Collider in RCT Subgroup Analysis

Well, I roll the rationale into the conclusion as follows:

a) if one decides the third variable is prognostic, then it cannot be either post-randomization nor be gating a post-randomization variable

b) a mediator is clearly a post-randomization variable - induced by the treatment (I call this an induced mediator)

c) an effect modifier is NOT a post-randomization variable but clearly gates such a variable (I call this a conditional mediator aka proxy for an induced mediator)

Now all three are prognostic for the outcome so they only differ by one of the above. My point is that each needs a specific analysis and that if a variable either gates a post-randomization variable or is a post-randomization variable then in both cases it is a mediator in some sense. So the first conclusion is that a variable can only be an effect modifier if it influences at least one mediator in a causally sufficient representation of all pathways from treatment to outcome. The second conclusion is that the mediator collides on sample selection with the treatment so mediators should not enter the outcome model to avoid bias.