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匹配

2021-06-28T14:28:08Z

PengWu：过匹配那儿不太熟悉，需进一步审阅

{{#seo:
|keywords=结构因果模型,因果科学,因果之梯
|description='通过在观察研究或准实验研究中比较已处理和未处理的单元，以评估处理的效果
}}
'''匹配(Matching)是'''在'''观察研究(Observational Study)'''或'''准实验研究(Quasi-experiment)'''（即'''处理(Treatment)'''是非随机分配的）中，通过比较已接受处理和未接受处理的个体(unit)，以评估处理效应的一种统计技术。匹配的目标是，对每个处理组的个体，找到一个(或多个)具有相似可观察特征的控制组(即未接受处理组)的个体，以评估处理效应。匹配方法通过将处理组个体与相似的控制组个体进行匹配，可以减少混杂因素带来的偏差，使得处理组与控制组的结果(outcome)可直接比较，从而估计出处理效应。<ref>{{cite journal | doi=10.2307/2529684 | last=Rubin | first=Donald B. | title=Matching to Remove Bias in Observational Studies | journal=Biometrics | volume=29 | issue=1 | year=1973 | pages=159–183 | jstor=2529684}}</ref><ref>{{cite journal | title=On Stratification, Grouping and Matching | last=Anderson | first=Dallas W. |author2=Kish, Leslie |author3=Cornell, Richard G. | journal=Scandinavian Journal of Statistics | volume=7 | issue=2 | year=1980 | pages=61–66 | jstor=4615774}}</ref><ref>{{cite journal | doi=10.2307/2530417 | title=Matching in Epidemiologic Studies: Validity and Efficiency Considerations | last=Kupper | first=Lawrence L. |author2=Karon, John M. |author3=Kleinbaum, David G. |author4=Morgenstern, Hal |author5= Lewis, Donald K. | journal=Biometrics | volume=37 | issue=2 | year=1981 | pages=271–291 | jstor=2530417 | pmid=7272415| citeseerx=10.1.1.154.1197 }}</ref>''' 倾向性得分匹配(Propensity Score Matching)'''，一种早期的匹配技术，是作为'''鲁宾因果模型(Rubin Causal Model''')'''<ref name="Rosenbaum Rubin">{{cite journal |last1=Rosenbaum |first1=Paul R. |last2=Rubin |first2=Donald B. |title=The Central Role of the Propensity Score in Observational Studies for Causal Effects |journal=Biometrika |year=1983 |volume=70 |issue=1 |pages=41–55 |doi=10.1093/biomet/70.1.41 |doi-access=free }}</ref>'''的一部分发展起来的，但已被证明会增加模型依赖性和偏差、降低估计效率和检验功效(power)，与其他匹配方法相比不再推荐使用。<ref>{{Cite journal|last1=King|first1=Gary|last2=Nielsen|first2=Richard|date=October 2019|title=Why Propensity Scores Should Not Be Used for Matching|url=https://www.cambridge.org/core/product/identifier/S1047198719000111/type/journal_article|journal=Political Analysis|language=en|volume=27|issue=4|pages=435–454|doi=10.1017/pan.2019.11|issn=1047-1987|doi-access=free}}</ref>

匹配由'''唐纳德•鲁宾(Donald Rubin)'''<ref name="Rosenbaum Rubin" />推动，在经济学中主要受到LaLonde<ref>{{cite journal | last = LaLonde | first = Robert J. | title = Evaluating the Econometric Evaluations of Training Programs with Experimental Data | journal = American Economic Review | volume = 76 | issue = 4 |year = 1986 | pages = 604–620 | jstor=1806062 }}</ref>的批评。LaLonde在一个实验中比较了处理效应估计和运用匹配方法产生的可比估计，表明匹配方法是有偏的。Dehejia和Wahba重新评估了LaLonde的批评，并指出匹配是一个很好的解决方案。<ref>{{cite journal | title = Causal Effects in Nonexperimental Studies: Reevaluating the Evaluation of Training Programs |first1 = R. H. |last1=Dehejia |first2 = S. |last2=Wahba |journal=Journal of the American Statistical Association |year=1999 |volume=94 |issue=448 |pages=1053–1062 |doi=10.1080/01621459.1999.10473858 |url = http://www.nber.org/papers/w6586.pdf }}</ref>政治学<ref>{{cite journal |last1=Arceneaux |first1=Kevin |first2=Alan S. |last2=Gerber |first3=Donald P. |last3=Green |year=2006 |title=Comparing Experimental and Matching Methods Using a Large-Scale Field Experiment on Voter Mobilization |journal=Political Analysis |volume=14 |issue=1 |pages=37–62 |doi=10.1093/pan/mpj001 }}</ref>和社会学期刊<ref>{{cite journal |last1=Arceneaux |first1=Kevin |first2=Alan S. |last2=Gerber |first3=Donald P. |last3=Green |year=2010 |title=A Cautionary Note on the Use of Matching to Estimate Causal Effects: An Empirical Example Comparing Matching Estimates to an Experimental Benchmark |journal=Sociological Methods & Research |volume=39 |issue=2 |pages=256–282 |doi=10.1177/0049124110378098}}</ref>上也提出了类似的批评。
== 分析 ==

