Inverse probability weighting

Inverse probability weighting is a statistical technique for calculating statistics standardized to a 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.^[1] There may be prohibitive factors barring researchers from directly sampling from the target population such as cost, time, or ethical concerns.^[2] 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.^[3] 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.^[4] With an estimate of the inclusion 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 underrepresented due to a large degree of missing data.

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 ${\displaystyle \{{\bigl (}X_{i},A_{i},Y_{i}{\bigr )}\}_{i=1}^{n}}$ drawn i.i.d from unknown distribution P, where

• ${\displaystyle X\in \mathbb {R} ^{p}}$ covariates
• ${\displaystyle A\in \{0,1\}}$
• ${\displaystyle Y\in \mathbb {R} }$ response
• We do not assume treatment randomly assigned completely at random

The goal is to estimate the potential outcome, ${\displaystyle Y^{*}{\bigl (}a{\bigr )}}$, 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: ${\displaystyle \mu _{a}=\mathbb {E} Y^{*}(a)}$. We want to estimate $\mu_a$ using observed data ${\displaystyle \{{\bigl (}X_{i},A_{i},Y_{i}{\bigr )}\}_{i=1}^{n}}$.

Estimator Formula

${\displaystyle {\hat {\mu }}_{a,n}^{IPWE}=n^{-1}\Sigma _{i=1}^{n}Y_{i}1_{A_{i}=a}/{\hat {p}}_{n}(A_{i}|X_{i})}$

Constructing the IPWE

1. ${\displaystyle \mu _{a}=\mathbb {E} \{Y1_{A=a}/p(A|X)\}}$ where ${\displaystyle p(a|x)=P(A=a,X=x)}$ $A$
2. Construct ${\displaystyle {\hat {p}}_{n}(a|x)}$ of ${\displaystyle p(a|x)}$ using logistic regression.
3. ${\displaystyle {\hat {\mu }}_{a,n}^{IPWE}=n^{-1}\Sigma _{i=1}^{n}Y_{i}1_{A_{i}=a}/{\hat {p}}_{n}(A_{i}|X_{i})}$

With mean of each treatment group compute, a statistical t-test or ANOVA test can be used to judge difference between group means and determine statistical significance of treatment effect.

Assumptions

1. Consistency: ${\displaystyle Y=Y^{*}(A)}$
2. No unmeasured confounders: ${\displaystyle \{Y^{*}(0),Y^{*}(1)\}\perp A|X}$
   • Treatment assignment is based solely on covariate data and independent of potential outcomes.
3. Positivity: ${\displaystyle P(A=a|X=x)>0}$ for all $a$ and $x$

Limitations

The Inverse Probability Weighted Estimator (IPWE) can be unstable if estimated propensities are small. If the probability of either treatment assignment is small than 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).^[5]

Estimator Formula

${\displaystyle {\hat {\mu }}_{a,n}^{AIPWE}={\frac {1}{n}}\sum _{k=1}^{n}{\Biggl (}{\frac {Y_{i}1_{A_{i}=a}}{{\hat {p}}_{n}(A_{i}|X_{i})}}-{\frac {1_{A_{i}=a}-{\hat {p}}_{n}(A_{i}|X_{i})}{{\hat {p}}_{n}(A_{i}|X_{i})}}{\hat {Q}}_{n}(X_{i},a){\Biggr )}}$

Constructing the AIPWE

1. Construct regression estimator ${\displaystyle {\hat {Q}}_{n}(x,a)}$ to predict outcome $Y$ based on covariates $X$ and treatment $A$
2. Construct propensity estimate ${\displaystyle {\hat {p}}_{n}(A_{i}|X_{i})}$
3. Combine in AIPWE to obtain ${\displaystyle {\hat {\mu }}_{a,n}^{AIPWE}}$

References

1. 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.
2. 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. PMID 19357328.
3. Horvitz, D. G.; Thompson, D. J. (1952). "A generalization of sampling without replacement from a finite universe". Journal of the American Statistical Association. 47: 663–685. doi:10.1080/01621459.1952.10483446.
4. Hernan, MA; Robins, JM (2006). "Estimating Causal Effects From Epidemiological Data". J Epi Comm. 60: 578–596. CiteSeerX 10.1.1.157.9366. doi:10.1136/jech.2004.029496.
5. Cao, Weihua; Tsiatis, Anastasios A.; Davidian, Marie (2016-12-04). "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. PMID 20161511.