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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。

为了形式化定义平均处理效应，我们定义了两个潜在的结果: <math>y_{0}(i)</math > 是个体 <math> i </math> 没有被处理时的结果变量的取值，<math> y _ {1}(i) </math> 是个体 <math> i </math> 被处理时的结果变量的取值。例如，<math>y_{0}(i)</math > 是个体 <math> i </math> 没有被注射研究药物时的健康状态，<math>y_{1}(i)</math > 是个体 <math> i </math> 被注射药物时的健康状态。

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  

:<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>

 

If we could observe, for each individual, <math>y_{1}(i)</math> and <math>y_{0}(i)</math> among a large representative sample of the population, we could estimate the ATE simply by taking the average value of <math>y_{1}(i)-y_{0}(i)</math> across the sample. However, we can not observe both <math>y_{1}(i)</math> and <math>y_{0}(i)</math> for each individual since an individual cannot be both treated and not treated. For example, in the drug example, we can only observe  <math>y_{1}(i)</math> for individuals who have received the drug and <math>y_{0}(i)</math> for those who did not receive it. This is the main problem faced by scientists in the evaluation of treatment effects and has triggered a large body of estimation techniques.

If we could observe, for each individual, <math>y_{1}(i)</math> and <math>y_{0}(i)</math> among a large representative sample of the population, we could estimate the ATE simply by taking the average value of <math>y_{1}(i)-y_{0}(i)</math> across the sample. However, we can not observe both <math>y_{1}(i)</math> and <math>y_{0}(i)</math> for each individual since an individual cannot be both treated and not treated. For example, in the drug example, we can only observe  <math>y_{1}(i)</math> for individuals who have received the drug and <math>y_{0}(i)</math> for those who did not receive it. This is the main problem faced by scientists in the evaluation of treatment effects and has triggered a large body of estimation techniques.

如果我们能观察到一个大型代表性样本中每个个体的<math> y _ {1}(i) </math> 和 <math> y _ {0}(i) </math> ，我们可以简单地通过取样本中 <math> y _ {1}(i)-y _ {0}(i) </math> 的平均值来估计平均治疗效果。然而，我们不能同时观察每个个体的<math> y _ {1}(i)、y _ {0}(i) </math>，因为每个个体不能同时被处理和不被处理。例如，在药物例子中，我们只能观察到个体接受过药物治疗的<math> y _ {1}(i) </math> 和个体未接受药物的 <math> y _ {0}(i) </math> 。这是研究学者在评估治疗效果时面临的主要问题，并因此引发了大量估计技术的研究。

如果我们能观察到一个大型代表性样本中每个个体的<math> y _ {1}(i) </math> 和 <math> y _ {0}(i) </math> ，我们可以简单地通过取样本中 <math> y _ {1}(i)-y _ {0}(i) </math> 的平均值来估计平均处理效应。然而，我们不能同时观察每个个体的<math> y _ {1}(i)、y _ {0}(i) </math>，因为每个个体不能同时被处理和不被处理。例如，在药物例子中，我们只能观察到个体接受过药物治疗的<math> y _ {1}(i) </math> 和个体未接受药物的 <math> y _ {0}(i) </math> 。这是研究者们在评估治疗效果时面临的主要问题，并因此引发了大量的与估计方法相关的研究。

== 估计 Estimation ==

== 估计 Estimation ==

Depending on the data and its underlying circumstances, many methods can be used to estimate the ATE. The most common ones are:

Depending on the data and its underlying circumstances, many methods can be used to estimate the ATE. The most common ones are:

根据数据及其潜在环境，可以使用许多方法来估计平均处理效应<math> \text{ATE} </math>。最常见方法是:

根据数据及其潜在环境的不同，我们可以使用许多方法来估计平均处理效应<math> \text{ATE} </math>。最常见方法包括:

* 自然实验 Natural Experiment  

* 自然实验 Natural Experiment  

Consider an example where all units are unemployed individuals, and some experience a policy intervention (the treatment group), while others do not (the control group). The causal effect of interest is the impact a job search monitoring policy (the treatment) has on the length of an unemployment spell: On average, how much shorter would one's unemployment be if they experienced the intervention? The ATE, in this case, is the difference in expected values (means) of the treatment and control groups' length of unemployment.

Consider an example where all units are unemployed individuals, and some experience a policy intervention (the treatment group), while others do not (the control group). The causal effect of interest is the impact a job search monitoring policy (the treatment) has on the length of an unemployment spell: On average, how much shorter would one's unemployment be if they experienced the intervention? The ATE, in this case, is the difference in expected values (means) of the treatment and control groups' length of unemployment.

考虑一个失业群体，对一些个体给与政策干预（处理组），其余的不做任何处理(控制组) 。现需要计算求职监控政策(干预)对失业期长短的影响: 平均来说，如果对个体进行求职监控（给与干预），失业期会缩短多少？在这种情况下，平均处理效应是处理组和对照组的失业时间长度的期望值(平均值)差异。

考虑一个失业群体，对其中一些个体给与政策干预（处理组），其余的不做任何处理（控制组）。现需要计算求职监控政策（干预）对失业期长短的影响: 平均来说，如果对个体进行求职监控（给与干预），失业期会缩短多少？在选择一种干预这种情况下，平均处理效应是处理组和对照组的失业时间长度的期望值（平均值）差异。

A positive ATE, in this example, would suggest that the job policy increased the length of unemployment. A negative ATE would suggest that the job policy decreased the length of unemployment. An ATE estimate equal to zero would suggest that there was no advantage or disadvantage to providing the treatment in terms of the length of unemployment. Determining whether an ATE estimate is distinguishable from zero (either positively or negatively) requires statistical inference.

A positive ATE, in this example, would suggest that the job policy increased the length of unemployment. A negative ATE would suggest that the job policy decreased the length of unemployment. An ATE estimate equal to zero would suggest that there was no advantage or disadvantage to providing the treatment in terms of the length of unemployment. Determining whether an ATE estimate is distinguishable from zero (either positively or negatively) requires statistical inference.

== Heterogenous treatment effects ==

== （异质处理效应 Heterogenous treatment effects） ==

Some researchers call a treatment effect "heterogenous" if it affects different individuals differently (heterogeneously).  For example, perhaps the above treatment of a job search monitoring policy affected men and women differently, or people who live in different states differently.

Some researchers call a treatment effect "heterogenous" if it affects different individuals differently (heterogeneously).  For example, perhaps the above treatment of a job search monitoring policy affected men and women differently, or people who live in different states differently.

一种异质处理效应的研究方法是将研究数据进行分组(例如，按照男、女性别，或者按区域) ，比较平均治疗效果在子组内的效应差异。每个子组的平均处理效应被称为“条件平均治疗效应”(Cnditional Average Treatment Effect,CATE) ，也就是说，每个子组的 平均处理效应被称为条件平均治疗效应，以子组内的成员为条件。

一种异质处理效应的研究方法是将研究数据进行分组(例如，按照男、女性别，或者按区域) ，比较平均处理效果在子组内的效应差异。每个子组的平均处理效应被称为'''<font color="#ff8000">“条件平均治疗效应”(Conditional Average Treatment Effect，CATE)</font>''' ，也就是说，每个子组的平均处理效应被称为条件平均治疗效应，以子组内的分类方式为条件。

There is some work on detecting heterogenous treatment effects using random forests.

There is some work on detecting heterogenous treatment effects using random forests.