# 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:

$$e(x) \sim Bern(0.5)$$ ?

• +1. Any answer please don't just give the definition of the propensity score as the "probability of program participation/exposure/receiving treatment" alongside a logistic regression. The question is more substantial than that! Commented Oct 22, 2018 at 23:18
• po asked what propensity score may look like, how is the question more substantial than a possible functional form of propensity score? Commented Oct 25, 2018 at 19:05

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.

• Essentially, the above is assuming a superpopulation and we are imagining draws from such a superpopulation and the propensity score as you have defined it is a statement made about the sampling process itself? Commented Oct 23, 2018 at 16:06
• I do not think there is superpopulation and sampling process in recruiting the patients in clinical trials. When the doctor find the qualified patient who signs consent, the patient will be assign to a treatment randomly according to plan. Basically, there is no situation that there are more qualified patients than you need and you must random select part of them. Commented Oct 24, 2018 at 0:19
• it's exactly because the sampling process is under defined that we need propensity score to help compare apples in treatment group to apples in control groups. propensity score is the probability of being treatment GIVEN the observed covariates (patient characteristics). Just because half of your samples were treated does not mean that they are comparable to the other half in control. Commented Oct 25, 2018 at 19:07

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.

• I don't understand how this is downvoted. Rosenbaum proposed propensity score matching to address nonrandomized or less-than-ideal randomized treatments in observational studies. Ratio of treatment to (treatment + placebo), as is mentioned in the other answer, must be based on the fact that the treatment is perfectly randomized. If you already have a perfectly randomized experiment to begin with, you do not need propensity score. In fact, Rosenbaum also proposed modeling propensity score using Logit in his 1983 paper (pp. 47). Logit is still widely used today. Commented Oct 23, 2018 at 13:25
• You still need propensity scores in randomized experiments, if you have adaptive randomization (i.e. the probability on ending up on the different treatments changes for each patient). One should not ignore that in the analysis of a randomized trial. Commented Oct 23, 2018 at 14:29
• To me that is a constraint on randomization, so yes, unless you have a perfectly randomized experiment, you will need some remedies, and matching is one of them, and propensity score is one way to match. But how does the other answer (percentage of treated units) address nonconstant treatment probability? Commented Oct 23, 2018 at 18:34