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I'm a bit confused about the various notations that are used to describe effects and coefficients in mixed models. I have 2 main effects and included the 2-way interaction effect in my model. The Wilkinson notation of that model is given with

$$ R \sim 1 + G + V + G{:}V + (1|I_{ID}) $$

with my two main effects $G \in \{G_1, G_2, ... G_N\}$ and $V \in \{V_1, V_2, ... V_M\}$.

Now I understand, the $:$ operator only includes the interactions, i.e. $G:M \in \{G_1V_1, G_1V_2, ... G_NV_M\}$ whereas the $*$ operator would also include the lower order effects. So $G*M$ would be equivalent to $G + M + G{:}M$.

Using Jamovi and GAMLj, I get omnibus test result that then include test-results for these effects: $G$, $V$, $G*V$. The last entry seems redundant because following the definition above, it should be $G{:}M$ instead.

The same is true for coefficients table. These then include entries like $G_1-G_2 * V_1-V_2$ which is the combination of the $G_1$ versus its reference $G_2$ and $V_1$ versus its reference $V_2$. Again I'm a bit confused why the $*$ operator is used here.

Does anybody know why the $*$ operator is used in these cases? Or is there a standard on how to use the two operators and what they actually denote?

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  • $\begingroup$ I suspect this is a software "infelicity" (i.e. someone might argue it's not actually a bug, and it's not worth getting hung up on whether it is or not). It would be good to provide a link: gamlj.github.io . I would suggest submitting an issue at github.com/gamlj/gamlj/issues $\endgroup$ – Ben Bolker Feb 1 at 18:07
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I don't know what Jamovi and GAMLj are, but I agree with you that, when you specify the main effects $G$ and $V$, it is superflous to write also write $G*V$ since $G*V$ literally means $G + V + G:V$, so that $G + V + G*V$ actually means $G + V + G + V + G:V$. All software that I am aware of would just ignore the extra main effects.

On that basis, I can't see any reason that the $∗$ operator is used in your case.

However, we should always remember that an interaction term is literally the multiplication of one variable by another, so if we were to define $*$ as the interaction, then it would make sense.

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