In a randomized trial, what is the propensity score? In Rosenbaum's 1983 paper, he states that "in a randomized trial, the propensity score is a known function so that there exists one accepted specification."
I am wondering what this specification is in closed form. If we let $e(x)$ be the propensity score, would it be:
$$
e(x) \sim Bern(0.5)
$$
?
 A: According to the definition: "probability of program participation/exposure/receiving treatment", we have
propensity score = the ratio of number of patients receive active treatment to total number of patients for patient who received active treatment.
propensity score = the ratio of number of patients receive placebo to total number of patients for patient who received placebo.
For example, if the design is # of patients in two treatments are equal, then propensity score is 0.5. If design is 1:2 (active:placebo), then propensity score for patients receiving active treatment is 1/3, and for the patients receiving placebo is 2/3.
A: In many applications, propensity score is modeled as a logit function of covariates, i.e.
$$e(X) = \dfrac{1}{1+\exp(-\beta^T X)}$$
which is estimated using observed covariates and treatment assignment (i.e. $T\in\{0,1\}$). Propensity score is hypothetically the probability of being assigned to treatment given observed covariates. You can choose other functional forms such as Probit, Tobit or else, but there is no science that favor one parametric form over another. The specification, once chosen, applies to all observational units.
