An alternative to direct matching or matching on propensity scores involves the use of the inverse of propensity scores in a weighted regression framework (Horvitz and Thompson (1952), known as inverse probability of treatment weighted (IPTW) regression where:
IPTW regression (with weights specified as above) specifically estimates the average treatment effect (ATE) (Astin, 2011):
ATE = E[Y1i-Y0i]
Inverse probability of treatment weighting (IPTW) uses weights derived from the propensity scores to create a pseudo population such that the distribution of covariates in the population are independent of treatment assignment. (Astin,2011). This is an appeal to the CIA and Rosenbaum and Rubin’s propensity score theorem discussed before. The weighting scheme essentially ‘weights up’ control units to look like treatment units (Stuart,2011).
Austin, P.(2011). An Introduction to Propensity Score Methods for Reducing the Effects of Confounding in Observational Studies. Multivariate Behav Research.May; 46(3): 399–424.
Horvitz D. G. & Thompson D. J.(1952) . A Generalization of Sampling Without Replacement From a Finite Universe. Journal of the American Statistical Association, Vol. 47, No. 260 (Dec., 1952), pp. 663- 685
Maciejewski, M. L. & Brookhart, M.A. (2011). Propensity score workshop . Retrieved January 19,2013. Website: http://ahrqplexnet.sharepointspace.com/Webinars/PS_webinar_followup.pdf
Stuart ,E.(2011).Propensity score methods for estimating causal effects: The why, when, and how . Johns Hopkins Bloomberg School of Public Health. Department of Mental Health. Department of Biostatistics. Retrieved January 19,2013. Website: www.biostat.jhsph.edu/estuart