# What is the difference between a Randomized Control Trial (RCT) and an Observation Study in terms of the propensity score?

I am wondering if there is a direct relation between a Randomized Control Trial (RCT) and an Observation Study in terms of the treatment assignment mechanism or the propensity score. It seems in an RCT, certain conditions are listed, such as the propensity score/assignment mechanism being between 0 and 1:

$$0 < P(Z_i|X_i) <1$$

where $Z_i$ is the treatment indicator taking on $1$ or $0$ and $X_i$ are the covariates.

Is there a succint relation between these and the RCT or observational study? Thanks.

• If it is RCT already why you use propensity score ? since in RCT you randomize subjects to treat and control group. You use it because you don't believe your randomization? I cannot understand the logic for a RCT to use propensity score adjustment(or matching) Jul 12 '17 at 0:51
• The reason I mention it is because I would like to understand if the RCT is a special case of the Observational Study and vice versa. For example, ideally I am hoping to see if there is something like, "Relax this assumption, then the observational study becomes an RCT" Jul 12 '17 at 1:46