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'''逆概率加权'''是一种统计技术，用于计算与收集数据的人群不同的伪总体（[[pseudo-population]]）的标准化统计数据。在应用中，抽样人群和目标推断人群（目标人群）不一致的研究设计是很常见的<ref name="refname2" />。可能有一些禁止性因素，如成本、时间或道德方面的考虑，使研究人员无法直接从目标人群中抽样<ref name="refname3" />。解决这个问题的方法是使用另一种设计策略，如分层抽样（[[stratified sampling]]）。如果应用得当，加权可以潜在地提高效率，减少非加权估计的偏差。

'''逆概率加权'''是一种统计技术，用于计算与收集数据的人群不同的伪总体（[[pseudo-population]]）的标准化统计数据。在应用中，抽样人群和目标推断人群（目标人群）不一致的研究设计是很常见的<ref>Robins, JM; Rotnitzky, A; Zhao, LP (1994). "Estimation of regression coefficients when some regressors are not always observed". Journal of the American Statistical Association. 89 (427): 846–866. doi:10.1080/01621459.1994.10476818.</ref>。可能有一些禁止性因素，如成本、时间或道德方面的考虑，使研究人员无法直接从目标人群中抽样<ref></ref>。解决这个问题的方法是使用另一种设计策略，如分层抽样（[[stratified sampling]]）。如果应用得当，加权可以潜在地提高效率，减少非加权估计的偏差。

当数据缺失的受试者不能被纳入主要分析时，逆概率加权也被用来解释缺失的数据<ref name="refname1" />。有了对抽样概率的估计，或该因素在另一次测量中被测量的概率，逆概率加权可以用来提高那些由于数据缺失程度大而代表性不足的受试者的权重。

当数据缺失的受试者不能被纳入主要分析时，逆概率加权也被用来解释缺失的数据<ref></ref>。有了对抽样概率的估计，或该因素在另一次测量中被测量的概率，逆概率加权可以用来提高那些由于数据缺失程度大而代表性不足的受试者的权重。

== 逆概率加权估计量(Inverse Probability Weighted Estimator, IPWE) ==

== 逆概率加权估计量(Inverse Probability Weighted Estimator, IPWE) ==

= 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>).

公式的后面重排有助于揭示基本思想：我们的估计器是基于使用模型的平均预测结果的(即<math>\frac{1}{n}\sum_{i=1}^n\Biggl(\hat{Q}_n(X_i,a)\Biggr)</math>)。然而，那么模型的残差就不会（在完整的治疗组<math>a</math>）大约为0。 我们可以通过增加模型的平均残差（<math>Q</math>）与结果（<math>Y</math>）的真实值的额外项来纠正这种潜在的偏差(即<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>). 因为我们有<math>Y</math>的缺失值，所以我们给予权重，以提高每个残差的相对重要性（这些权重是基于看到每个个体观测值的反倾向性，也就是逆概率）。(参见文献<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>的第10页).

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.: \frac{1}{n}\sum_{i=1}^n\Biggl(\hat{Q}_n(X_i,a)\Biggr)). 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.: \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)). 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 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. link for the paper).

这种估计器的“双重稳健”效益来自这样一个事实，即两个模型中的一个已经被正确指定，估计器是无偏的(即可能是<math>\hat{Q}_n(X_i,a)</math>或<math>\hat{p}_{n}(A_{i}|X_{i})</math>, 或两者都是)。这是因为如果结果模型被很好地指定，那么它的残差将大约为0(不管每个残差将得到多少权重)。如果模型是有偏差的，但是加权模型是很好地指定的，那么偏差将被加权平均的残差很好地估计(并修正)<ref name = "kang2007"/><ref>Kim, Jae Kwang, and David Haziza. "Doubly robust inference with missing data in survey sampling." Statistica Sinica 24.1 (2014): 375-394. [https://lib.dr.iastate.edu/cgi/viewcontent.cgi?article=1110&context=stat_las_pubs link to the paper]</ref><ref>Seaman, Shaun R., and Stijn Vansteelandt. "Introduction to double robust methods for incomplete data." Statistical science: a review journal of the Institute of Mathematical Statistics 33.2 (2018): 184. [https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5935236/ link to the paper]</ref>。

