I am trying to understand how it is justified to use "with replacement" methods in propensity score matching. In the literature there are many statements like this:
However, situations arise when there are not enough controls in the overlapping region to fully provide one match per treated unit. In this case it can help to use some control observations as matches for more than one treated unit. This approach is often called matching with replacement, a term which commonly refers to with one-to-one matching but could generalize to multiple control matches for each control. Such strategies can create better balance, which should yield estimates that are closer to the truth on average. Once such data are incorporated into a regression, however, the multiple matches reduce to single data points, which suggests that matching with replacement has limitations as a general strategy.
- How is the bold sentence above proven? I thought we were trying to find the truth, not cook the books that give us what we want.
- How, even philosophically, can this possibly be justified? If this is real data, can you go around making clones and claiming the new set means anything? With replacement methods in Monte Carlo world make sense; we are taking limits and we want to preserve uniformity in our sample space - I can't see how that works here.
Any insight would be appreciated.