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The inverse probability weighting estimator can be used to demonstrate causality when the researcher cannot conduct a controlled experiment but has observed data to model. Because it is assumed that the treatment is not randomly assigned, the goal is to estimate the counterfactual or potential outcome if all subjects in population were assigned either treatment.

The inverse probability weighting estimator can be used to demonstrate causality when the researcher cannot conduct a controlled experiment but has observed data to model. Because it is assumed that the treatment is not randomly assigned, the goal is to estimate the counterfactual or potential outcome if all subjects in population were assigned either treatment.

Suppose observed data are <math>\{\bigl(X_i,A_i,Y_i\bigr)\}^{n}_{i=1}</math> drawn [[Independent and identically distributed random variables|i.i.d (independent and identically distributed)]] from unknown distribution P, where

Suppose observed data are <math>\{\bigl(X_i,A_i,Y_i\bigr)\}^{n}_{i=1}</math> drawn [[Independent and identically distributed random variables|i.i.d (independent and identically distributed)]] from unknown distribution P, where

= = = = = = = # mu { a } = mathbb { e } frac { mathbf {1}{ a = a } y }{ p (a | x)}其中 p (a | x) = frac { p (a = a，x = x)}{ p (x = x)}}{ p (x = x)}} # construct hat { p }{ n }(a | x)或 p (a | x)使用任意模型(通常是 Logit模型模型) # 帽子{ mu } ^ { IPWE } _ { a，n } = sum ^ { n } _ { i = 1} frac { y { i }1 _ { a _ { i } = a }{ n hat { p } _ { n }(a _ { i } | x { i })计算每个治疗组的平均值,方差分析和统计 t 检验可以用来判断治疗效果的差异，并确定治疗效果的统计显著性。

= = = = = = = # mu { a } = mathbb { e } frac { mathbf {1}{ a = a } y }{ p (a | x)}其中 p (a | x) = frac { p (a = a，x = x)}{ p (x = x)}}{ p (x = x)}} # construct hat { p }{ n }(a | x)或 p (a | x)使用任意模型(通常是 Logit模型模型) # 帽子{ mu } ^ { IPWE } _ { a，n } = sum ^ { n } _ { i = 1} frac { y { i }1 _ { a _ { i } = a }{ n hat { p } _ { n }(a _ { i } | x { i })计算每个治疗组的平均值,方差分析和统计 t 检验可以用来判断治疗效果的差异，并确定治疗效果的统计显著性。

==== Assumptions ====

==== 假设 ====

# Consistency: <math>Y = Y^{*}(A)</math>

# Consistency: <math>Y = Y^{*}(A)</math>

# No unmeasured confounders: <math>\{Y^{*}(0), Y^{*}(1)\} \perp A|X</math>  

# No unmeasured confounders: <math>\{Y^{*}(0), Y^{*}(1)\} \perp A|X</math>  

= = = = 极限 = = = = = 反概率加权估计量(IPWE)在估计倾向较小时可能不稳定。如果任一处理分配的概率很小，那么 Logit模型模型可能在尾部附近变得不稳定，导致 IPWE 也变得不稳定。

= = = = 极限 = = = = = 反概率加权估计量(IPWE)在估计倾向较小时可能不稳定。如果任一处理分配的概率很小，那么 Logit模型模型可能在尾部附近变得不稳定，导致 IPWE 也变得不稳定。

== Augmented Inverse Probability Weighted Estimator (AIPWE) ==

== 增广逆概率加权估计器 ==

An alternative estimator is the augmented inverse probability weighted estimator (AIPWE) combines both the properties of the regression based estimator and the inverse probability weighted estimator. It is therefore a 'doubly robust' method in that it only requires either the propensity or outcome model to be correctly specified but not both. This method augments the IPWE to reduce variability and improve estimate efficiency. This model holds the same assumptions as the Inverse Probability Weighted Estimator (IPWE).<ref>{{Cite journal|last1=Cao|first1=Weihua|last2=Tsiatis|first2=Anastasios A.|last3=Davidian|first3=Marie|author3-link= Marie Davidian |year=2009|title=Improving efficiency and robustness of the doubly robust estimator for a population mean with incomplete data|journal=Biometrika|volume=96|issue=3|pages=723–734|doi=10.1093/biomet/asp033|issn=0006-3444|pmc=2798744|pmid=20161511}}</ref>

An alternative estimator is the augmented inverse probability weighted estimator (AIPWE) combines both the properties of the regression based estimator and the inverse probability weighted estimator. It is therefore a 'doubly robust' method in that it only requires either the propensity or outcome model to be correctly specified but not both. This method augments the IPWE to reduce variability and improve estimate efficiency. This model holds the same assumptions as the Inverse Probability Weighted Estimator (IPWE).<ref>{{Cite journal|last1=Cao|first1=Weihua|last2=Tsiatis|first2=Anastasios A.|last3=Davidian|first3=Marie|author3-link= Marie Davidian |year=2009|title=Improving efficiency and robustness of the doubly robust estimator for a population mean with incomplete data|journal=Biometrika|volume=96|issue=3|pages=723–734|doi=10.1093/biomet/asp033|issn=0006-3444|pmc=2798744|pmid=20161511}}</ref>

An alternative estimator is the augmented inverse probability weighted estimator (AIPWE) combines both the properties of the regression based estimator and the inverse probability weighted estimator. It is therefore a 'doubly robust' method in that it only requires either the propensity or outcome model to be correctly specified but not both. This method augments the IPWE to reduce variability and improve estimate efficiency. This model holds the same assumptions as the Inverse Probability Weighted Estimator (IPWE).

