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Originating from early statistical analysis in the fields of agriculture and medicine, the term "treatment" is now applied, more generally, to other fields of natural and social science, especially [[psychology]], [[political science]], and [[economics]] such as, for example, the evaluation of the impact of public policies. The nature of a treatment or outcome is relatively unimportant in the estimation of the ATE—that is to say, calculation of the ATE requires that a treatment be applied to some units and not others, but the nature of that treatment (e.g., a pharmaceutical, an incentive payment, a political advertisement) is irrelevant to the definition and estimation of the ATE.

Originating from early statistical analysis in the fields of agriculture and medicine, the term "treatment" is now applied, more generally, to other fields of natural and social science, especially [[psychology]], [[political science]], and [[economics]] such as, for example, the evaluation of the impact of public policies. The nature of a treatment or outcome is relatively unimportant in the estimation of the ATE—that is to say, calculation of the ATE requires that a treatment be applied to some units and not others, but the nature of that treatment (e.g., a pharmaceutical, an incentive payment, a political advertisement) is irrelevant to the definition and estimation of the ATE.

'''<font color="#ff8000">处理(Treatment)</font>'''一词起源于农业和医药领域的早期统计分析，现在被更广泛地用于自然科学和社会科学的其他领域，尤其是心理学、政治科学和经济学，例如评价公共政策的影响。试验中的处理或'''<font color="#ff8000">结果 (Outcome)</font>'''的具体内容在评估平均处理效应时相对而言并不重要，也就是说，平均处理效应估算要求对某些个体进行处理，但不处理其他个体，但处理具体内容(例如药物、奖励性支付、政治广告)与平均处理效应的定义和估计无关。

'''<font color="#ff8000">处理 (Treatment)</font>'''一词起源于农业和医药领域的早期统计分析，现在被更广泛地用于自然科学和社会科学的其他领域，尤其是心理学、政治科学和经济学，例如评价公共政策的影响。试验中的处理或'''<font color="#ff8000">结果 (Outcome)</font>'''的具体内容在评估平均处理效应时相对而言并不重要，也就是说，平均处理效应估算要求对某些个体进行处理，但不处理其他个体，但处理具体内容(例如药物、奖励性支付、政治广告)与平均处理效应的定义和估计无关。

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>

”处理效应”一词是指某一特定处理或干预(如给予某种药物)对结果变量(如病人的康复)的'''<font color="#ff8000">因果影响 (Causal Effect)</font>'''。在因果关系的 Neyman-Rubin“潜在结果框架”中，处理效应被定义为每个独立个体的两个“潜在结果”，如果该个体给与处理，就会显现一种结果; 如果该个体不给予处理，就会显现出另一种结果。“处理效果”是这两种潜在结果之间的差异。然而，这种个体水平的处理效果是不可观察的，因为每个独立个体只能接受处理或不接受处理，但不能同时接受两者。随机分配需要确保给处理组的个体和对照组的个体在大量迭代实验上是服从同分布。事实上，两组中的个体在协变量和潜在结果上的分布是相同的。因此，处理个体之间的平均结果是控制个体的平均结果的反事实。这两个平均值之间的差异是平均处理效应 ，这是不可观测到的个体层面的处理效果的中心趋势的估计。如果样本是从总体中随机构成，那么'''<font color="#ff8000">样本平均处理效应 (Sample Average Treatment Effect， SATE)</font>'''也是'''<font color="#ff8000">总体平均处理效应 (Population Average Treatment Effect，PATE)</font>'''的估计值。

“处理效应”一词是指某一特定处理或干预(如给予某种药物)对结果变量(如病人的康复)的'''<font color="#ff8000">因果影响 (Causal Effect)</font>'''。在因果关系的 Neyman-Rubin“潜在结果框架”中，处理效应被定义为每个独立个体的两个“潜在结果”，如果该个体给与处理，就会显现一种结果; 如果该个体不给予处理，就会显现出另一种结果。“处理效果”是这两种潜在结果之间的差异。然而，这种个体水平的处理效果是不可观察的，因为每个独立个体只能接受处理或不接受处理，但不能同时接受两者。随机分配需要确保给处理组的个体和对照组的个体在大量迭代实验上是服从同分布。事实上，两组中的个体在协变量和潜在结果上的分布是相同的。因此，处理个体之间的平均结果是控制个体的平均结果的反事实。这两个平均值之间的差异是平均处理效应 ，这是不可观测到的个体层面的处理效果的中心趋势的估计。如果样本是从总体中随机构成，那么'''<font color="#ff8000">样本平均处理效应 (Sample Average Treatment Effect， SATE)</font>'''也是'''<font color="#ff8000">总体平均处理效应 (Population Average Treatment Effect，PATE)</font>'''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结果以及所有协变量在处理组和对照组中的等价分布，但是在观察性研究中，情况并非如此。在观察性研究中，处理组个体并不是随机分配和受人为控制（这句话翻译错了），因此处理单位的分配可能取决于未观测到或不可观测的因素。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) ，但是任何关于平均处理效应的估计都可能与不可观察因素混淆，这些因素影响了哪些单位接受了处理，哪些单位没有接受处理。

虽然实验确保了潜在结果以及所有协变量在处理组和对照组中的等价分布，但是在观察性研究中，情况并非如此。在观察性研究中，处理组和对照组个体并不服从随机分布，因此处理个体可能取决于未观测到或不可观测的因素。观察到的因素可以在统计学上加以控制 (如通过回归或匹配) ，但是任何关于平均处理效应的估计都可能与不可观察因素混淆，这些因素影响了哪些单位接受了处理，哪些单位没有接受处理。

== 形式化定义 Formal definition ==

== 形式化定义 Formal definition ==