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Let's say there's one DV (Y) and three IVs (X1, X2, X3), and among IVs, X1 is a dummy variable. In the regression model without interaction terms, the results can be represented like this:

Y ~ X1 + X2 + X3

X1 : non-significant
X2 : significant
X3 : significant

In this case, is it meaningful to check some interaction terms (e.g. X1 $\cdot$ X2 or X1 $\cdot$ X3)? At first I thought I don't have to because the main effect of X1 indicates non-significance. But I'm afraid I'm missing something important.

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Yes, interaction coefficients are still meaningful to estimate when they involve variables with insignificant main effects. An example of a meaningful interaction would occur if, in the absence of a main effect (e.g., if $\beta$X1$=0$), a moderator variable produced a large difference in the slope for X1 at different levels of the moderator (e.g., if $\beta$X1$=.5$ when X2 $>\mu+1\ SD$, and $\beta$X1$=-.5$ when X2 $<\mu-1\ SD)$.

What's generally important is that you still estimate the main effects of those variables in your model anyway. Don't remove the simple X1 term if you're going to add either of those interaction terms you mention that involve the same variable, even if you know its $\beta$ won't differ significantly from zero. An insignificant but nonzero effect is still different from no effect whatsoever. It's important to freely estimate that main effect for the sake of your interactive model, if not necessarily for the sake of your theory.

It's worth noting that some exceptions to the above may exist (see Including the interaction but not the main effects in a model), but you'd want to be sure to know what you're doing (and you'd probably want to be a lot more specific about it here) before omitting main effects in the presence of an interaction. Odder cases involving interactions and semi-arbitrary hypothesis rejection thresholds may occur (e.g., What if interaction wipes out my direct effects in regression?), but if they do, try to remember how limited your interpretation of the change in the $p$ value should be, especially if it's just dancing around your threshold without really changing a great deal (see Geoff Cumming's "Dance of the p values" on YouTube).

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