Should a variable one is interested in examining for effect modification be used in a propensity model to balance covariates? Given I have a binary outcome (Y), binary exposure (A), binary potential effect modifier (Z), and a bunch of covariates (C1-C20).
We would like to examine for effect modification of Z on the treatment effect of A on Y. The treated vs. not treated have different characteristics and we plan to either weight or match treated/not treated using a propensity score.
My question is whether the effect modifier should be included in the propensity model (P(A|C1-C20,Z))? Eeren et al. (2015) seem to indicate no, but I've found little else to back that up. If not, what is the best way to test for effect modification, while still balancing covariates using a propensity score without the effect modifier (P(A|C1-C20))? Possible methods I see:

*

*Make two propensity models within each effect modifier group. Then look at treatment effect within each of these subgroups.

*Make single propensity model excluding effect modifier. Then include effect modifier into effectiveness model as an interaction term with treatment.

Any suggestions with references to published papers would be appreciated.

The specific answer is no, Z should not be included in the creation of a PS. Also, if you have a bunch of covariates c1 - c20, perhaps a DAG can clarify the minimal sufficient adjustment set for estimating the direct effect of A on Y and then perhaps covariate adjustment might be possible. PS methods are agnostic to both HTE and effect modification and are best avoided if feasible. A logistic regression with A, Z and A#Z balanced for potential covariates is still not very useful because there is no way to determine if a potential non-null product term in this GLM is due to an artifact of the sample (overfitting) or to true effect modification by Z. See this paper
