How can we visualize multiple regression with 3 or more continuous variables or with categorical variables I'm trying to understand multiple regression visually.
So far I understand that when you have two continuous variables explaining y, this will take the shape of a plane (see the image).

What happens if you have 3 continuous variables? Does this take the shape of a cube or rectangular prism? What about 4 continuous variables? I don't think we can visualize 4 dimensions? As a size note, I am aware that I can use Level plots. But I still wanted to build from the plane example.
Also, how can we imagine adding a categorical variable? How would that effect the plane.
Using the example from the image bellow, we could add the "make of the car" to the model.
"Make" could be a vector contaning ("Toyota", "Subaru", "Chevrolet")
So we have MPG~Weight+Horsepower+Make.
Would "Make" be a new axis? Or would it instead be a group of points. For example Toyota could be red points, Subaru = blue points, and Chevrolet being black points?
Thank you!

I have used matrix plots before, but I also used another way. I made a regression plot between the measured variable and the main input variable. Then I took the residuals, and plotted those as a function of the second most important variable. And so on.
I feel this is a nice graphical representation of how much “value” you get from each step in making the model more complicated. I personally find 3D plots of any kind difficult to read and interpret, so my choice is very much dependent on my personal aesthetic preferences.
I obviously do present the regression results for the “all at once”: THAT is the analysis, everything else is pretty pictures for illustrative purposes.
Main Trouble with this approach is that a lot of people pretty much ignore entire sections as soon as the word “residual” comes up. It’s up to me to educate these people ;)