当感兴趣的结果是二元变量时，分析匹配后的数据的最常用工具是条件Logistic回归模型，因为它可以处理'''任意多个层，连续或二元处理变量（或自变量）'''，并且可以控制协变量。在特定情况下，一些常用的检验方法可直接使用，如：'''配对差异检验(paired difference test)'''、 McNemar 检验和 Cochran-Mantel-Haenzel 检验等。

当感兴趣的结果是连续的，通过对'''平均处理效应(Average Treatment Effect)'''进行估计。

匹配也可用于在进行其他技术分析（例如回归分析）之前“预处理”样本。<ref>{{cite journal |last1=Ho |first1=Daniel E. |first2=Kosuke |last2=Imai |first3=Gary |last3=King |first4=Elizabeth A. |last4=Stuart |year=2007 |title=Matching as Nonparametric Preprocessing for Reducing Model Dependence in Parametric Causal Inference |journal=Political Analysis |volume=15 |issue=3 |pages=199–236 |doi=10.1093/pan/mpl013 |doi-access=free }}</ref>

== 过匹配 ==
过匹配是对表面是中介变量、实际上是暴露的结果进行匹配。如果中介变量本身是分层的，则很可能引致一种暴露与疾病的令人费解的关系。<ref name=marsh/>过匹配因此导致统计偏误。<ref name=marsh>{{cite journal |title=Removal of radiation dose response effects: an example of over-matching |last1=Marsh |first1=J. L. |last2=Hutton |first2=J. L. |last3=Binks |first3=K. |year=2002 |journal=British Medical Journal |volume=325 |issue=7359 |pages=327–330 |pmid=12169512 |doi=10.1136/bmj.325.7359.327 |pmc=1123834}}</ref>

例如，在估计体外受精（IVF）后的围产期死亡率和出生体重时，按妊娠期和/或多胎数来匹配对照组就是过度匹配，因为IVF本身会增加早产和多胎的风险。<ref>{{cite journal |title=The danger of overmatching in studies of the perinatal mortality and birthweight of infants born after assisted conception |last1=Gissler |first1=M. |last2=Hemminki |first2=E. |journal=Eur J Obstet Gynecol Reprod Biol |year=1996 |volume=69 |issue=2 |pages=73–75 |pmid=8902436 |doi=10.1016/0301-2115(95)02517-0}}</ref>

它可以被看作是一个降低研究外部效度的抽样偏误，因为相比一般人群，对照组在暴露方面变得更类似于病例。

== 另见 ==
* [[倾向得分匹配]]

==参考文献 ==
{{reflist|30em}}

== 进一步阅读 ==

*{{cite book |last1=Angrist |first1=Joshua D. |last2=Pischke |first2=Jörn-Steffen |chapter=Regression Meets Matching |title=Mostly Harmless Econometrics: An Empiricist's Companion |publisher=Princeton University Press |year=2009 |isbn=978-0-691-12034-8 |pages=69–80 }}

