Just for example: Say I am trying to measure the impact of pills, exercise, and vaccines on a dependent variable y. For simplicity, pills, exercise and vaccines are binomial so either 0 or 1. However, I want all three of them to act as treatments so it is possible that observations can have any combinations os these treatments. I have been trying to run a Propensity Score Analysis to run this but was not sure if this is possible? Does anyone know any tutorial or reference where I can learn more about it. Almost everything I have read online only looks at one treatment with multivalues (0,1,2, or any number) but I want all three to count as treatments. So, my control groups would have to have matching that takes into account the possibility that different observations can have any combinations of these treatments. Thanks in advance!!
Interesting question that I hadn't thought about before. The most similar application I can think of is this paper using a control function estimator for two (mutually exclusive) treatments.
In practice the simplest way to proceed might be doing three separate propensity score analyses (controlling for the other endogenous variables). My intuition is that this would overlook any difference between the marginal and joint distributions of the underlying unobservables... although even if you knew the joint distribution accounting for the unobservables in practice might be tricky since the interactions are probably very correlated with unobserved determinants of health (in the example you mention).
As an aside, in my opinion propensity score designs feel the most credible when at least some of the variation in propensity comes from truly exogenous variables.
Could you define a new treatment variable that takes 8 values?
E.g. $T=0$ if all three treatments are 0.
$T=1$ if pills=1 and the other two are 0.
$T=2$ if exercise = 1 and the other two are 0, ...
$T=8$ if all three are 1?
I've never thought about propensity score matching with a multivalued treatment, but that seems to be a thing.