I am working with a team of researchers looking to select a control group out of a large population (150,000) to compare with a relatively small treated group (~900). We plan on using a propensity score to match similar treated members to potential control group members, but also exactly match on other factors (i.e. gender). That is, for a group of criteria that are relevant for matching, we have a portion that will be included in the propensity score, and the remainder will be matched exactly. So I have two questions:
Should we create the propensity score (logistic regression) before exactly matching on certain criteria, and use the score as just one matching criterion along with the exact criteria, OR should the scores be created for each group only after exactly matching on certain criteria?
If the propensity score is formed before exact matching takes place, should the exact matching criteria also be in the propensity score? This question arose when members on the team expressed concern that the significance of the exact matching criteria might be higher than the other criteria included in the propensity score. SAS examples of the PSMATCH procedure also show that exactly matched criteria are included in the PS model in their examples (here: https://documentation.sas.com/?docsetId=statug&docsetTarget=statug_psmatch_examples04.htm&docsetVersion=14.3&locale=en)
Any further insight, readings, or suggestions would be helpful. Thanks.