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Formally it has been written as [Y_i1, Y_i0] ⊥ Tx_i, 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]).

Formally it has been written as [Y_i1, Y_i0] ⊥ Tx_i, 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[Y_i1, Y_i0] ⊥ Tx_i ，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，但如果我们为“已实现”的真实处理结果添加下标，为“理想”（潜在）世界的处理结果添加上标，就会变得很清楚。

其数学形式可记为[Y_i1, Y_i0] ⊥ Tx_i ，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]; 可视化帮助可参考:[https://drive.google.com/open?id=1nLHHH0il225LIy33nRiH3ZfgoX1_-_V9 potential outcomes simplified]).

换句话说，我们可以忽略人们是如何在一种情况下和另一种情况下结束生命的，而把他们的潜在结果看作是可以交换的。虽然这看起来很厚，但是如果我们为“理想”(潜在)世界添加“已实现”的下标和上标就变得很清楚了(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).

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:Y₁¹/*Y₀¹是潜在结果Y，如果个体被处理(上角标为 ¹) ，那么实际上它们是(Y₁¹, 下角标也为 ₁) ，而不是(*Y₀¹:: * 表示这个值是无法实现或不可观测的，即''完全''与事实相反或称为反事实（counterfactual, CF）)。

所以:如果个体被处理(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：* 表示这个值是无法实现或不可观测的，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).

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，*Y₁⁰/Y₀⁰是个体未被处理 (上角标为 ⁰)的潜在结果Y，当现实中它们是(*Y₁⁰, 下角标为 ₁),或实际上不是 (Y₀⁰).

同样，当现实中它们是(*Y₁⁰, 下角标为 ₁),或实际上不是 (Y₀⁰)时，表个体未被处理 (上角标为 ⁰)，对应的潜在结果Y为*Y₁⁰/Y₀⁰。

 

对于相同的条件分配，每个潜在结果(PO)中只有一个是实际发生的，而另一个不会发生，因此当我们试图估计治疗效果时，我们需要用可观测值(或估计值)来代替完全相反的结果。当可忽略性/外生性成立时，如人们被随机分配治疗与否，我们可以用可观察到的*''Y''₀¹’替换‘Y₁¹,而 *Y₁⁰ 与其对应的''Y''₀⁰，不在 Y_i的个体水平，而是 E[''Y''_''i''¹ – ''Y''_''i''⁰]平均层面而言在 ，这样的平均值时，正是因果治疗效应(TE)试图恢复的结果。

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Y_i，而是从平均角度出发，如 E[''Y''_''i''¹ – ''Y''_''i''⁰]，这正是人们试图计算的因果处理效应（TE）。

Because of the ‘consistency rule’, the potential outcomes are the values actually realized, so we can write Y_i⁰ = Y_i0⁰ and Y_i¹ = Y_i1¹ (“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.&nbsp;872). Hence TE = E[Y_i¹ – Y_i⁰] = E[Y_i1¹ – Y_i0⁰].

Because of the ‘consistency rule’, the potential outcomes are the values actually realized, so we can write Y_i⁰ = Y_i0⁰ and Y_i¹ = Y_i1¹ (“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.&nbsp;872). Hence TE = E[Y_i¹ – Y_i⁰] = E[Y_i1¹ – Y_i0⁰].

Now, by simply adding and subtracting the same fully counterfactual quantity *Y₁⁰ we get:

Now, by simply adding and subtracting the same fully counterfactual quantity *Y₁⁰ we get:

E[Y_i1¹ – Y_i0⁰] = E[Y_i1¹ –*Y₁⁰  +*Y₁⁰ - Y_i0⁰] = E[Y_i1¹ –*Y₁⁰] + E[*Y₁⁰ - Y_i0⁰] = ATT + {Selection Bias},

E[Y_i1¹ – Y_i0⁰] = E[Y_i1¹ –*Y₁⁰  +*Y₁⁰ - Y_i0⁰] = E[Y_i1¹ –*Y₁⁰] + E[*Y₁⁰ - Y_i0⁰] = 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.  

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.  

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.

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.

 

无论是普通的还是条件性的可忽略性，都意味着这种选择偏差可以被忽略，因此人们可以得到(或估计)因果效应。

可忽略性，无论是普通的还是条件性的，都意味着这种选择偏差可以被忽略，因此人们可以恢复(或估计)因果效应。

 

 

 

 

 

 

类别: 因果推理

类别: 因果推理