Formal Definition
A propensity score is the probability of a unit (e.g., person, classroom, school) being assigned to a particular treatment given a set of observed covariates. Propensity scores are used to reduce selection bias by equating groups based on these covariates.
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:
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.
If unconfoundedness holds, then
Pearl (2000) has shown that a simple graphical criterion called backdoor provides an equivalent definition of unconfoundedness.
Read more about this topic: Propensity Score Matching
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