An alternative estimator is the augmented inverse probability weighted estimator (AIPWE) combines both the properties of the regression based estimator and the inverse probability weighted estimator. It is therefore a 'doubly robust' method in that it only requires either the propensity or outcome model to be correctly specified but not both. This method augments the IPWE to reduce variability and improve estimate efficiency. This model holds the same assumptions as the Inverse Probability Weighted Estimator (IPWE).

= = 增广逆概率加权估计(AIPWE) = = 另一种估计是增广逆概率加权估计(AIPWE) ，它综合了基于回归的估计和逆概率加权估计的性质。因此，这是一个双重稳健的方法，因为它只需要正确指定倾向或结果模型，而不是两者都要求。这种方法增强了 IPWE，减少了变异性，提高了估计效率。该模型与逆概率加权估计(IPWE)具有相同的假设条件。

另一种估计是增广逆概率加权估计(Augmented Inverse Probability Weighted Estimator，AIPWE) ，它综合了基于回归的估计和逆概率加权估计的性质。因此，这是一个双重稳健的方法，因为它只需要正确指定倾向或结果模型，而不是两者都要求。这种方法增强了 IPWE，减少了变异性，提高了估计效率。该模型与逆概率加权估计(IPWE)具有相同的假设条件。

=== Estimator Formula ===

=== Estimator Formula ===

=== Interpretation and "double robustness" ===

=== Interpretation and "double robustness" ===

= = = 解释和“双重稳健性”= = =  

= 解释和“双重稳健性” =  

The later rearrangement of the formula helps reveal the underlying idea: our estimator is based on the average predicted outcome using the model (i.e.: <math>\frac{1}{n}\sum_{i=1}^n\Biggl(\hat{Q}_n(X_i,a)\Biggr)</math>). However, if the model is biased, then the residuals of the model will not be (in the full treatment group a) around 0. We can correct this potential bias by adding the extra term of the average residuals of the model (Q) from the true value of the outcome (Y) (i.e.: <math>\frac{1}{n}\sum_{i=1}^n\frac{1_{A_{i}=a}}{\hat{p}_{n}(A_{i}|X_{i})}\Biggl(Y_{i} - \hat{Q}_n(X_i,a)\Biggr)</math>). Because we have missing values of Y, we give weights to inflate the relative importance of each residual (these weights are based on the inverse propensity, a.k.a. probability, of seeing each subject observations) (see page 10 in <ref name = "kang2007">Kang, Joseph DY, and Joseph L. Schafer. "Demystifying double robustness: A comparison of alternative strategies for estimating a population mean from incomplete data." Statistical science 22.4 (2007): 523-539. [https://projecteuclid.org/journals/statistical-science/volume-22/issue-4/Demystifying-Double-Robustness--A-Comparison-of-Alternative-Strategies-for/10.1214/07-STS227.full link for the paper]</ref>).

The later rearrangement of the formula helps reveal the underlying idea: our estimator is based on the average predicted outcome using the model (i.e.: <math>\frac{1}{n}\sum_{i=1}^n\Biggl(\hat{Q}_n(X_i,a)\Biggr)</math>). However, if the model is biased, then the residuals of the model will not be (in the full treatment group a) around 0. We can correct this potential bias by adding the extra term of the average residuals of the model (Q) from the true value of the outcome (Y) (i.e.: <math>\frac{1}{n}\sum_{i=1}^n\frac{1_{A_{i}=a}}{\hat{p}_{n}(A_{i}|X_{i})}\Biggl(Y_{i} - \hat{Q}_n(X_i,a)\Biggr)</math>). Because we have missing values of Y, we give weights to inflate the relative importance of each residual (these weights are based on the inverse propensity, a.k.a. probability, of seeing each subject observations) (see page 10 in <ref name = "kang2007">Kang, Joseph DY, and Joseph L. Schafer. "Demystifying double robustness: A comparison of alternative strategies for estimating a population mean from incomplete data." Statistical science 22.4 (2007): 523-539. [https://projecteuclid.org/journals/statistical-science/volume-22/issue-4/Demystifying-Double-Robustness--A-Comparison-of-Alternative-Strategies-for/10.1214/07-STS227.full link for the paper]</ref>).

==See also==

==See also==

* [[Propensity score matching]]

* 倾向评分匹配（[[Propensity score matching]]）

= 参考文献 =

 

 

 

 

= = 参考文献 = =

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[[Category:Survey methodology]]

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