Mediation using PROCESS - Add covariates to model with Y only, or both M and Y? I am conducting a mediation analysis (model 4) in SPSS using the PROCESS macro by Hayes.
I have cross-sectional survey data and want to test the following relation:


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*Childhood adversity (X) -> Current loneliness (M) -> Current psychosis (Y)


I want to add three covariates to the analysis: age/sex/depressed feelings. 
PROCESS gives me the option to add these covariates in models of both M and Y, and Y only (and M only, but not interested in that). I know for sure that I want to add these covariates to the model with Y, as I think that the covariates might influence the relation between X and Y, and between M and Y. However, I could also think of some reasons why they might influence the relation between X and M – I am not necessarily interested in this relation, but they could influence the estimate of the mediation effect.
My question is: should I only add my covariates to the model with Y as this was my a priori reason to include the covariates, or should I also add them to the model with M as there might be some relationship there which would be good to control for? (I.e., are there good statistical reasons to include covariates to both M and Y models, rather than Y only?)
 A: You should definitely add (some of) those covariates to the X -> M relationship. You ideally want to control for all possible confounders of the X -> M relationship to estimate the X -> M path ina na unbiased way. You clearly have not collected enough variables to do that (e.g., SES, family structure, neighborhood characteristics, etc.), but you can get closer to a correct path by including the variables that you have that make sense. (Note: without controlling for these variables, you cannot estimate the effect of childhood adversity on loneliness; the association between the two could be due to any of the confounding factors I mentioned or others. I would be very hesitant about interpreting your findings as mediation rather than association). 
Sex may cause both adversity and loneliness. Current age does not cause childhood adversity, but it acts as a proxy for year born, which might cause childhood adversity. Either way, both of these variables are fine to include in X -> M. Depressed feelings (if measured currently) cannot cause childhood adversity; rather, adversity causes depressed feelings. This would be inappropriate to include in the X -> M model. Rather, it's pretty clear that depressed feelings could be another potential mediator between childhood adversity and current psychosis or even a consequence of current psychosis. Where it belongs in the model is unclear, and how you include can wildly change your estimates and make them uninterpretable. See my answer here for why this is such a problem.
Basically, and I know this is a larger conclusion than what you asked for, you have not collected enough data or the right kind of data to perform the analysis you described and interpret any of the relationships as mediation. You have not collected enough variables to isolate the effects. You have not clearly delineated the temporal order of your variables (i.e., depressed feelings, loneliness, and psychosis are all concurrently measured so the causal directions are entirely unclear). Even the associations you find may not be validly interpretable as partial associations if you condition on outcomes.
