# 平均因果效应

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The **average treatment effect** (**ATE**) is a measure used to compare treatments (or interventions) in randomized experiments, evaluation of policy interventions, and medical trials. The ATE measures the difference in mean (average) outcomes between units assigned to the treatment and units assigned to the control. In a randomized trial (i.e., an experimental study), the average treatment effect can be estimated from a sample using a comparison in mean outcomes for treated and untreated units. However, the ATE is generally understood as a causal parameter (i.e., an estimate or property of a population) that a researcher desires to know, defined without reference to the study design or estimation procedure. Both observational studies and experimental study designs with random assignment may enable one to estimate an ATE in a variety of ways.

The average treatment effect (ATE) is a measure used to compare treatments (or interventions) in randomized experiments, evaluation of policy interventions, and medical trials. The ATE measures the difference in mean (average) outcomes between units assigned to the treatment and units assigned to the control. In a randomized trial (i.e., an experimental study), the average treatment effect can be estimated from a sample using a comparison in mean outcomes for treated and untreated units. However, the ATE is generally understood as a causal parameter (i.e., an estimate or property of a population) that a researcher desires to know, defined without reference to the study design or estimation procedure. Both observational studies and experimental study designs with random assignment may enable one to estimate an ATE in a variety of ways.

平均治疗效果(ATE)是在随机试验、政策干预评估和医学试验中用于比较治疗(或干预)的一种方法。ATE 测量分配给治疗的单位和分配给控制的单位之间的平均(平均)结果的差异。在随机试验(即实验研究)中，平均治疗效果可以通过比较治疗单位和未治疗单位的平均结果从样本中估计出来。然而，ATE 通常被理解为研究人员希望知道的一个因果参数(即一个总体的估计或特性) ，定义时不参考研究设计或估计过程。随机分配的观察性研究和实验性研究设计都可以使人们以多种方式估计自动测试设备。

## General definition

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.

”治疗”一词起源于农业和医药领域的早期统计分析，现在更广泛地用于自然和社会科学的其他领域，特别是心理学、政治科学和经济学，例如评价公共政策的影响。治疗或结果的性质相对而言并不重要ーー也就是说，计算治疗能力需要某种治疗适用于某些单位，而不是其他单位，但该治疗的性质(例如药物、奖励性支付、政治广告)与治疗能力的定义和估计无关。

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 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 distributions of covariates and potential outcomes. Thus the average outcome among the treatment units serves as a 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.^{[1]} If a sample is randomly constituted from a population, the sample ATE (abbreviated SATE) is also an estimate of the population ATE (abbreviated PATE).^{[2]}

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 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 distributions of covariates and potential outcomes. Thus the average outcome among the treatment units serves as a 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. If a sample is randomly constituted from a population, the sample ATE (abbreviated SATE) is also an estimate of the population ATE (abbreviated PATE).

”治疗效果”一词是指某一特定治疗或干预(例如，给予某种药物)对某一结果变量(例如，病人的健康)的因果影响。在因果关系的 Neyman-Rubin“潜在结果框架”中，治疗效果被定义为每个单元的两个“潜在结果”每个单位都有一个结果，如果该单位暴露于治疗，这个结果就会显现; 如果该单位暴露于控制，另一个结果就会显现。“治疗效果”是这两种潜在结果之间的差异。然而，这种个体水平的治疗效果是不可观察的，因为个体单位只能接受治疗或控制，但不能同时接受两者。随机分配给处理确保分配给处理的单元和分配给控制的单元是相同的(经过大量的实验迭代)。事实上，两组中的单位在协变量和潜在结果上的分布是相同的。因此，治疗单位之间的平均结果与控制单位之间的平均结果相反。这两个平均值之间的差异是 ATE，这是一个估计的中心趋势的分布不可观测的个人水平的治疗效果。如果样本是从总体中随机构成的，那么样本 ATE (缩写为 sat)也是总体 ATE (缩写为 PATE)的估计值。

While an experiment ensures, in 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 or matching), but any estimate of the ATE could be confounded by unobservable factors that influenced which units received the treatment versus the control.

