I'm currently having a debate with a friend about the moderation effect in regression analysis, and I'd like to get some clarity on this. Let's say we're interested in empirically testing whether there's a relationship between X and Y, and we hypothesize that this effect might depend on a third variable, M (a moderator). My question is: when formulating our hypothesis for why the impact of X on Y depends on M, is it also necessary to argue for a direct relationship between M and Y? My understanding is that moderation focuses on how M alters the X-Y relationship, not necessarily requiring M to have a direct effect on Y. However, my friend insists that in order to introduce a moderator, there must always be a direct relationship between M and Y. Is it standard practice to require this direct M-Y relationship, or is it sufficient to hypothesize the moderating effect without such a requirement?
-
1$\begingroup$ To know whether something is 'standard practice' or not it would help to give information about which is the field in question. For instance, in many fields one would rather talk about 'mediators' and 'effect modifiers' rather than 'moderators'. $\endgroup$– KukuCommented Sep 9 at 13:06
-
$\begingroup$ Thank you, the research field is economics $\endgroup$– user49942Commented Sep 9 at 13:17
3 Answers
I'm not an economist (perhaps terminology is different in that field), but in general Wikipedia says:
the effect of a moderating variable is characterized statistically as an interaction; that is, a categorical (e.g., sex, ethnicity, class) or continuous (e.g., age, level of reward) variable that is associated with the direction and/or magnitude of the relation between dependent and independent variables.
As an example, Wikipedia continues:
for a response $Y$ and two variables $x_1$ and moderating variable $x_2$: $$ Y = b_0 + b_1 x_1 + b_2 x_2 + b_3 (x_1 \times x_2) + \epsilon .$$
From the perspective of the regression model, the predictor of interest $x_1$ and the moderator $x_2$ are indistinguishable. The effect on outcome $Y$ of a change in moderator $x_2$ depends on the value of $x_1$, just as the value of the moderator $x_2$ affects the association between $Y$ and the predictor of interest $x_1$.
As the answer from @DavidB notes, it would be possible to have no (partial) correlation between $Y$ and $x_2$ depending on the predictor values and the association between $x_1$ and $x_2$ in the data set. From the modeling perspective, however, the moderator $x_2$ has as much of a "direct relationship" with outcome as does $x_1$. Causal interpretation (which might be hiding in your definition of "direct relationship") is another matter.
If you happen to find that the $b_2$ coefficient is "insignificant" in your model summary, don't jump to the conclusion that $x_2$ has no "direct relationship" with outcome. When there's an interaction, the value (and thus the "significance" of its difference from 0) of a predictor's individual coefficient depends on how its interacting predictors are coded. Change the coding of $x_1$ and you will change the "significance" of $b_2$. This answer provides a worked-through example.
You are correct: "moderation focuses on how M alters the X-Y relationship, not necessarily requiring M to have a direct effect on Y." For a very simple example, see Figure 5 at https://www.yellowbrickstats.com/partial.htm.
There is no single rule here. Essentially, if M and Y are not correlated, then you are hypothesizing an interaction where the simple slopes intersect somewhere around the mean value of X (perhaps not exactly the mean, if X and M are still correlated). See Figure 1A here: https://journals.sagepub.com/doi/10.1177/25152459231187531.
The larger question is: given the field you are in and the specific substantive research area you are investigating, does it make sense that M would have no effect on Y? I think many people (myself included) would expect M and Y to be correlated for many research questions, but there is no statistical reason why they need to be.
