可忽略性

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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.

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个特征，即数据收集的方法(以及缺失数据的性质)不依赖于缺失的数据。缺失数据机制，例如处理分配或调查抽样策略是”可忽略的”，如果缺失数据矩阵表明哪些变量是观察到的或缺失的，它独立于缺失数据条件的观察到的数据。

This idea is part of the Rubin Causal Inference Model, developed by Donald Rubin in collaboration with 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,^[1] 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 ^[2] Rubin and Rosenbaum rather define strongly ignorable treatment assignment which is a stronger condition, mathematically formulated as [math]\displaystyle{ (r_1,r_0) \perp \!\!\!\perp z \mid v ,\quad 0\lt \operatorname{pr}(z=1)\lt 1 \quad \forall v }[/math], where [math]\displaystyle{ r_t }[/math] is a potential outcome given treatment [math]\displaystyle{ t }[/math], [math]\displaystyle{ v }[/math] is some covariates and [math]\displaystyle{ z }[/math] is the actual treatment.

This idea is part of the Rubin Causal Inference Model, developed by Donald Rubin in collaboration with 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, 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 Rubin and Rosenbaum rather define strongly ignorable treatment assignment which is a stronger condition, mathematically formulated as [math]\displaystyle{ (r_1,r_0) \perp \!\!\!\perp z \mid v ,\quad 0\lt \operatorname{pr}(z=1)\lt 1 \quad \forall v }[/math], where [math]\displaystyle{ r_t }[/math] is a potential outcome given treatment [math]\displaystyle{ t }[/math], [math]\displaystyle{ v }[/math] is some covariates and [math]\displaystyle{ z }[/math] is the actual treatment.

这个想法是鲁宾因果推理模型的一部分，由唐纳德 · 鲁宾和保罗 · 罗森鲍姆在20世纪70年代早期合作开发。在那个时期，他们的文章的确切定义是不同的。鲁宾在1978年的一篇文章中讨论了可忽略的分配机制，这种机制可以理解为个体被分配到治疗小组的方式，这与数据分析无关，因为关于个体的所有记录都是记录在案的。后来，在1983年鲁宾和罗森鲍姆更愿意定义强可忽略的治疗分配，这是一个更强的条件，数学公式为 < math > (r _ 1，r _ 0) perp！\！\！对于所有的病人来说，数学是一个潜在的治疗结果，数学是一些协变量，数学是实际的治疗方法。

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

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标准，称为“后门”(back-door) ，它包含可忽略性并识别实现此条件的协变量集。

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.^[3]

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.

可忽略性(更好地称为外生性)简单地意味着，当涉及到潜在结果时，我们可以忽略一个人如何最终处于一个群体中而非另一个群体中(“处理过的”Tx = 1，或“控制过的”Tx = 0)。它也被称为不混淆，选择的可观察的，或没有遗漏的变量偏见。

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 David Freedman; a visual can help here: potential outcomes simplified).

在形式上，它被写成[ y i 1，y i 0]⊥ Tx i ，或者用文字来说，人们的潜在 y 结果我已经治疗或不治疗不取决于他们是否真的被(可观察的)治疗。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(由 https://www.cambridge.org/core/books/statistical-models-and-causal-inference/7ce8d4957ff6e9615aaac4128fa8246e David Freedman 提出的符号; 一个视觉可以在这里帮助: [ https://drive.google.com/open?id=1nlhhh0il225liy33nrih3zfgox1_-_v9潜在结果的简化])。

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: y 1 1 /* y 0 < 1 是潜在的 y 结果，如果人被处理(上标 1 ) ，而实际上它们是(y 1 1 ，下标 1 >) ，或不是(* y 0 < > < 1 </> : * 这个数量是不可能实现或观察到的，或完全与事实或事实相反，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，如果未经治疗(上标 < 上标 > 0 </>) ，实际上可能发生(* y < 上标 > 1 < > < 上标 > 0 ，下标 < 上标 > 1 < 0 ，< 上标 > 1 ，< 上标 < 上标 > 1 > > > ，或者实际上不发生(y < 上标 > 0 < 0 > < 0 </>)。

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’ *Y₀¹ with its observable counterpart Y₁¹, and *Y₁⁰ with its observable counterpart Y₀⁰, not at the individual level Y_i’s, but when it comes to averages like E[Y_i¹ – Y_i⁰], which is exactly the causal treatment effect (TE) one tries to recover.

对于相同的条件分配，每个潜在结果(PO)中只有一个可以实现，而另一个则不能，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对应的 y 1 1 替换 y 0 1 ,而 y 1 0 与其对应的 y 0 0 ，不在个体水平 y i >’s，而在 e [ y i 1 >-y i 0 > 这样的平均值时，正是因果治疗效应(TE)试图恢复的结果。

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”,^[4] p. 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”, p. 872). Hence TE = E[Y_i¹ – Y_i⁰] = E[Y_i1¹ – Y_i0⁰].

由于“一致性规则”，潜在的结果是实际实现的价值，因此我们可以写 y i 0 < 0 和 y i 1 = y < 1 (“一致性规则指出，假设个体在某种条件下实现的潜在结果恰恰是该个体所经历的结果”，p. 872)。因此，TE = e [ y i 1 -y i 0 ] = e [ y i 1 -y i0 0 ]。

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

现在，通过简单的加减相同的完全反事实量 * y 1 0 我们得到:

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 i 1 1 -y > i 0 0 ] = e [ y i 1 1 -* y 1 0 + y 1 0 > 0 > > y > i 0 > > 0 > 0 > 0 > > > 0 > > </0 </> > ] = e[ y i 1 1 -* y 1 0 ]+ e [ * y 1 0 -y i0 0 ] = ATT + {选择偏差} ,

where ATT = average treatment effect on the treated ^[5] 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 and the second term is the bias introduced when people have the choice to belong to either the ‘treated’ or the ‘control’ group.

其中 ATT = 治疗组的平均治疗效果，第二项是当人们可以选择属于治疗组或对照组时引入的偏倚。

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.

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。

References

↑ Rubin, Donald (1978). "Bayesian Inference for Causal Effects: The Role of Randomization". The Annals of Statistics. 6 (1): 34–58. doi:10.1214/aos/1176344064.
↑ Rubin, Donald B.; Rosenbaum, Paul R. (1983). "The Central Role of the Propensity Score in Observational Studies for Causal Effects". Biometrika. 70 (1): 41–55. doi:10.2307/2335942. JSTOR 2335942.
↑ Yamamoto, Teppei (2012). "Understanding the Past: Statistical Analysis of Causal Attribution". Journal of Political Science. 56 (1): 237–256. doi:10.1111/j.1540-5907.2011.00539.x. hdl:1721.1/85887.
↑ Pearl, Judea (2010). "On the consistency rule in causal inference: axiom, definition, assumption, or theorem?". Epidemiology. 21 (6): 872–875. doi:10.1097/EDE.0b013e3181f5d3fd. PMID 20864888.
↑ Imai, Kosuke (2006). "Misunderstandings between experimentalists and observationalists about causal inference". Journal of the Royal Statistical Society, Series A (Statistics in Society). 171 (2): 481–502. doi:10.1111/j.1467-985X.2007.00527.x.

可忽略性

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