While an experiment ensures, in 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 or matching), but any estimate of the ATE could be 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，所以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所混淆，这些因素影响了哪些单位接受了治疗，哪些单位接受了对照。

## Formal definition

In order to define formally the ATE, we define two potential outcomes : [math]\displaystyle{ y_{0}(i) }[/math] is the value of the outcome variable for individual [math]\displaystyle{ i }[/math] if they are not treated, [math]\displaystyle{ y_{1}(i) }[/math] is the value of the outcome variable for individual [math]\displaystyle{ i }[/math] if they are treated. For example, [math]\displaystyle{ y_{0}(i) }[/math] is the health status of the individual if they are not administered the drug under study and [math]\displaystyle{ 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]\displaystyle{ y_{0}(i) }[/math] is the value of the outcome variable for individual [math]\displaystyle{ i }[/math] if they are not treated, [math]\displaystyle{ y_{1}(i) }[/math] is the value of the outcome variable for individual [math]\displaystyle{ i }[/math] if they are treated. For example, [math]\displaystyle{ y_{0}(i) }[/math] is the health status of the individual if they are not administered the drug under study and [math]\displaystyle{ 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 < math > y _ {0}(i) </math > 就是他们的健康状况，如果他们被治疗，那么 < math > y _ {1}(i) </math > 就是健康状况。

The treatment effect for individual [math]\displaystyle{ i }[/math] is given by [math]\displaystyle{ 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]\displaystyle{ i }[/math] is given by [math]\displaystyle{ 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

- [math]\displaystyle{ \text{ATE} = \frac{1}{N}\sum_i (y_{1}(i)-y_{0}(i)) }[/math]

[math]\displaystyle{ \text{ATE} = \frac{1}{N}\sum_i (y_{1}(i)-y_{0}(i)) }[/math]

{ n } sum i (y _ {1}(i)-y _ {0}(i)) </math >

where the summation occurs over all [math]\displaystyle{ N }[/math] individuals in the population.

where the summation occurs over all [math]\displaystyle{ N }[/math] individuals in the population.

这里的总和发生在所有的人口中。

If we could observe, for each individual, [math]\displaystyle{ y_{1}(i) }[/math] and [math]\displaystyle{ 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]\displaystyle{ y_{1}(i)-y_{0}(i) }[/math] across the sample. However, we can not observe both [math]\displaystyle{ y_{1}(i) }[/math] and [math]\displaystyle{ 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]\displaystyle{ y_{1}(i) }[/math] for individuals who have received the drug and [math]\displaystyle{ 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]\displaystyle{ y_{1}(i) }[/math] and [math]\displaystyle{ 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]\displaystyle{ y_{1}(i)-y_{0}(i) }[/math] across the sample. However, we can not observe both [math]\displaystyle{ y_{1}(i) }[/math] and [math]\displaystyle{ 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]\displaystyle{ y_{1}(i) }[/math] for individuals who have received the drug and [math]\displaystyle{ 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，对于每个个体，在一个大的有代表性的人口样本中，对于每个个体，< math > y _ {1}(i) </math > 和 < math > y _ {0}(i) </math > ，我们可以简单地通过取样本 < math > y _ {1}(i)-y _ {0}(i) </math > 的平均值来估计 ATE。然而，我们不能同时观察每个个体的数学和数学，因为个体既不能治疗也不能不治疗。例如，在药物的例子中，我们只能观察到已经接受药物的个体和未接受药物的个体的 < math > y _ {1}(i) </math > 。这是科学家在评价治疗效果时面临的主要问题，并引发了大量的估计技术。

## 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:

根据数据及其基本情况，可以使用许多方法来估计 ATE。最常见的是:

## An example

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.

考虑一个例子，其中所有单位都是失业的个人，一些单位经历了政策干预(治疗组) ，而其他单位没有(对照组)。利息的因果效应是找工作监督政策(治疗)对失业期限的影响: 平均来说，如果一个人经历了干预，他的失业期会缩短多少？在这种情况下，ATE 是治疗组和对照组失业时间长度的期望值(平均值)的差异。

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.

