The basis of my question is whether it makes sense, given you have a set of propensity scores, to stratify the data based on the propensity scores and on a second variable which was included in the propensity score model.
For example, suppose I have a set of propensity scores based on the following model: Y~X+Z.
Using this, I attain propensity scores and stratify the scores into 5 separate quintiles, and then proceed with some analysis within each quintile, the idea being that within the quintiles the observations are similar, so there is a weak form of matching going on.
However, suppose I also wanted the matching to be very close along the Z dimension. Could I stratify the dataset not only on the propensity score, but also on Z? Or if I wanted to do this, should I first remove Z from the propensity score model (even though Z is highly predictive of Y, the treatment variable in this example)?