Inverse probability weighting
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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 In statistics, stratified sampling is a method of sampling from a population which can be partitioned into subpopulations. In statistical surveys, when subpopulations within an overall population vary, it could be advantageous to sample each s ...
. 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 In statistics, the Horvitz–Thompson estimator, named after Daniel G. Horvitz and Donovan J. Thompson, is a method for estimating the total and mean of a pseudo-population in a stratified sample. Inverse probability weighting is applied to ac ...
of the mean. When the
sampling probability In statistics, in the theory relating to sampling from finite populations, the sampling probability (also known as inclusion probability) of an element or member of the population, is its probability of becoming part of the sample during the dra ...
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 Marginal structural models are a class of statistical models used for causal inference in epidemiology. Such models handle the issue of time-dependent confounding in evaluation of the efficacy of interventions by inverse probability weighting for ...
, the standardized mortality ratio, and the
EM algorithm EM, Em or em may refer to: Arts and entertainment Music * EM, the E major musical scale * Em, the E minor musical scale * Electronic music, music that employs electronic musical instruments and electronic music technology in its production * Ency ...
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 In statistics, missing data, or missing values, occur when no data value is stored for the variable in an observation. Missing data are a common occurrence and can have a significant effect on the conclusions that can be drawn from the data. Mis ...
.


Inverse Probability Weighted Estimator (IPWE)

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 \^_ drawn i.i.d (independent and identically distributed) from unknown distribution P, where * X \in \mathbb^ covariates * A \in \ are the two possible treatments. * Y \in \mathbb response * We do not assume treatment is randomly assigned. The goal is to estimate the potential outcome, Y^\bigl(a\bigr), that would be observed if the subject were assigned treatment a. Then compare the mean outcome if all patients in the population were assigned either treatment: \mu_ = \mathbbY^(a). We want to estimate \mu_a using observed data \^_.


Estimator Formula

\hat^_ = \frac\sum^_Y_ \frac


Constructing the IPWE

# \mu_ = \mathbb\left frac \right/math> where p(a, x) = \frac # construct \hat_(a, x) or p(a, x) using any propensity model (often a logistic regression model) # \hat^_ = \sum^_\frac With the mean of each treatment group computed, a statistical t-test or ANOVA test can be used to judge difference between group means and determine statistical significance of treatment effect.


Assumptions

Recall the joint probability model (X,A,Y)\sim P for the covariate X, action A, and response Y. If X and A are known as x and a, respectively, then the response Y(X=x, A=a)=Y(x, a) has the distribution \begin Y(x, a) \sim \frac. \end We make the following assumptions. * (A1) Consistency: Y = Y^(A) * (A2) No unmeasured confounders: \ \perp A, X. More formally, for each bounded and measurable functions f and g, \begin \qquad \mathbb_\left \, X\right=\mathbb_\left \, X\right\, \mathbb_\left \, X\right \endThis means that treatment assignment is based solely on covariate data and independent of potential outcomes. * (A3) Positivity: P(A=a, X=x)=\mathbb_ \, X=x>0 for all a and x.


