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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)?

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You do not need to remove Z from the propensity score model. Stratifying on a variable has approximately the same effect of controlling for it in regression (except of course you don't need to make assumptions about functional form). So doing as you proposed would be fine, and no different from "doubly robust" effect estimation (e.g., where the outcome model includes covariates already used in the PS model). What you're proposing is also approximately isomorphic to propensity score within subgroups, where each subgroup is a quantile of Z. Propensity scores in subgroups are discussed in Green & Stuart (2014)

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