Propensity score
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In the analysis of treatment effects, suppose that we have a binary treatment T, an outcome Y, and background variables X. The propensity score is defined as the conditional probability of treatment given background variables:
The propensity score was introduced by Rosenbaum and Rubin (1983) to provide an alternative method for estimating treatment effects when treatment assignment is not random, but can be assumed to be unconfounded. Let Y(0) and Y(1) denote the potential outcomes under control and treatment, respectively. Then treatment assignment is (conditionally) unconfounded if treatment is independent of potential outcomes conditional on X. This can be written compactly as
where 
denotes statistical independence.
Rosenbaum and Rubin showed that if unconfoundedness holds, then
[edit] References
Rosenbaum, P. R., and Rubin, D. B., (1983), "The Central Role of the Propensity Score in Observational Studies for Causal Effects," Biometrika 70, 41-55.
