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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.

为了正式定义 ATE，我们定义了两个潜在的结果: < math > y _ {0}(i) </math > 是个体 < math > i </math > 如果他们没有被处理，< math > y _ {1}(i) </math > 是个体 < math > i </math > 的结果变量的值。例如，如果他们没有被研究中的药物治疗，那么“数学”就是他们的健康状况，而“数学”就是他们被治疗时的健康状况。

为了正式定义 ATE，我们定义了两个潜在的结果: <math>y_{0}(i)</math > 是个体 <math> i </math> 如果他们没有被处理，<math> y _ {1}(i) </math > 是个体 <math> i </math> 的结果变量的值。例如，如果他们没有被研究中的药物治疗，那么“数学”就是他们的健康状况，而“数学”就是他们被治疗时的健康状况。

The treatment effect for individual <math>i</math> is given by <math>y_{1}(i)-y_{0}(i)=\beta(i)</math>. In the general case, there is no reason to expect this effect to be constant across individuals. The average treatment effect is given by  

The treatment effect for individual <math>i</math> is given by <math>y_{1}(i)-y_{0}(i)=\beta(i)</math>. In the general case, there is no reason to expect this effect to be constant across individuals. The average treatment effect is given by  

个体 < math > i </math > 的治疗效果由 < math > y _ {1}(i)-y _ {0}(i) = beta (i) </math > 给出。在一般情况下，没有理由期望这种影响在个体之间是恒定的。平均处理效果由

个体 <math> i </math> 的治疗效果由 <math> y_{1}(i)-y_{0}(i) = beta (i) </math> 给出。在一般情况下，没有理由期望这种影响在个体之间是恒定的。平均处理效果由