双重稳健估计器的偏差被称为'''二阶偏差'''，它取决于差分<math>\frac{1}{\hat{p}_{n}(A_{i}|X_{i})} - \frac{1}{{p}_{n}(A_{i}|X_{i})}</math>和差分<math>\hat{Q}_n(X_i,a) - Q_n(X_i,a)</math>的乘积。这个特性使我们在样本容量足够大的情况下，通过使用机器学习估计器(而不是参数模型)来降低双重稳健估计器的总体偏差<ref>Hernán, Miguel A., and James M. Robins. "Causal inference." (2010): 2. [https://cdn1.sph.harvard.edu/wp-content/uploads/sites/1268/2021/03/ciwhatif_hernanrobins_30mar21.pdf link to the book] - page 179</ref>。

 

The "doubly robust" benefit of such an estimator comes from the fact that it's sufficient for one of the two models to be correctly specified, for the estimator to be unbiased (either <math>\hat{Q}_n(X_i,a)</math> or <math>\hat{p}_{n}(A_{i}|X_{i})</math>, or both). This is because if the outcome model is well specified then its residuals will be around 0 (regardless of the weights each residual will get). While if the model is biased, but the weighting model is well specified, then the bias will be well estimated (And corrected for) by the weighted average residuals.<ref name = "kang2007"/><ref>Kim, Jae Kwang, and David Haziza. "Doubly robust inference with missing data in survey sampling." Statistica Sinica 24.1 (2014): 375-394. [https://lib.dr.iastate.edu/cgi/viewcontent.cgi?article=1110&context=stat_las_pubs link to the paper]</ref><ref>Seaman, Shaun R., and Stijn Vansteelandt. "Introduction to double robust methods for incomplete data." Statistical science: a review journal of the Institute of Mathematical Statistics 33.2 (2018): 184. [https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5935236/ link to the paper]</ref>

 

The "doubly robust" benefit of such an estimator comes from the fact that it's sufficient for one of the two models to be correctly specified, for the estimator to be unbiased (either \hat{Q}_n(X_i,a) or \hat{p}_{n}(A_{i}|X_{i}), or both). This is because if the outcome model is well specified then its residuals will be around 0 (regardless of the weights each residual will get). While if the model is biased, but the weighting model is well specified, then the bias will be well estimated (And corrected for) by the weighted average residuals.Kim, Jae Kwang, and David Haziza. "Doubly robust inference with missing data in survey sampling." Statistica Sinica 24.1 (2014): 375-394. link to the paperSeaman, Shaun R., and Stijn Vansteelandt. "Introduction to double robust methods for incomplete data." Statistical science: a review journal of the Institute of Mathematical Statistics 33.2 (2018): 184. link to the paper

 

 

The bias of the doubly robust estimators is called a '''second-order bias''', and it depends on the product of the difference <math>\frac{1}{\hat{p}_{n}(A_{i}|X_{i})} - \frac{1}{{p}_{n}(A_{i}|X_{i})}</math> and the difference <math>\hat{Q}_n(X_i,a) - Q_n(X_i,a)</math>. This property allows us, when having a "large enough" sample size, to lower the overall bias of doubly robust estimators by using [[machine learning]] estimators (instead of parametric models).<ref>Hernán, Miguel A., and James M. Robins. "Causal inference." (2010): 2. [https://cdn1.sph.harvard.edu/wp-content/uploads/sites/1268/2021/03/ciwhatif_hernanrobins_30mar21.pdf link to the book] - page 179</ref>

 

 

==参见==

==参见==

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