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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，但是在观察性研究中，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ATE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处理，哪些单位不接受处理。

== Formal definition ==

== Formal definition ==

In order to define formally the ATE, we define two potential outcomes : <math>y_{0}(i)</math> is the value of the outcome variable for individual <math>i</math> if they are not treated, <math>y_{1}(i)</math> is the value of the outcome variable for individual <math>i</math> if they are treated. For example, <math>y_{0}(i)</math>  is the health status of the individual if they are not administered the drug under study and <math>y_{1}(i)</math> is the health status if they are administered the drug.

In order to define formally the ATE, we define two potential outcomes : <math>y_{0}(i)</math> is the value of the outcome variable for individual <math>i</math> if they are not treated, <math>y_{1}(i)</math> is the value of the outcome variable for individual <math>i</math> if they are treated. For example, <math>y_{0}(i)</math>  is the health status of the individual if they are not administered the drug under study and <math>y_{1}(i)</math> is the health status if they are administered the drug.