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Is it ever possible for a variable to be a moderator and partial mediator?

I have a hypothetical example, which I think shows this can be the case. But I might be making a mistake.

X = Socioeconomic status Y = Number of participant's friends who are obese Z = Weight of participant

Partial mediation: high X is associated with lower Y, and Y is associated with higher Z. Therefore, some of the effect of X and Z is the result of differences in Y between the levels of X.

In other words, higher SES people have less friends who are obese. Those with less friends who are obese have a lower weight. Therefore, the effect of SES on weight is partially caused by differences in the number of friends who are obsese between SES groups.

Moderation: effect of Y on Z is stronger for low X.

In other words, having friends who are obese causes larger weight in participants from lower SES groups, maybe because low SES friends put greater peer pressure on the participant to overeat.

So can a variable be both a moderator and partial mediator?

If so, how would I identify if it is? (eg. in the case above)

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Yes, standard causal mediation approaches consider this possibility. See Imai, Keele, & Tingley (2010) or valeri & VanderWeele (2013) for discussion. These approaches decompose the total effect in variation mediation/moderation components that can be used to describe the phenomenon you're describing.

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