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第26行： − 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> 没有被处理的结果变量的取值，<math> y _ {1}(i) </math> 是个体 <math> i </math> 被处理的结果变量的取值。例如，<math>y_{0}(i)</math > 是个体 <math> i </math> 没有被注射研究药物的个体健康状态，<math>y_{1}(i)</math > 是个体 <math> i </math> 被注射药物的健康状态。+ − +
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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.
为了正式定义平均处理效应，我们定义了两个潜在的结果: <math>y_{0}(i)</math > 是个体 <math> i </math> 没有被处理的结果变量的取值，<math> y _ {1}(i) </math> 是个体 <math> i </math> 被处理的结果变量的取值。例如，<math>y_{0}(i)</math > 是个体 <math> i </math> 没有被注射研究药物的健康状态，<math>y_{1}(i)</math > 是个体 <math> i </math> 被注射药物的健康状态。
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> 的治疗效果定义为 <math> y_{1}(i)-y_{0}(i) = beta (i) </math> 。在一般情况下，这种治疗影响在个体之间是不一样的。平均处理效果定义为
个体 <math> i </math> 的处理效应定义为 <math> y_{1}(i)-y_{0}(i) = \beta (i) </math> 。在一般情况下，这种治疗影响在个体之间是不一样的。平均处理效果定义为
:<math>\text{ATE} = \frac{1}{N}\sum_i (y_{1}(i)-y_{0}(i))</math>
:<math>\text{ATE} = \frac{1}{N}\sum_i (y_{1}(i)-y_{0}(i))</math>