更改

'''Inverse probability weighting''' is a statistical technique for calculating statistics standardized to a pseudo-population different from that in which the data was collected. Study designs with a disparate sampling population and population of target inference (target population) are common in application. There may be prohibitive factors barring researchers from directly sampling from the target population such as cost, time, or ethical concerns. A solution to this problem is to use an alternate design strategy, e.g. stratified sampling. Weighting, when correctly applied, can potentially improve the efficiency and reduce the bias of unweighted estimators.

'''Inverse probability weighting''' is a statistical technique for calculating statistics standardized to a pseudo-population different from that in which the data was collected. Study designs with a disparate sampling population and population of target inference (target population) are common in application<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>. There may be prohibitive factors barring researchers from directly sampling from the target population such as cost, time, or ethical concerns<ref>Breslow, NE; Lumley, T; et al. (2009). "Using the Whole Cohort in the Analysis of Case-Cohort Data". Am J Epidemiol. 169 (11): 1398–1405. doi:10.1093/aje/kwp055. PMC 2768499. <nowiki>PMID 19357328</nowiki>.</ref>. A solution to this problem is to use an alternate design strategy, e.g. stratified sampling. Weighting, when correctly applied, can potentially improve the efficiency and reduce the bias of unweighted estimators.

One very early weighted estimator is the Horvitz–Thompson estimator of the mean. When the sampling probability is known, from which the sampling population is drawn from the target population, then the inverse of this probability is used to weight the observations. This approach has been generalized to many aspects of statistics under various frameworks. In particular, there are weighted likelihoods, weighted estimating equations, and weighted probability densities from which a majority of statistics are derived. These applications codified the theory of other statistics and estimators such as marginal structural models, the standardized mortality ratio, and the EM algorithm for coarsened or aggregate data.

One very early weighted estimator is the Horvitz–Thompson estimator of the mean<ref>Horvitz, D. G.; Thompson, D. J. (1952). "A generalization of sampling without replacement from a finite universe". ''Journal of the American Statistical Association''. '''47''' (260): 663–685. doi:10.1080/01621459.1952.10483446</ref>. When the sampling probability is known, from which the sampling population is drawn from the target population, then the inverse of this probability is used to weight the observations. This approach has been generalized to many aspects of statistics under various frameworks. In particular, there are weighted likelihoods, weighted estimating equations, and weighted probability densities from which a majority of statistics are derived. These applications codified the theory of other statistics and estimators such as marginal structural models, the standardized mortality ratio, and the EM algorithm for coarsened or aggregate data.

Inverse probability weighting is also used to account for missing data when subjects with missing data cannot be included in the primary analysis. With an estimate of the sampling probability, or the probability that the factor would be measured in another measurement, inverse probability weighting can be used to inflate the weight for subjects who are under-represented due to a large degree of missing data.

Inverse probability weighting is also used to account for missing data when subjects with missing data cannot be included in the primary analysis<ref>Hernan, MA; Robins, JM (2006). "Estimating Causal Effects From Epidemiological Data". ''J Epidemiol Community Health''. '''60''' (7): 578–596. CiteSeerX 10.1.1.157.9366. doi:10.1136/jech.2004.029496. PMC 2652882. <nowiki>PMID 16790829</nowiki></ref>. With an estimate of the sampling probability, or the probability that the factor would be measured in another measurement, inverse probability weighting can be used to inflate the weight for subjects who are under-represented due to a large degree of missing data.

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

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

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)<ref>Cao, Weihua; Tsiatis, Anastasios A.; Davidian, Marie (2009). "Improving efficiency and robustness of the doubly robust estimator for a population mean with incomplete data". ''Biometrika''. '''96''' (3): 723–734. doi:10.1093/biomet/asp033. ISSN 0006-3444. PMC 2798744. <nowiki>PMID 20161511</nowiki></ref>.

=== Estimator Formula ===

=== Estimator Formula ===

With the following notations:

With the following notations:

# is an indicator function if subject i is part of treatment group a (or not).

# [[文件:Wiki-AIPWE-2.png]]is an indicator function if subject i is part of treatment group a (or not).

# Construct regression estimator  to predict outcome  based on covariates  and treatment , for some subject i. For example, using ordinary least squares regression.

# Construct regression estimator  [[文件:Wiki-AIPWE-3.png]]to predict outcome Y based on covariates X and treatment A , for some subject i. For example, using ordinary least squares regression.

# Construct propensity (probability) estimate . For example, using logistic regression.

# Construct propensity (probability) estimate [[文件:Wiki-AIPWE-4.png]]. For example, using logistic regression.

# Combine in AIPWE to obtain  

# Combine in AIPWE to obtain[[文件:Wiki-AIPWE-5.png]]

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

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

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.: ). 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.: ). 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 ).

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.: [[文件:Wiki-AIPWE-6.png]]). 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.:[[文件:Wiki-AIPWE-7.png]] ). 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=":0">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</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  or , 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.

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  [[文件:Wiki-AIPWE-3.png]]or [[文件:Wiki-AIPWE-4.png]], 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=":0" /><ref>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 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. link to the paper</ref>

The bias of the doubly robust estimators is called a '''second-order bias''', and it depends on the product of the difference  and the difference . 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).

The bias of the doubly robust estimators is called a '''second-order bias''', and it depends on the product of the difference [[文件:Wiki-AIPWE-8.png]] and the difference [[文件:Wiki-AIPWE-9.png]]. 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. link to the book - page 170</ref>

== 参考文献 ==

== 参考文献 ==