==编者推荐==
===集智课程===
====[https://campus.swarma.org/course/1858 计量经济学因果分析工具在快手中的应用]====
在产品迭代和公司决策中，我们常常关心A如何影响B，在不方便使用A/B实验的场景下，我们可以用因果分析的方法，结合观测数据来回答这个问题。主讲人会介绍在快手常用的因果分析计量经济学方法（包括工具变量法、匹配法、双重差分法、合成控制法等），因果分析和机器学习结合的前沿方法（矩阵补全法、基于决策树/随机森林的异质性的因果效应估计等），以及这些方法如何与业务实践相结合。

 
===集智文章===
*[https://swarma.org/?p=25679 Physics Reports最新综述：解决物理、生物、经济中的稳定匹配]

----
本中文词条由[[用户: Frombtotop|Frombtotop]]翻译，[[用户:薄荷|薄荷]]编辑，欢迎在讨论页面留言。

'''本词条内容源自wikipedia及公开资料，遵守 CC3.0协议。'''

[[Category: 偏误]]
[[Category: 实验设计]]
[[Category:医学统计]]
[[Category: 观察性研究]]
[[Category:抽样技术]]

匹配

2021-06-28T14:01:18Z

PengWu：需进一步修改

{{#seo:
|keywords=结构因果模型,因果科学,因果之梯
|description='通过在观察研究或准实验研究中比较已处理和未处理的单元，以评估处理的效果
}}
'''匹配(Matching)是'''在'''观察研究(Observational Study)'''或'''准实验研究(Quasi-experiment)'''（即'''处理(Treatment)'''是非随机分配的）中，通过比较已接受处理和未接受处理的个体(unit)，以评估处理效应的一种统计技术。匹配的目标是，对每个处理组的个体，找到一个(或多个)具有相似可观察特征的控制组(即未接受处理组)的个体，以评估处理效应。通过将处理组个体与相似的控制组个体进行匹配，匹配后的处理组个体与控制组个体的结果可直接比较，从而可以减少混杂效应带来的偏差，评估处理效应。<ref>{{cite journal | doi=10.2307/2529684 | last=Rubin | first=Donald B. | title=Matching to Remove Bias in Observational Studies | journal=Biometrics | volume=29 | issue=1 | year=1973 | pages=159–183 | jstor=2529684}}</ref><ref>{{cite journal | title=On Stratification, Grouping and Matching | last=Anderson | first=Dallas W. |author2=Kish, Leslie |author3=Cornell, Richard G. | journal=Scandinavian Journal of Statistics | volume=7 | issue=2 | year=1980 | pages=61–66 | jstor=4615774}}</ref><ref>{{cite journal | doi=10.2307/2530417 | title=Matching in Epidemiologic Studies: Validity and Efficiency Considerations | last=Kupper | first=Lawrence L. |author2=Karon, John M. |author3=Kleinbaum, David G. |author4=Morgenstern, Hal |author5= Lewis, Donald K. | journal=Biometrics | volume=37 | issue=2 | year=1981 | pages=271–291 | jstor=2530417 | pmid=7272415| citeseerx=10.1.1.154.1197 }}</ref>''' 倾向性得分匹配(Propensity Score Matching)'''，一种早期的匹配技术，是作为'''鲁宾因果模型(Rubin Causal Model''')'''<ref name="Rosenbaum Rubin">{{cite journal |last1=Rosenbaum |first1=Paul R. |last2=Rubin |first2=Donald B. |title=The Central Role of the Propensity Score in Observational Studies for Causal Effects |journal=Biometrika |year=1983 |volume=70 |issue=1 |pages=41–55 |doi=10.1093/biomet/70.1.41 |doi-access=free }}</ref>'''的一部分发展起来的。它已被证明会增加模型依赖性、偏差、降低估计的效率和检验的功效('''power) '''，与其他匹配方法相比不再推荐使用。<ref>{{Cite journal|last1=King|first1=Gary|last2=Nielsen|first2=Richard|date=October 2019|title=Why Propensity Scores Should Not Be Used for Matching|url=https://www.cambridge.org/core/product/identifier/S1047198719000111/type/journal_article|journal=Political Analysis|language=en|volume=27|issue=4|pages=435–454|doi=10.1017/pan.2019.11|issn=1047-1987|doi-access=free}}</ref>

