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When running a regression with a categorical variable as the independent variable, the regression essentially picks one of the levels to leave out and runs all the other levels together.

There is no difference between doing this and dichotomizing each variable level and running all these binary variables, leaving out the variable of choice to run against.

My question therefore is whether running a regression with a categorical variable is a multiple regression by definition and should be labelled as such?

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You could argue for either:

Yes

Regression with a categorical variable consisting of $k>2$ categories is identical to multiple regression using $k-1$ dummy variables. You have multiple explanatory variables, so you are doing multiple regression.

No

Multiple regression comes with a variety of phenomena that do not exist in simple linear regression, for example:

Interaction between different categorical variables is possible, but a single categorical variable usually refers to mutually exclusive classes. Colinearity can technically occur (i.e. the dummy variable trap), but that would be due to poor encoding, and not actual redundancy among explanatory variables. Model selection ends with the choice of including an intercept, which usually isn't a difficult choice.

Since none of these things are relevant for linear regression with a single categorical variable, it would seem equally befitting to consider it a form of simple linear regression.

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