## When you have an interaction, which variable moderates which?

I was talking recently with a colleague about interpreting moderator effects, and the question came up: when you have a 2-way interaction between A and B, how do you decide whether to say that A moderates B versus B moderates A?

Mathematically, of course, A*B = B*A, so the underlying math is indifferent. I was schooled in the Baron and Kenny approach to moderation and mediation. I’ve never found any hard and fast rules in any of Kenny’s writing on the subject (if I’ve missed any, please let me know in the comments section). B&K talk about the moderator moderating the “focal” variable, and I’ve always taken that to be an interpretive choice by the researcher. If the researcher’s primary goal is to understand how A affects Y, and in the researcher’s mind B is some other interesting variable across which the A->Y relationship might vary, then B is the moderator. And vice versa. And to me, it’s entirely legitimate to talk about the same analysis in different ways — it’s a framing issue rather than a deep substantive issue.

However, my colleague has been trying to apply Kraemer et al.’s “MacArthur framework” and has been running into some problems. One of the MacArthur rules is that the variable you call the moderator (M) is the one that comes first, since (in their framework) the moderator always temporally precedes the treatment (T). But in my colleague’s study the ordering is not clear. (I believe that in my colleague’s study, the variables in question meet all of Kraemer’s other criteria for moderation — e.g., they’re uncorrelated — but they were measured at the same timepoint in a longitudinal study. Theoretically it’s not clear which one “would have” come first. Does it come down to which one came first in the questionnaire packet?)

I’ll admit that I’ve looked at Kraemer et al.’s writing on mediation/moderation a few times and it’s never quite resonated with me — they’re trying to make hard-and-fast rules for choosing between what, to me, seem like 2 legitimate alternative interpretations. (I also don’t really grok their argument that a significant interaction can sometimes be interpreted as mediation — unless it’s “mediated moderation” in Kenny-speak — but that’s a separate issue.) I’m curious how others deal with this issue…