匹配由'''唐纳德•鲁宾(Donald Rubin)'''<ref name="Rosenbaum Rubin" />推动，在经济学中主要受到LaLonde<ref>{{cite journal | last = LaLonde | first = Robert J. | title = Evaluating the Econometric Evaluations of Training Programs with Experimental Data | journal = American Economic Review | volume = 76 | issue = 4 |year = 1986 | pages = 604–620 | jstor=1806062 }}</ref>的批评。LaLonde在一个实验中比较了处理效应估计和运用匹配方法产生的可比估计，表明匹配方法是有偏的。Dehejia和Wahba重新评估了LaLonde的批评，并指出匹配是一个很好的解决方案。<ref>{{cite journal | title = Causal Effects in Nonexperimental Studies: Reevaluating the Evaluation of Training Programs |first1 = R. H. |last1=Dehejia |first2 = S. |last2=Wahba |journal=Journal of the American Statistical Association |year=1999 |volume=94 |issue=448 |pages=1053–1062 |doi=10.1080/01621459.1999.10473858 |url = http://www.nber.org/papers/w6586.pdf }}</ref>政治学<ref>{{cite journal |last1=Arceneaux |first1=Kevin |first2=Alan S. |last2=Gerber |first3=Donald P. |last3=Green |year=2006 |title=Comparing Experimental and Matching Methods Using a Large-Scale Field Experiment on Voter Mobilization |journal=Political Analysis |volume=14 |issue=1 |pages=37–62 |doi=10.1093/pan/mpj001 }}</ref>和社会学期刊<ref>{{cite journal |last1=Arceneaux |first1=Kevin |first2=Alan S. |last2=Gerber |first3=Donald P. |last3=Green |year=2010 |title=A Cautionary Note on the Use of Matching to Estimate Causal Effects: An Empirical Example Comparing Matching Estimates to an Experimental Benchmark |journal=Sociological Methods & Research |volume=39 |issue=2 |pages=256–282 |doi=10.1177/0049124110378098}}</ref>上也提出了类似的批评。
== 分析 ==

当感兴趣的结果是二元变量时，分析匹配数据最常用的工具是条件Logistic回归模型，因为它可以处理'''任意大小的层次和连续或二元处理变量（自变量）strata of arbitrary size and continuous or binary treatments (predictors)''' ，并且可以控制协变量。在特定情况下，可以使用''' 配对差异检验 paired difference test'''、 McNemar 检验和 Cochran-Mantel-Haenzel 检验等更简单的检验。

当感兴趣的结果是连续的，对'''平均处理效应 Average Treatment Effect '''进行估计。

匹配也可用于在通过其他技术分析之前（例如回归分析）“预处理”样本。<ref>{{cite journal |last1=Ho |first1=Daniel E. |first2=Kosuke |last2=Imai |first3=Gary |last3=King |first4=Elizabeth A. |last4=Stuart |year=2007 |title=Matching as Nonparametric Preprocessing for Reducing Model Dependence in Parametric Causal Inference |journal=Political Analysis |volume=15 |issue=3 |pages=199–236 |doi=10.1093/pan/mpl013 |doi-access=free }}</ref>

== 过匹配 ==
过匹配是对表面是中介变量、实际上是暴露的结果进行匹配。如果中介变量本身是分层的，则很可能引致一种暴露与疾病的令人费解的关系。<ref name=marsh/>过匹配因此导致统计偏误。<ref name=marsh>{{cite journal |title=Removal of radiation dose response effects: an example of over-matching |last1=Marsh |first1=J. L. |last2=Hutton |first2=J. L. |last3=Binks |first3=K. |year=2002 |journal=British Medical Journal |volume=325 |issue=7359 |pages=327–330 |pmid=12169512 |doi=10.1136/bmj.325.7359.327 |pmc=1123834}}</ref>

例如，在估计体外受精（IVF）后的围产期死亡率和出生体重时，按妊娠期和/或多胎数来匹配对照组就是过度匹配，因为IVF本身会增加早产和多胎的风险。<ref>{{cite journal |title=The danger of overmatching in studies of the perinatal mortality and birthweight of infants born after assisted conception |last1=Gissler |first1=M. |last2=Hemminki |first2=E. |journal=Eur J Obstet Gynecol Reprod Biol |year=1996 |volume=69 |issue=2 |pages=73–75 |pmid=8902436 |doi=10.1016/0301-2115(95)02517-0}}</ref>

