Inverse probability weighting

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. μ ^ a , n I P W E = n − 1 Σ i = 1 n Y i 1 A i = a / p ^ n ( A i = a | X i ) {displaystyle {hat {mu }}_{a,n}^{IPWE}=n^{-1}Sigma _{i=1}^{n}Y_{i}1_{A_{i}=a}/{hat {p}}_{n}(A_{i}=a|X_{i})} μ ^ a , n A I P W E = 1 n ∑ i = 1 n ( Y i 1 A i = a p ^ n ( A i | X i ) − 1 A i = a − p ^ n ( A i | X i ) p ^ n ( A i | X i ) Q ^ n ( X i , a ) ) {displaystyle {hat {mu }}_{a,n}^{AIPWE}={frac {1}{n}}sum _{i=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 )}} 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. 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. 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.