处理效应

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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.

平均处理效应(Average Treatment Effect, ATE)是在随机实验、政策干预评估和医学实验中用于比较治疗或干预的一种测量方法。平均处理效应测量分配给处理单位和控制单位之间的平均 Mean结果的差异。在随机实验或者实验研究中，平均处理效应可以通过比较样本在处理单元和未处理单元的平均结果进行估计获得。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属性) ，定义时不参考试验设计或估计过程。观察性研究和随机赋值的实验性研究设计可能使以多种方式估计平均处理效应。

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.

”处理”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。处理或结果在评估ATE时相对而言并不重要，也就是说，ATE的计算要求对某些单元进行处理，但不处理其他单元，但治疗的性质(例如药物、奖励性支付、政治广告)与处理的定义和估计无关。

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]

”处理效果”一词是指某一特定处理或干预(例如，给予某种药物)对有关结果变量(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，处理效果被定义为每个个体单元的两个“潜在结果”，如果该个体单元给与处理，就会显现一种结果; 如果该个体单元不给予处理，就会显现出另一个结果。“处理效果”是这两种潜在结果之间的差异。然而，这种个体水平的处理效果是不可观察的，因为个体单位只能接受处理或不接受处理，但不能同时两者。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(经过大量的迭代实验)。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，处理单元之间的平均结果是控制单元的平均结果的反事实。这两个平均值之间的差异是 ATE，这是不可观测到的个人水平的处理效果的中心趋势的估计。如果样本是从总体中随机构成的，那么样本 ATE (缩写为SQTE)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.

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

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]\displaystyle{ y_{0}(i) }[/math] 是个体 [math]\displaystyle{ i }[/math] 没有被处理的结果变量的取值，[math]\displaystyle{ y _ {1}(i) }[/math] 是个体 [math]\displaystyle{ i }[/math] 被处理的结果变量的取值。例如，[math]\displaystyle{ y_{0}(i) }[/math] 是个体 [math]\displaystyle{ i }[/math] 没有被注射研究药物的个体健康状态，[math]\displaystyle{ y_{1}(i) }[/math] 是个体 [math]\displaystyle{ i }[/math] 被注射药物的健康状态。

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]\displaystyle{ i }[/math] 的治疗效果定义为 [math]\displaystyle{ 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]\displaystyle{ \text{ATE} = \frac{1}{N}\sum_i (y_{1}(i)-y_{0}(i)) }[/math]

这里对所有的群体中的N个个体进行了求和。

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.

如果我们能观察到一个大型代表性样本中每个个体的[math]\displaystyle{ y _ {1}(i) }[/math] 和 [math]\displaystyle{ y _ {0}(i) }[/math] ，我们可以简单地通过取样本中 [math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ y _ {1}(i) }[/math] 和个体未接受药物的 [math]\displaystyle{ 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。

Estimation

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ATE。最常见方法是:

自然实验 Natural experiment

双重差分模型 Difference in differences

断点回归设计 Regression discontinuity design

倾向评分匹配 Propensity score matching

工具变量估计 Instrumental variables estimation

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.

考虑一个例子，所有单元都是失业个体，给一些个体给与政策干预（干预组），其余的不给予干预(治疗组) 。需要计算的因果效应是工作监督政策(治疗)对失业期的影响: 平均来说，如果监督个体寻找工作（给与干预），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.

在这个例子中，正ATE意味着就业政策延长了失业期，负ATE表明就业政策缩短了失业期。取值为零的ATE表明，提供就业政策对失业期并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)需要统计推断。

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 并不表明治疗对任何特定个体是有益的或者有害的。因此，平均治疗效果忽略了治疗效果分布。即使平均效应是正的，群体的部门个体也可能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.

一些研究人员称治疗效果“异质性”，如果治疗对不同个体的影响是不同的。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，或者对生活在不同地区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.

看待异质治疗效果的一种方法是将研究数据进行分组(例如，男性和女性，或者按地区) ，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，以子组内的成员为条件。

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，可能没有足够的数据来正确判断在子组上的影响。

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.

有一些利用随机森林检测异质处理效果相关工作。

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