它可以被看作是一个降低研究外部效度的抽样偏误，因为相比一般人群，对照组在暴露方面变得更类似于病例。

== 另见 ==
* [[倾向得分匹配]]

==参考文献 ==
{{reflist|30em}}

== 进一步阅读 ==

*{{cite book |last1=Angrist |first1=Joshua D. |last2=Pischke |first2=Jörn-Steffen |chapter=Regression Meets Matching |title=Mostly Harmless Econometrics: An Empiricist's Companion |publisher=Princeton University Press |year=2009 |isbn=978-0-691-12034-8 |pages=69–80 }}

==编者推荐==
===集智课程===
====[https://campus.swarma.org/course/1858 计量经济学因果分析工具在快手中的应用]====
在产品迭代和公司决策中，我们常常关心A如何影响B，在不方便使用A/B实验的场景下，我们可以用因果分析的方法，结合观测数据来回答这个问题。主讲人会介绍在快手常用的因果分析计量经济学方法（包括工具变量法、匹配法、双重差分法、合成控制法等），因果分析和机器学习结合的前沿方法（矩阵补全法、基于决策树/随机森林的异质性的因果效应估计等），以及这些方法如何与业务实践相结合。

 
===集智文章===
*[https://swarma.org/?p=25679 Physics Reports最新综述：解决物理、生物、经济中的稳定匹配]

----
本中文词条由[[用户: Frombtotop|Frombtotop]]翻译，[[用户:薄荷|薄荷]]编辑，欢迎在讨论页面留言。

'''本词条内容源自wikipedia及公开资料，遵守 CC3.0协议。'''

[[Category: 偏误]]
[[Category: 实验设计]]
[[Category:医学统计]]
[[Category: 观察性研究]]
[[Category:抽样技术]]

可忽略性

2021-06-04T12:20:34Z

PengWu：

{{#seo:
|keywords=实验设计，因果推断
|description=实验设计，因果推断，可忽略性
}}

In [[statistics]], '''ignorability''' is a feature of an [[experiment design]] whereby the method of data collection (and the nature of missing data) do not depend on the missing data. A missing data mechanism such as a treatment assignment or survey sampling strategy is "ignorable" if the missing data matrix, which indicates which variables are observed or missing, is independent of the missing data conditional on the observed data.

在[[统计学]]中，'''可忽略性'''是[[实验设计]]的一种特征，即数据收集方式(以及缺失数据的性质)不依赖于缺失数据。若在给定已观测数据的条件下，表示哪些变量被观测到或缺失的缺失数据指示矩阵与缺失数据独立，则称该数据缺失机制（例如处理分配或抽样调查策略）是“可忽略的”。

This idea is part of the [[Rubin Causal Model|Rubin Causal Inference Model]], developed by [[Donald Rubin]] in collaboration with [[Paul R. Rosenbaum|Paul Rosenbaum]] in the early 1970s. The exact definition differs between their articles in that period. In one of Rubins articles from 1978 Rubin discuss ''ignorable assignment mechanisms'',<ref name="rubin78">{{cite journal |last1=Rubin |first1=Donald |title=Bayesian Inference for Causal Effects: The Role of Randomization |journal=The Annals of Statistics |date=1978 |volume=6 |issue=1 |pages=34–58|doi=10.1214/aos/1176344064 |doi-access=free }}</ref> which can be understood as the way individuals are assigned to treatment groups is irrelevant for the data analysis, given everything that is recorded about that individual. Later, in 1983 <ref>{{cite journal |last1=Rubin |first1=Donald B. |last2=Rosenbaum |first2=Paul R. |title=The Central Role of the Propensity Score in Observational Studies for Causal Effects |journal=Biometrika |date=1983 |volume=70 |issue=1 |pages=41–55 |doi=10.2307/2335942 |jstor=2335942 |doi-access=free }}</ref> Rubin and Rosenbaum rather define ''strongly ignorable treatment assignment'' which is a stronger condition, mathematically formulated as <math>(r_1,r_0) \perp \!\!\!\perp z \mid v ,\quad 0<\operatorname{pr}(z=1)<1 \quad \forall v</math>, where <math>r_t</math> is a potential outcome given treatment <math>t</math>, <math>v</math> is some covariates and <math>z</math> is the actual treatment.

