What is the right approach when there is a variable that does not affect participation but effects the outcome measure? I have data set of the health outcome of a treatment and control group. I have data on 3 covariates. 2 demographic and one is a baselines score. I believe that base line score did not affect the participation decision. Hence including it estimating the propensity score will be wrong. But measuring ATT by ignoring it will be ignoring data.
You do propensity score matching on the covaritates to create a quasi-randomized study. You only need to match on the covariates that are different in the study groups (treatment vs. control) and affect the treatment decision (as the study is NOT randomized which is pracitically done by a fair coin toss or binary algorithm (0 or 1)). As a matter of fact the propensity score is defined as the conditional probability of treatment given confounding covariates:
propscore(x) = Pr(T=1 | X=x) (1)
Let O(C) and O(T) represent the potential outcomes under control and treatment. Then treatment assignment is conditionally unconfounded if potential outcomes are statistically independent of treatment conditional on confounding covariates X. This can be written as
O(C), O(T) ⊥ T│X (2)
where ⊥ denotes statistical independence. My question to you is: How do you know that the baseline covariate did not affect treatment decision? I don't understand how you can be sure that this is the case. I also don't think that estimating the propensity score using the baseline covariate would be wrong. So my recommendation is: Use the baseline covariate in the propensity score matching process. As you already mentioned, there's a general principle in statistics: The more data the better (I call it the TMDTB axiom). It would be interesting to know on which scale the covariates are and what exactly they are. Then I could give you some more tips. I hope this helps a bit.
According to Cuong (2013)'s "Which covariates should be controlled in propensity score matching?..." all those variables should be part of the psm calculation which affect program participation and outcome but not those which only affect outcome but not program participation.