一个积极的 ATE，在这个例子中，意味着就业政策延长了失业的时间。一个消极的自动终端设备将表明就业政策缩短了失业的时间。一个等于零的劳动能力估计数表明，就失业时间长短而言，提供这种待遇没有任何好处或不利之处。判断一个 ATE 估计值是否可以区分为零(正的或负的)需要推论统计学。

Because the ATE is an estimate of the average effect of the treatment, a positive or negative ATE does not indicate that any particular individual would benefit or be harmed by the treatment. Thus the average treatment effect neglects the distribution of the treatment effect. Some parts of the population might be worse off with the treatment even if the mean effect is positive.

Because the ATE is an estimate of the average effect of the treatment, a positive or negative ATE does not indicate that any particular individual would benefit or be harmed by the treatment. Thus the average treatment effect neglects the distribution of the treatment effect. Some parts of the population might be worse off with the treatment even if the mean effect is positive.

因为 ATE 是对治疗的平均效果的估计，一个正的或负的 ATE 并不表明任何特定的个人会受益或受到治疗的伤害。因此，平均治疗效果忽略了治疗效果的分布。即使平均效应是正面的，一部分人口可能会因为这种治疗而变得更糟。

## 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.

一些研究人员称治疗效果“异质性”，如果它影响不同的个人(异质性)。例如，上面提到的求职监控政策对男性和女性的影响是不同的，或者对生活在不同州的人的影响是不同的。

One way to look for heterogeneous treatment effects is to divide the study data into subgroups (e.g., men and women, or by state), and see if the average treatment effects are different by subgroup. A per-subgroup ATE is called a "conditional average treatment effect" (CATE), i.e. the ATE conditioned on membership in the subgroup.

One way to look for heterogeneous treatment effects is to divide the study data into subgroups (e.g., men and women, or by state), and see if the average treatment effects are different by subgroup. A per-subgroup ATE is called a "conditional average treatment effect" (CATE), i.e. the ATE conditioned on membership in the subgroup.

寻找异质治疗效果的一种方法是将研究数据分为子组(例如，男性和女性，或按州) ，看看平均治疗效果是否因子组而异。每个子组的 ATE 被称为“条件平均治疗效应”(CATE) ，也就是说，每个子组的 ATE 被称为条件平均治疗效应。自动选举委员会的条件是成员在该小组。

A challenge with this approach is that each subgroup may have substantially less data than the study as a whole, so if the study has been powered to detect the main effects without subgroup analysis, there may not be enough data to properly judge the effects on subgroups.

A challenge with this approach is that each subgroup may have substantially less data than the study as a whole, so if the study has been powered to detect the main effects without subgroup analysis, there may not be enough data to properly judge the effects on subgroups.

这种方法的一个挑战是，每个分组的数据可能比整个研究少得多，所以如果这项研究在没有进行分组分析的情况下就能检测出主要的影响，那么就可能没有足够的数据来正确判断对分组的影响。

There is some work on detecting heterogenous treatment effects using random forests.^{[3]}^{[4]}

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

在利用随机森林检测异质处理效果方面有一些工作。

## References

- ↑ Holland, Paul W. (1986). "Statistics and Causal Inference".
*J. Amer. Statist. Assoc.***81**(396): 945–960. doi:10.1080/01621459.1986.10478354. JSTOR 2289064. - ↑ Imai, Kosuke; King, Gary; Stuart, Elizabeth A. (2008). "Misunderstandings Between Experimentalists and Observationalists About Causal Inference".
*J. R. Stat. Soc. Ser. A*.**171**(2): 481–502. doi:10.1111/j.1467-985X.2007.00527.x. - ↑ https://arxiv.org/abs/1510.04342
- ↑ https://www.markhw.com/blog/causalforestintro

## Further reading

- Wooldridge, Jeffrey M. (2013). "Policy Analysis with Pooled Cross Sections".
*Introductory Econometrics: A Modern Approach*. Mason, OH: Thomson South-Western. pp. 438–443. ISBN 978-1-111-53104-1.

Category:Estimation theory

类别: 参数估测

Category:Medical statistics

类别: 医学统计

Category:Experiments

分类: 实验

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