这个想法是20世纪70年代早期[[Donald Rubin]]和[[Paul R. Rosenbaum|Paul Rosenbaum]] 合作提出的[[鲁宾因果推理模型]]的一部分。但那时，他们文章中可忽略性的确切定义不同。1978年鲁宾在一篇文章中讨论了''可忽略的分配机制''<ref name="rubin78">{{cite journal |last1=Rubin |first1=Donald |title=Bayesian Inference for Causal Effects: The Role of Randomization |journal=The Annals of Statistics |date=1978 |volume=6 |issue=1 |pages=34–58|doi=10.1214/aos/1176344064 |doi-access=free }}</ref> ,其可理解为将个体分配到处理组的方式与数据分析无关，因为已经记录了有关该个体的所有信息。后来，在 1983 年，Rubin 和 Rosenbaum 更确切地定义了“处理分配的强可忽略性”<ref>{{cite journal |last1=Rubin |first1=Donald B. |last2=Rosenbaum |first2=Paul R. |title=The Central Role of the Propensity Score in Observational Studies for Causal Effects |journal=Biometrika |date=1983 |volume=70 |issue=1 |pages=41–55 |doi=10.2307/2335942 |jstor=2335942 |doi-access=free }}</ref>，这是一个更强的假设条件，数学上表示为<math>(r_1,r_0) \perp \!\!\!\perp z \mid v ,\quad 0<\operatorname{pr}(z=1)<1 \quad \forall v</math>，其中<math>r_t</math>是给定处理状态 <math>t</math>下的潜在结果，<math>v</math> 是协变量，<math>z</math> 是实际的处理状态。

Pearl [2000] devised a simple graphical criterion, called ''back-door'', that entails ignorability and identifies sets of covariates that achieve this condition.

Pearl [2000]设计了一个简单的图形准则，称为“后门”(back-door) ，它需要可忽略性并能识别满足后门准则条件的协变量集。

Ignorability (better called exogeneity) simply means we can ignore how one ended up in one vs. the other group (‘treated’ Tx = 1, or ‘control’ Tx = 0) when it comes to the potential outcome (say Y). It was also called unconfoundedness, selection on the observables, or no omitted variable bias.<ref>{{cite journal|last1=Yamamoto|first1=Teppei|title=Understanding the Past: Statistical Analysis of Causal Attribution|journal=Journal of Political Science|date=2012|volume=56|issue=1|pages=237–256|doi=10.1111/j.1540-5907.2011.00539.x|hdl=1721.1/85887}}</ref>

可忽略性(称为外生性更好)其简明含义是，当涉及潜在结果（Y）时，我们可以忽略一个人是怎样最终处于一个群体中而非另一个群体中(“处理组”Tx = 1，或“控制组”Tx = 0)。它也被称为无混淆杂性、基于可观测变量的选择或无遗漏变量偏差<ref>{{cite journal|last1=Yamamoto|first1=Teppei|title=Understanding the Past: Statistical Analysis of Causal Attribution|journal=Journal of Political Science|date=2012|volume=56|issue=1|pages=237–256|doi=10.1111/j.1540-5907.2011.00539.x|hdl=1721.1/85887}}</ref>。

Formally it has been written as [Yi1, Yi0] ⊥ Txi, or in words the potential Y outcome of person ''i'' had they been treated or not does not depend on whether they have really been (observable) treated or not. We can ignore in other words how people ended up in one vs. the other condition, and treat their potential outcomes as exchangeable. While this seems thick, it becomes clear if we add subscripts for the ‘realized’ and superscripts for the ‘ideal’ (potential) worlds (notation suggested by [https://www.cambridge.org/core/books/statistical-models-and-causal-inference/7CE8D4957FF6E9615AAAC4128FA8246E David Freedman]; a visual can help here: [https://drive.google.com/open?id=1nLHHH0il225LIy33nRiH3ZfgoX1_-_V9 potential outcomes simplified]).