Formal derivation

Under the assumptions (A1)-(A3), we will derive the following identities \begin \mathbb\left Y^(a) \right = \mathbb_\left Y(X,a)\right = \mathbb_\left \frac \right \qquad \cdots \cdots (*) \end The first equality follows from the definition and (A1). For the second equality, first use the iterated expectation to write \begin \mathbb_\left Y(X,a)\right= \mathbb_\left \,_X_\right\right.html"_;"title="Y(X,a)_\,.html"_;"title="\mathbb_\left[_Y(X,a)_\,">\,_X_\right\right">Y(X,a)_\,.html"_;"title="\mathbb_\left[_Y(X,a)_\,">\,_X_\right\right \end By_(A3),_\mathbb_ \,_X_\right\right.html"_;"title="Y(X,a)_\,.html"_;"title="\mathbb_\left[_Y(X,a)_\,">\,_X_\right\right">Y(X,a)_\,.html"_;"title="\mathbb_\left[_Y(X,a)_\,">\,_X_\right\right \end By_(A3),_\mathbb_[\mathbf(A=a)\,">\,_X>0__almost_surely._Then_using_(A2),_note_that___ \begin \mathbb_\left \,_X_\right\right.html"_;"title="Y(X,a)_\,.html"_;"title="\mathbb_\left[_Y(X,a)_\,">\,_X_\right\right">Y(X,a)_\,.html"_;"title="\mathbb_\left[_Y(X,a)_\,">\,_X_\right\right \end By_(A3),_\mathbb_[\mathbf(A=a)\,">\,_X>0__almost_surely._Then_using_(A2),_note_that___ \begin \mathbb_\left[_Y(X,a)_\,">\,_X_\right&=__\frac_\\ &=_\frac_\qquad_\cdots_\cdots_(A2)_\\ &=_\mathbb_\left[_\frac_\,\bigg.html" ;"title="Y(X,a)_\,.html" ;"title="mathbf(A=a)\,.html" ;"title="Y(X,a)_\,">\,_X_\right\right.html" ;"title="Y(X,a)_\,.html" ;"title="\mathbb_\left[ Y(X,a) \,">\, X \right\right">Y(X,a)_\,.html" ;"title="\mathbb_\left[ Y(X,a) \,">\, X \right\right \end By (A3), \mathbb_[\mathbf(A=a)\,">\, X>0 almost surely. Then using (A2), note that \begin \mathbb_\left[ Y(X,a) \,">\, X \right&= \frac \\ &= \frac \qquad \cdots \cdots (A2) \\ &= \mathbb_\left[ \frac \,\bigg">\, X\right \qquad \cdots \cdots (\text) \end Hence integrating out the last expression with respect to X and noting that Y(X,a) \mathbf(A=a) = Y(X,A)\mathbf(A=a)= Y\,\mathbf(A=a) almost surely, the second equality in (*) follows.


Limitations

The Inverse Probability Weighted Estimator (IPWE) can be unstable if estimated propensities are small. If the probability of either treatment assignment is small, then the logistic regression model can become unstable around the tails causing the IPWE to also be less stable.


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).


Estimator Formula

\begin \hat^_ &= \frac \sum_^n\Biggl(\frac - \frac\hat_n(X_i,a)\Biggr) \\ &= \frac \sum_^n\Biggl(\fracY_ - (1-\frac)\hat_n(X_i,a)\Biggr) \\ &= \frac\sum_^n\Biggl(\hat_n(X_i,a)\Biggr) + \frac\sum_^n\frac\Biggl(Y_ - \hat_n(X_i,a)\Biggr) \end With the following notations: # 1_ is an indicator function if subject i is part of treatment group a (or not). # Construct regression estimator \hat_n(x,a) to predict outcome Y based on covariates X and treatment A, for some subject i. For example, using
ordinary least squares In statistics, ordinary least squares (OLS) is a type of linear least squares method for choosing the unknown parameters in a linear regression model (with fixed level-one effects of a linear function of a set of explanatory variables) by the prin ...
regression. # Construct propensity (probability) estimate \hat_n(A_i, X_i). For example, using
logistic regression In statistics, the logistic model (or logit model) is a statistical model that models the probability of an event taking place by having the log-odds for the event be a linear function (calculus), linear combination of one or more independent var ...
. # Combine in AIPWE to obtain \hat^_


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.: \frac\sum_^n\Biggl(\hat_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\sum_^n\frac\Biggl(Y_ - \hat_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
/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_n(X_i,a) or \hat_(A_, X_), 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 bias of the doubly robust estimators is called a second-order bias, and it depends on the product of the difference \frac - \frac and the difference \hat_n(X_i,a) - Q_n(X_i,a). This property allows us, when having a "large enough" sample size, to lower the overall bias of doubly robust estimators by using
machine learning Machine learning (ML) is a field of inquiry devoted to understanding and building methods that 'learn', that is, methods that leverage data to improve performance on some set of tasks. It is seen as a part of artificial intelligence. Machine ...
estimators (instead of parametric models).Hernán, Miguel A., and James M. Robins. "Causal inference." (2010): 2
link to the book
- page 170


See also

*
Propensity score matching In the statistical analysis of observational data, propensity score matching (PSM) is a statistical matching technique that attempts to estimate the effect of a treatment, policy, or other intervention by accounting for the covariates that predic ...


References

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