Propensity score matching in a case-control study: can the number of matches per patient be correlated with a matching variable? Say I am trying to investigate the association between antidepressants and achieving recovery in a depression programme. I have an observational cohort of depressed patients enrolled in the programme between the ages of 20-90, and I wish to match patients by a series of depression measures (QoL surveys etc), the presence of several chronic conditions (which affect severity of depression), medications affecting mood, as well as age, sex, and a bunch of other demographics that might confound results. I wish to use a propensity score to match patients with a caliper width of 0.25. Cases (recoverees) can be matched with up to 5 controls.
I would assume that the older a patient gets, the more complex and variable their morbidity/depression/treatment profiles are, and thus the harder it is to find matches within the caliper width, i.e. I would expect fewer matches, on average, for cases aged 80-90 years old compared to 20-30 years. In which case, you would end up with controls which are on average younger than cases. Is there anything stopping PSM correlating the number of matches a case is expected to get with a matching variable, resulting in an imbalance of the said variable within the matched set?
My second question, if this can occur, is: does it even matter, if there truly aren't the matches available and it's not some fault in study design? Or should you do something about it (e.g. lower the limit from 1:5 matching to, say, 1:3, if older people are roughly getting 3 matched controls).
(Yes you could probably simplify matching variables to a single metric that summarises morbidity severity, but in this hypothetical argument I want to assume that it's important to match on individual morbidity, so that older patients truly are harder to find matches for)
 A: There is precedent for what you are describing, which is called "variable ratio" matching, in which the matching ratio (the number of matched controls per treated unit) varies systematically. This is described in Ming and Rosenbaum (2000). They observe exactly what you do, which is that treated units with lower propensity scores are easier to match and therefore should be given more matches than units with higher propensity scores. They derive the optimal matching ratio for each unit based on the rank of its propensity score. Some units receive one match, and some receive more than one match, in a way governed by the derived formula. This is a way to increase the size of the matched sample without forcing low quality matches on units for whom it is hard to match. Variable ratio matching is implemented in the MatchIt package.
That said, I don't see why you need a method like this when the caliper should do the job of restricting the number of matches anyway. If only 3 control units are within a treated unit's caliper, then it will only receive 3 matches. You can place a caliper on any matching variable; it doesn't have to be on the propensity score.
It's important to analyze variable ratio matched data correctly (whether it comes from a variable ratio matching procedure or calipers causing units to have different numbers of matches). To do so, you need to compute matching weights for each control unit that are equal to the inverse of the number of control units that are matched to the same treated unit that the control unit in question is matched to. That is, any control units in a 3:1 match should receive a weight of 1/3, and any control units in a 5:1 match should receive a weight if 1/5.
Note that this answer refers specifically to cohort studies; I'm not sure if case-control studies operate in a meaningfully different way.