其数学形式可记为：[Yi1, Yi0] ⊥ Txi ；或者用文字表述为：个体“i”是否接受处理的潜在结果Y并不取决于他们是否真的（可观测到的）接受处理。换句话说，个体最终是通过什么方式处于一种与另一种处理状态我们是可忽略的，并将其潜在结果视为等价可交换的。虽然这看起来很复杂，但如果用下标表示“已实现”的真实处理状态，用上标表示“理想”（潜在）世界的处理状态，就会变得很清楚。(符号的提出可参考[https://www.cambridge.org/core/books/statistical-models-and-causal-inference/7CE8D4957FF6E9615AAAC4128FA8246E David Freedman]; 可视化帮助文档可参考:[https://drive.google.com/open?id=1nLHHH0il225LIy33nRiH3ZfgoX1_-_V9 potential outcomes simplified]).

So: Y11/*Y01 are potential Y outcomes had the person been treated (superscript 1), when in reality they have actually been (Y11, subscript 1), or not (*Y01: the * signals this quantity can never be realized or observed, or is ''fully'' contrary-to-fact or counterfactual, CF).

所以，如果个体接受处理(上角标为 1)，其对应的潜在结果Y为Y11/*Y01，实际上它们可观测的结果是(Y11, 下角标也为 1) ，而不是*Y01。注意：* 表示这个值是无法获取或不可观测的，即''完全与事实相反''或称为反事实（counterfactual, CF)。

Similarly, *Y10/Y00 are potential Y outcomes had the person not been treated (superscript 0), when in reality they have been (*Y10, subscript 1), or not actually (Y00).

同样，如果个体未接受处理(上角标为 0), 其对应的潜在结果Y为*Y10/Y00。在现实中它们是(Y00)，而不是(*Y10。

Only one of each potential outcome (PO) can be realized, the other cannot, for the same assignment to condition, so when we try to estimate treatment effects, we need something to replace the fully contrary-to-fact ones with observables (or estimate them). When ignorability/exogeneity holds, like when people are randomized to be treated or not, we can ‘replace’ *''Y''01 with its observable counterpart Y11, and *Y10 with its observable counterpart ''Y''00, not at the individual level Yi’s, but when it comes to averages like E[''Y''''i''1 – ''Y''''i''0], which is exactly the causal treatment effect (TE) one tries to recover.

对于相同的处理分配条件，每个潜在结果(PO)中只有一个是实际发生可观测的，而另一个不会发生也无法观测，所以当我们尝试估计处理效应时，需要用可观测值(或估计值)来替代无法观测的反事实结果。当可忽略性/外生性成立时，例如个体是否接受处理是随机的，此时可利用已观测的 Y11'替换'*''Y''01，利用已观测的 Y00'替换'*''Y''10，不是个人层面的Yi，而是从平均角度出发，如 E[''Y''''i''1 – ''Y''''i''0 ]，这正是大家尝试获取的因果处理效应（TE）。

Because of the ‘consistency rule’, the potential outcomes are the values actually realized, so we can write Yi0 = Yi00 and Yi1 = Yi11 (“the consistency rule states that an individual’s potential outcome under a hypothetical condition that happened to materialize is precisely the outcome experienced by that individual”,<ref>{{cite journal|last1=Pearl|first1=Judea|title=On the consistency rule in causal inference: axiom, definition, assumption, or theorem?|journal=Epidemiology|date=2010|volume=21|issue=6|pages=872–875|doi=10.1097/EDE.0b013e3181f5d3fd|pmid=20864888}}</ref> p. 872). Hence TE = E[Yi1 – Yi0] = E[Yi11 – Yi00].

由于“一致性准则”，潜在结果可利用实际观测值表示：Yi0 = Yi00 ； Yi1 = Yi11（“一致性准则指出，个体的潜在结果正是该个体的实际产生结果<ref>{{cite journal|last1=Pearl|first1=Judea|title=On the consistency rule in causal inference: axiom, definition, assumption, or theorem?|journal=Epidemiology|date=2010|volume=21|issue=6|pages=872–875|doi=10.1097/EDE.0b013e3181f5d3fd|pmid=20864888}}</ref> p. 872)。所以，TE = E[Yi1 – Yi0] = E[Yi11 – Yi00]。

