Is Propensity Score Matching the Correct Tool to Match Cohorts by Disease and not Exposure? For a set of patients that all underwent procedure A, is it valid to use propensity score matching to compare 2 subsets in which the patients have disease D (D+) to those that do not have disease D (D-), with the goal of comparing outcomes between these subgroups? 
From looking at the medical literature, it seems that you should only use propensity scores to compare by treatment, but a quick search on google scholar shows cases where researchers use propensity scores to match by disease. 
Clarification on this is much appreciated.
 A: If you are trying to isolate the disease's effect by creating a matched cohort where the distribution of all other covariates that impact the outcome is similar between the D+ group and D- group, then yes. Propensity scores are a good tool for constructing such a cohort.
You can think of the term exposure as a loose term if that helps. In terms of causal inference, it is cleaner to think of balancing the covariate distribution for an exposure like smoking rather than a disease like COPD because COPD is tied to a myriad of factors that would be difficult to balance. For example trying to balance smoking history in those with COPD and those without would be difficult without unattainably detailed information on a lifetime of smoking. Simply balancing "ever smoked" would leave a lot of room for residual confounding. That said, the primary goal of trying to isolate one factor for examination by balancing the distributions of all the other factors remains the same. 
And as always, make sure to thoroughly check the balance of your cohort after matching. The dataset needs to be rich enough to find those subjects to match.
