How do we report a regression with categorical * continouns interaction? I was wondering how would be the best way to report this model ?
### This model is from Winter (2019:139)
# Center SER: mutate(SER_c = SER - mean(SER, na.rm = T)
# Model:

lm(Iconicity ~ SER_c * POS)

term  estimate
intercept 0.66
SER_c     0.11
POSVerb   0.72
SER_c:POSVerb 0.50

Obs: SER_c is a centered continuos predictor and POS is a categorical predictor with 2 levels (Nouns and Verbs)


*

*1 I know that I cannot talk about main effects because B1 and B2 are simple effects, so would it be sth like:

"a regression model predicts an effect of SER (b = 0.11, p < 0.5) and POS (POS's beta is a composed beta between B1 + B3, so I dont know what I should report here?) as well as a significant interaction between the variables (b3 = 0.50, p < 0.05). Iconicity (the Y variable) is 0.66 for NOUNS w/ AVERAGE SER and 1.38 for VERBS w/ AVERAGE SER and the positive interaction indicates that Verbs are more iconic (the Y variable is iconicity) than Nouns for average SER words"

*

*2 Also, I can talk about the interaction effect as a moderator/control, cant I? Like this:

"we have controlled for SER in our model"

*

*This is the paper in which Winter (the book's author) reports these results, but authors didn't include the interaction in the final version, so I'm a bit confused on how to report that

It is my first time reporting a centered lm with a significant interaction between a centered continuos variable and a categorical variable with two levels, I wanna get it right. I'd really appreciate some advice on that! Thanks in advance!
This post was helpful, but I couldn't make it to my example

Centering (more generally, standardizing) predictors can make the regression coefficient harder to interpret. This seems to happen in your example because nouns and verbs have different mean SER.
I didn't know what SER is (and I still don't), so I was looking at (Winter 2017) to figure it out. In a great example of reproducibility, the data and the code for that analysis are available on GitHub.
This means I was able to reproduce your regression.
#>   (Intercept)          SER_c        POSVerb  SER_c:POSVerb  
#>        0.6642         0.1182         0.7237         0.5084

But first I wanted to look at the distribution of SER for nouns and verbs.

You can see from the histogram that verbs tend to have lower SER than nouns. We can also compute the average SER by part of speech (POS).
#> # A tibble: 2 × 3
#>   POS       n `mean(SER)`
#>   <chr> <int>       <dbl>
#> 1 Noun   1106        3.42
#> 2 Verb    373        2.96

So, while SER = 0 might be arbitrary, so may be the average SER of verbs and nouns (in this dataset). The (unweighted) sample mean SER is a 3:1 combination of the mean noun SER (3.42) and the mean verb SER (2.96):
(1106 * 3.42 + 373 * 2.96) / (1106 + 373) = 3.3

This average is hard to interpret. It follows that the concepts of "NOUNS w/ AVERAGE SER" and "VERBS w/ AVERAGE SER" are also hard to interpret.
Instead I would report the model as two regression lines, one for nouns and one for verbs. Here is the model overlaid on top of the data.

Appendix
Since POS is categorical, the model Iconicity ~ SER * POS is equivalent to a separate regression line for each category. (The errors about the regression lines have the same variance.) The interaction term is a specific parametrization which indicates that the regression slope is different for nouns and verbs.
$$
\begin{aligned}
Y 
&= \beta_0 + \beta_1\text{SER}
+ \beta_2\operatorname{Is}\left\{\text{Verb}\right\}
+ \beta_3\text{SER} \times \operatorname{Is}\left\{\text{Verb}\right\}
\\
&= 
\operatorname{Is}\left\{\text{Noun}\right\}
\color{red}{\Big[\beta_0 + \beta_1\text{SER}\Big]}
+ \operatorname{Is}\left\{\text{Verb}\right\}
\color{blue}{\Big[(\beta_0+\beta_2) + (\beta_1+\beta_3)\text{SER}\Big]} \\
&=
\operatorname{Is}\left\{\text{Noun}\right\}
\color{red}{\Big[\beta_0 + \beta_1\text{SER}\Big]}
+ \operatorname{Is}\left\{\text{Verb}\right\}
\color{blue}{\Big[(\tilde{\beta}_2 + \tilde{\beta}_3\text{SER}\Big]} \\
\end{aligned}
$$
B. Winter, M. Perlman, L. K. Perry, and G. Lupyan. Which words are most iconic?: Iconicity in english sensory words. Interaction Studies, 18(3):443–464, 2017.