Now, by simply adding and subtracting the same fully counterfactual quantity *Y10 we get:
E[Yi11 – Yi00] = E[Yi11 –*Y10 +*Y10 - Yi00] = E[Yi11 –*Y10] + E[*Y10 - Yi00] = ATT + {Selection Bias},
where ATT = average treatment effect on the treated <ref>{{cite journal|last1=Imai|first1=Kosuke|title=Misunderstandings between experimentalists and observationalists about causal inference|journal=Journal of the Royal Statistical Society, Series A (Statistics in Society)|date=2006|volume=171|issue=2|pages=481–502|doi=10.1111/j.1467-985X.2007.00527.x|url=http://nrs.harvard.edu/urn-3:HUL.InstRepos:4142695}}</ref> and the second term is the bias introduced when people have the choice to belong to either the ‘treated’ or the ‘control’ group.

现在，我们通过简单的加减相同的完全反事实量 *Y10 得到:

E[Yi11 – Yi00] = E[Yi11 –*Y10 +*Y10 - Yi00] = E[Yi11 –*Y10] + E[*Y10 - Yi00] = ATT + {选择性偏差},

其中，第一项 ATT = 处理组的平均处理效应<ref>{{cite journal|last1=Imai|first1=Kosuke|title=Misunderstandings between experimentalists and observationalists about causal inference|journal=Journal of the Royal Statistical Society, Series A (Statistics in Society)|date=2006|volume=171|issue=2|pages=481–502|doi=10.1111/j.1467-985X.2007.00527.x|url=http://nrs.harvard.edu/urn-3:HUL.InstRepos:4142695}}</ref>，第二项是当个体可选择属于“处理”组或“控制”组而非完全随机分配时引入的偏差。

Ignorability, either plain or conditional on some other variables, implies that such selection bias can be ignored, so one can recover (or estimate) the causal effect.

无论是普通的还是在给定一些变量条件下的可忽略性，都意味着这种选择偏差可以被忽略或消除，因此人们可以得到(或估计)因果效应。

==参考文献 ==

*{{cite paper |title=Bayesian Inference for Causal Effects: The Role of Randomization |first=Rubin|last=Donald |journal= The Annals of Statistics |year=1978 |volume=6(1) |pages=34–58|doi=10.1214/aos/1176344064}}
*{{cite paper |arxiv=1109.2143 |title=The Central Role of the Propensity Score in Observational Studies for Causal Effects |first=Rubin|last=Donald B.|first2=Rosenbaum|last2=Paul R.|journal= Biometrika |year=1983 |volume=70(1) |pages=41–55|doi=10.2307/2335942}}
*{{cite paper |title=The Central Role of the Propensity Score in Observational Studies for Causal Effects |first=Yamamoto|last=Teppei|journal= Journal of Political Science|year=1983 |volume= 56 (1) |pages= 237–256|doi=10.1111/j.1540-5907.2011.00539.x}}
*{{cite paper |title=On the consistency rule in causal inference: axiom, definition, assumption, or theorem? |first=Pearl|last=Judea |journal= Epidemiology|year=2010 |volume= 21 (6) |pages= 872–875|doi=10.1097/EDE.0b013e3181f5d3fd}}
*{{cite paper |title=Misunderstandings between experimentalists and observationalists about causal inference |first=Imai|last=Kosuke |journal= Epidemiology|year=2006 |volume= 171 (2) |pages= 481–502|doi=10.1111/j.1467-985X.2007.00527.x}}

== 推荐阅读 ==

*{{cite book |first=Andrew |last=Gelman |author-link=Andrew Gelman |first2=John B. |last2=Carlin |first3=Hal S. |last3=Stern |first4=Donald B. |last4=Rubin |title=Bayesian Data Analysis |publisher=Chapman & Hall/CRC |location=New York |year=2004 }}

*{{cite book |first=Judea |last=Pearl |title=Causality: Models, Reasoning, and Inference |url=https://archive.org/details/causalitymodelsr0000pear |url-access=registration |publisher=Cambridge University Press |year=2000 }}

*{{cite paper |arxiv=1109.2143 |title=Ignorability in Statistical and Probabilistic Inference |first=Manfred |last=Jaeger |journal=Journal of Artificial Intelligence Research |year=2011 |volume=24 |pages=889–917 |doi=10.1613/jair.1657 |bibcode=2011arXiv1109.2143J }}

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