制的单元是相同的(经过大量的实验迭代)。事实上，两组中的单位在协变量和潜在结果上的分布是相同的。因此，治疗单位之间的平均结果与控制单位之间的平均结果相反。这两个平均值之间的差异是 ATE，这是一个估计的中心趋势分布的不可观测的个人水平的治疗效果。如果样本是从总体中随机构成的，那么样本 ATE (缩写为 sat)也是总体 ATE (缩写为 PATE)的估计值。

While an [[experiment]] ensures, in [[Law of large numbers|expectation]], that potential outcomes (and all covariates) are equivalently distributed in the treatment and control groups, this is not the case in an [[observational study]]. In an observational study, units are not assigned to treatment and control randomly, so their assignment to treatment may depend on unobserved or unobservable factors. Observed factors can be statistically controlled (e.g., through [[regression analysis|regression]] or [[Matching (statistics)|matching]]), but any estimate of the ATE could be [[confounding|confounded]] by unobservable factors that influenced which units received the treatment versus the control.

While an [[experiment]] ensures, in [[Law of large numbers|expectation]], that potential outcomes (and all covariates) are equivalently distributed in the treatment and control groups, this is not the case in an [[observational study]]. In an observational study, units are not assigned to treatment and control randomly, so their assignment to treatment may depend on unobserved or unobservable factors. Observed factors can be statistically controlled (e.g., through [[regression analysis|regression]] or [[Matching (statistics)|matching]]), but any estimate of the ATE could be [[confounding|confounded]] by unobservable factors that influenced which units received the treatment versus the control.

虽然实验确保了潜在的结果(以及所有的协变量)在治疗组和对照组中是等价分布的，但是在观察性研究的情况并非如此。在观察性研究中，治疗单位并不是随机分配和控制的，因此治疗单位的分配可能取决于未观测或不可观测的因素。观察到的因素可以在统计学上加以控制(例如，通过回归或匹配) ，但是任何关于自动测试的估计都可能被不可观察的因素混淆，这些因素影响了哪些单位接受了治疗，哪些单位。

虽然实验确保了潜在的结果(以及所有的协变量)在治疗组和对照组中是等价分布的，但是在观察性研究的情况并非如此。在观察性研究中，治疗单位并不是随机分配和控制的，因此治疗单位的分配可能取决于未观测或不可观测的因素。观察到的因素可以在统计学上加以控制(例如，通过回归或匹配) ，但是任何关于自动测试的估计都可能被不可观察的因素混淆，这些因素影响了哪些单位接受了治疗，哪些单位。

== Formal definition ==

== Formal definition ==