# Mixed Model / Wilkinson Notation for Interaction Effects

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?

• 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 – Ben Bolker Feb 1 at 18:07

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.