Categorical variables

I have a simple question that I cannot figure out. For a multilevel categorical independent variables (let's say four levels), what is the difference in modelling the variable as 3 dummy variables in comparison with coding it as four levels within one variable in a regression model? Are they equivalent and is there an advantage or disadvantage to either method (is the regression equation the same?). Thanks for your responses.


Not a robit
Curious how many obs are within each group? MLM control for both within and between group variance. Not controlling for it can increase type I errors. MLM also allow you to model random intercepts for groups and add random effect for groups.
I mean excluding random vs fixed effects, I was wondering whether there is a difference between using three dummy variables when you input into statistical software versus if you specify four groups and let's say set one of the groups as the reference?