Stratification and subsampling
Stratification in PSM involves dividing a study population into strata or subclasses based on their propensity scores. This involves dividing the study sample into several strata or subclasses based on the distribution of their propensity scores. For example, quintiles are commonly used, dividing the sample into multiple (five) groups with approximately equal numbers of individuals. The aim is to balance the covariates within each stratum between the treatment and control groups. By doing so, the method attempts to mimic the conditions of a randomized controlled trial, where treatment assignment would be independent of these covariates.
Theory
After stratifying the data, within each stratum, we can estimate treatment effects independently. Assuming the potential outcome framework, let 
and 
denote the potential outcomes under treatment and control, respectively. The ATE within a stratum can be calculated as follows:
(3...
