Main effect vs simple effect In the following linear model:
$y = \beta_0 +\beta_1 x_1 +\beta_2 x_2 + \beta_3x_1x_2 + \epsilon$
what are called $x_i$? The term $x_1x_2$ is the interaction. In ANOVA, $x_i$ are named "main effects", but in regression I saw a paper mentioning that it is wrong in the context of regression (even though I see it all the times in papers and on CV) and they should be called "simple effects" or "simple main effects". Is it really wrong? Would it be really better to call them "conditional relationship" or something similar?
 A: I try to avoid the term "main effect" when there's an interaction, or at least wrap it in quotes if I feel forced to use it. The reason is perhaps mostly psychological, but avoiding the term can help avoid a lot of confusion.
The problem is that, with an interaction term involving a predictor, changing the reference value of its interacting predictor can change its own apparent "main effect"! See this page for a worked-through example. In your equation, if $\beta_3$ is non-zero, the value of $\beta_1$ will change if you change the centering of $x_2$ (continuous case) or change the $x_2$ reference level (categorical case).
I have frequently answered questions on this site about mysterious unexpected "lack of significance" of "main effects" known to be "important" when they are involved in interactions. The confusion always results from an attempt to interpret the "significance" of a "main effect" coefficient as representing the "importance" of a predictor that is involved in an interaction. If there's an interaction, evaluating the "importance" of the predictor must take into account its interaction coefficients too.
Why should something that sounds so definitive as a "main effect" of one predictor depend on how you have handled another predictor? The term "simple effect" sounds appropriately less definitive.
